Paper Title: A Search for High-Threshold Qutrit Magic State Distillation Routines.






































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theoremTheorem

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1cm







Department of Physics and Computer Science, Dayalbagh Educational Institute, Agra, India






§ ABSTRACT 
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We carry out a search for high-threshold magic state distillation routines for a highly-symmetric qutrit magic state known as the strange state. Our search covers a large class of [[n,1]]_3 qutrit stabilizer codes with up to 23 qutrits, and   is facilitated by a theorem that relates the distillation performance of a qudit stabilizer code to its weight-enumerators. We could not find any code with n<23 qutrits that distills the strange state with better than linear noise suppression, other than the 11-qutrit Golay code. However, for n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic. 














§ INTRODUCTION


Contextuality was identified as a necessary and possibly-sufficient condition for universal quantum computing in <cit.>. The argument of <cit.> is based on magic state distillation <cit.> for qudits of odd-prime dimension, and later extended to qudits of arbitrary odd dimensions in <cit.>, and continuous variable systems in <cit.>.[There are certain subtleties associated with state-independent contextuality for qubits and qudits of even dimension – for simplicity, we focus exclusively on qudits of odd-prime dimension in this paper.] The authors of these works showed that qudit states that do not exhibit contextuality with respect to stabilizer measurements have a non-negative discrete Wigner function <cit.>. The set of such states is known as the Wigner polytope. Because Clifford unitaries and stabilizer measurements are efficiently simulable for states in the Wigner polytope <cit.>, they thus cannot be distilled into pure magic states. 

In order to demonstrate that contextuality is not only necessary but also sufficient for qutrit quantum computation, one must demonstrate that a supply of qudits that do exhibit contextuality with respect to stabilizer measurements may be used to achieve universal quantum computation. In the language of magic state distillation, this translates into the question, can any qudit mixed state outside the Wigner polytope be distilled into a pure magic state?

For qudits of odd prime dimension p, the Wigner polytope is a convex polytope with p^2 facets that lives in the p^2-1 dimensional space of qudit density matrices.  In <cit.>, it was shown that no finite magic state distillation routine can distill all states that lie outside one of the faces of the Wigner polytope[Some distillation routines that distill qutrit states up to one of the hyperedges of the Wigner polytope were found in <cit.>.], generalizing the analogous result for qubit states that lie outside the stabilizer polytope <cit.>. However, the possibility remains that a sequence of magic state distillation routines, based on stabilizer codes of increasing length n may distill states arbitrarily close to a face of the Wigner polytope. Is there any evidence that such a sequence of magic state distillation routines exist?

The problem simplifies if one focuses on qutrits. There exists a qutrit magic state, first identified by Howard and van Dam <cit.>, sometimes known in the literature in the qutrit strange state |S⟩ <cit.>, that lies directly above one of the facets of the Wigner polytope[See <cit.> for another application of the strange state.]. As discussed in <cit.>, noisy |S⟩ states can be twirled via Clifford unitaries to lie on a line connecting a pure |S⟩ state to the maximally mixed state:

    ρ̂(ϵ)=(1-ϵ) |S⟩⟨S| + ϵ1/3,

with all noise converted to depolarizing noise, parameterized by ϵ. Any state ρ̂(ϵ) for ϵ <3/4 lies outside the Wigner polytope and exhibits contextuality with respect to stabilizer measurements. We then ask whether or not there exists a family of n-to-1 magic state distillation routines that distill the strange state with a threshold approaching ϵ = 3/4 as n →∞? [An additional consideration is the increase in overhead cost of distillation as n →∞.]

At the time <cit.> was published, and for several years thereafter, while some qutrit and qudit magic state distillation routines had been proposed <cit.>, no magic state distillation routine that distilled the Howard van Dam strange state was known. It was later discovered that an 11-qutrit CSS code based on the ternary Golay code can distill the |S⟩ state, with a threshold of ϵ_* = 0.38 <cit.>. Do there exist any other qutrit stabilizer codes that distill the strange state? If so, how do their thresholds compare to that of the 11-qutrit Golay code?[Recently, <cit.> reported the existence of 13-qutrit and 29-qutrit codes that could distill the qutrit strange state with high thresholds. However, we attempted to reproduce these results and found that neither of the codes reported in <cit.> are able to distill the qutrit strange state. See Appendix <ref>.] In this paper, we carry out a computational search over reasonably small qutrit error-correcting codes to help answer these questions. 

To facilitate our search, we prove a simple theorem connecting the performance of a stabilizer code for qudit magic state distillation its complete weight enumerator. For distillation routines for the qutrit strange state, this formula simplifies and depends only on the stabilizer code's simple weight enumerator. 

Our search is limited by practical considerations, as well as the existing classifications of qutrit error-correcting  codes in the literature – namely, the classification of qutrit stabilizer states in <cit.> and a classification of self-orthogonal classical ternary codes available on <cit.>. We carried out a search over all [[n,1]]_3 stabilizer codes with n ≤ 9, and a search over all [[11,1]]_3 stabilizer codes that can be obtained from a [[12,0,6]]_3 stabilizer state. For such codes, we demand transversality of a particular single-qutrit gate (the square of the qutrit Hadamard gate), which allows us to restrict our search to projection onto the trivial syndrome of each stabilizer code. We also searched over all [[n,1]]_3 CSS codes for odd n ≤ 23 that possess a complete set of transversal single-qudit Clifford gates. 

We found that none of the codes we searched with n<23 could distill the |S⟩ state with better-than-linear[We also found a few 9 and 11 qutrit codes that could distill the state with linear noise suppression.] noise suppression, other that the 11-qutrit Golay code of <cit.>. However, for n=23, we found over 600 CSS codes that could distill the strange state with cubic noise suppression – which is approximately 1/3 of all the codes we could construct from the ternary self-orthogonal codes listed in <cit.> – suggesting that for large codes, magic state distillation is somewhat generic. None of these 23-qutrit codes, however, had a threshold that exceeds that of the 11-qutrit Golay code.   

Let us point out that the new distillation routines we find here are mainly of theoretical interest. The success probabilities are quite low, and far better yields are obtained via triorthogonal codes <cit.> (see <cit.> for constructions of qutrit and qudit triorthogonal codes). Nevertheless, the CSS codes we study have a complete set of transversal Clifford gates, and may turn out to be useful for fault-tolerant quantum computation in other settings.  

Our paper is organized as follows. In section <ref> we briefly review some background material. In section <ref> we derive a relation between weight-enumerators and the performance of magic state distillation routines. In section <ref> we describe our search space and the results. In section <ref> we conclude with some brief discussion.In addition, in appendix <ref> we present a useful lemma, and in appendix <ref> we study a two qutrit codes, conjectured in <cit.> to distill the |S⟩ state, that we find do not distill the |S⟩ state.



§ PRELIMINARIES


Here we review some well-done definitions and results concerning stabilizer codes for qudits of dimension p where p is any odd prime.



 §.§ The Heisenberg-Weyl displacement group


The qudit version of the Pauli group is generated by

    X̂ = ∑_k |k+1⟩⟨k|,   Ẑ = ∑_k ω^k |k⟩⟨k|,
 and multiplication by ω= e^2π i/p, and is known as the Heisenberg-Weyl displacement group.  The conventions for Heisenberg Weyl displacement operators we will follow are those in <cit.>, 

    D̂(u,v) = ω^-2^-1 u vX̂^u Ẑ^v.

n-qudit Heisenberg Weyl-displacement operators are denoted as

    D̂(u⃗,v⃗) = D̂(u_1,v_1) ⊗D̂(u_2,v_2) ⊗…⊗D̂(u_n,v_n),

where u⃗=(u_1, u_2, …, u_n) and v⃗=(v_1, v_2, …, v_n).

It is convenient to combine u⃗ and v⃗ into a symplectic vector χ=(u⃗,v⃗), and write D̂(χ) =D̂(u⃗,v⃗). Then 

    D̂(χ)D̂(χ') = ω^2^-1  [χ, χ']  D_χ+χ',

where 

    [χ, χ']= u⃗·v⃗'- u⃗' ·v⃗

is the symplectic inner product (and · is the usual dot product). For commuting Heisenberg-Weyl operators [χ,χ'] vanish.


Note that [χ, χ]=0, so D_χ^m=D_m χ.

We arrange these into p+1 rays,

    D̂(0,1), D̂(1,0), D̂(1,1), …, D̂(1,p-1).

The set of p+1 bases formed by the eignespaces of these operators form a set of mutually unbiased bases.




 §.§ Qudit stabilizer codes

An [[n,k]]_p stabilizer code is defined by the simultaneous eigenspace of n-k independent commuting n-qudit Heisenberg-Weyl displacement operators, and has dimension p^k. We will assume that each of these operators are of the form given in (<ref>), with no additional overall phase. As such, to completely specify the stabilizer code, we also have to specify the eigenvalues of each stabilizer, which are of the form ω^a, where a can be taken to be an element of  ℤ_p. In this paper we will restrict our attention to codes defined by the 1=ω^0 eigenspace of each operator, for reasons we explain below. The set of commuting operators {D̂(u⃗_1, v⃗_1), …, D̂(u⃗_n-k, v⃗_n-k) } can be specified by a symplectic matrix

    H = ( [   u⃗_1   v⃗_1;   u⃗_2   v⃗_2;      ⋮      ⋮; u⃗_n-k v⃗_n-k ]),

such that the symplectic inner product of any two rows vanishes. These operators generate an abelian group of order p^n-k, which we refer to as the stabilizer group 𝒮. Because 𝒮 is abelian, there are no overall phases arising from multiplication and 𝒮 consists of all Heisenberg-Weyl Displacement operators of the form D̂(χ) where χ is in the row-span of H.

A stabilizer code can equivalently be thought of as an additive classical code over GF(p^2), that is self-orthogonal under the Hermitian inner product <cit.>. 

We use the notation N(𝒮) to denote all n-qudit Heisenberg-Weyl displacement operators that commute with 𝒮. 𝒮⊆ N(𝒮). In general, N(𝒮) will not form an abelian group, and multiplication of two different elements may give rise to Heisenberg-Weyl operators with overall phases. It is nevertheless useful to define N(𝒮)^* as N(𝒮) modulo overall phases (i.e., identify Heisenberg-Weyl operators that differ by an overall phase), and specify N(𝒮)^* as the row-span of a symplectic matrix. In this way N(𝒮)^* can be thought of as a classical additive code in GF(p^2), denoted as 𝒮^⊥, that is the set of all codewords orthogonal to 𝒮 under the Hermitian inner product. We use the notation 𝒮^⊥ and N(𝒮)^* interchangeably. 

Clearly, 𝒮⊂𝒮^⊥. Any L ∈𝒮^⊥ defines a coset of 𝒮 in 𝒮^⊥ which is of the form[In this expression, addition means addition as symplectic vectors (or equivelently, vectors in GF(p^2)^n), or multiplication as Heisenberg-Weyl displacement operators.] 𝒮 + L = { L + M | M ∈𝒮}.  All stabilizers in the same coset will commute with each other. 







 §.§ Weight enumerators

Let us briefly review the definition of the weight enumerator of a stabilizer code. 

Let 𝒮 denote the stabilizer group of an [[n,k,d]]_3 stabilizer code. The projector onto the codespace of 𝒮 with trivial syndrome can be written explicitly as,

    Π̂^0_𝒮 = 1/p^n-k∑_M ∈𝒮 M.


Let 𝒮̃ be a either a stabilizer code 𝒮, or one of its cosets in 𝒮^⊥. The complete weight enumerator of 𝒮̃ is a function of p^2 formal variables {y_ij}, for i, j = 0, …, p-1, defined as follows:

    w(𝒮̃; ω_ij ) = ∑_(x⃗|z⃗) ∈𝒮̃∏_i=1^n y_x_i,x_j


The simple weight enumerator of 𝒮̃ to be a function of two formal variables x and y, defined as follows:

    w(𝒮̃; x, y) =  w(𝒮̃; y_ij(x, y) ).
 
where 

    y_ij(x,y) =  x     (i,j)=(0,0) 
    
        y     (i,j) ≠ (0,0).

If we further set x=y=1, then the simple weight enumerator becomes equal to |𝒮̃|. These definitions coincide with the standard definitions of the simple and complete weight enumerators of a classical error-correcting code over the finite field GF(p^2). 





A MacWilliams identity relates w(𝒮;1,z) to w(𝒮^⊥; 1,z) <cit.>. This is:

    w(S^⊥;1,z) = (1+(p^2-1)z)^n/p^n-k w(S;1,1-z/1+(p^2-1)z).

It is conventional to define A_𝒮(z)=w(𝒮;1,z) and B_𝒮(z)=w(𝒮^⊥; 1,z). 
A_𝒮(z)-B_𝒮(z) is a polynomial in z with non-negative coefficients and is the simple weight enumerator of the set of all logical operators for the stabilizer code 𝒮. The lowest power of z that appears in A(z)-B(z) is z^d, where d is the distance of the code.



 §.§ Discrete Wigner functions

A discrete phase space formalism for qudits was formulated in <cit.>, that we briefly review here. 

Singe-qudit phase point operators are defined as,

    Â(0,0) = 1/p∑_x,zD̂(u,v),   Â(u,v) =    D̂(u,v)Â(0,0) D̂(u,v)^†.

These are normalized so that Â(u,v)=1.
Multi-qudit phase point operators are defined as 

    Â(u⃗,v⃗)=⊗_i Â(u_i,v_i),

and we will use the notation Â(0,0)^⊗ n = Â(0,0) ⊗Â(0,0) ⊗…⊗Â(0,0).

Any qudit density matrix ρ̂ can be written as a linear combination of the phase point operators with real, but possibly negative, coefficients. These coefficients define the discrete Wigner function of the qudit state:

    ρ̂ = ∑_u,v W( ρ̂; u,v)Â(u,v),    W( ρ̂; u, v) = 1/pρ̂Â(u,v)

Note that ρ̂ = ∑_u,v W( ρ̂; u,v) =1, and the discrete Wigner function defines a quasi-probability distribution.

The Clifford group acts covariantly on the discrete Wigner function, for qudits of odd prime dimension. Heisenberg-Weyl displacement operators act as translations in discrete phase, and a general Clifford unitary can be represented as a symplectic rotation followed by a translation. See <cit.>, and references therein, for more details. This allows one to use the discrete Wigner function to define an efficient classical simulation for Clifford unitaries and stabilizer measurements acting on qudit states with non-negative Wigner functions <cit.>, and develop resource theories of magic for qudits of odd prime dimension – see <cit.>.



§ DISTILLATION AND WEIGHT ENUMERATORS


In the magic state model of fault-tolerant quantum computing <cit.>, one begins with a quantum computer that can initialize qudits in the computational basis, perform Clifford unitaries, and carry out stabilizer measurements. These operations are conventionally assumed to be noise-free for simplicity. To obtain universal quantum computing we supplement this quantum computer with the ability to initialize ancilla qudits in certain non-stabilizer states known as magic states, which are noisy. A magic state distillation protocol is a way to distill an arbitrarily pure magic state from many noisy magic states using only Clifford unitaries and stabilizer measurements. For the purposes of determining the best attainable threshold <cit.>, any magic state distillation protocol for the qutrit strange state  can be thought of as a procedure that projects n noisy qudits onto the codespace of an [[n,k]]_p stabilizer code – if the projection is successful, one decodes the resulting qudits to obtain k hopefully-less-noisy magic states.[Sometimes, i.e., in the case of triorthogonal codes, error-correction is also possible prior to decoding, which can result in a substantial reduction in overhead. However, this is not possible for distillation routines for the qutrit strange state.] 





 §.§ A general formulation in terms of complete weight enumerators

Here we provide a general formulation for qudit magic state distillation in the language complete weight enumerators. These results follow in part from the formulation in <cit.>. Let us also mention that a similar formulation in terms of signed-weight enumerators for qubit magic state distillation was given in <cit.>.

Let 𝒮 be a group of commuting Heisenberg-Weyl operators acting on n qudits. (Recall that, each element of 𝒮 is assumed to be of the form given in equation (<ref>), with no additional overall phase.) The projector onto the joint eigenspace of the code, with eigenvalues 1=ω^0, is given by,

    Π̂^0_𝒮 = 1/|𝒮 |∑_M ∈𝒮 M.


lemmaLemma

    Let ρ̂ be a  single qudit mixed state described by the Wigner function W(ρ̂; x,z). Then the probability for successful projection onto the eigenspace of 𝒮 with trivial syndrome is given by the complete weight enumerator of 𝒮^⊥, w( 𝒮^⊥; { W(ρ̂;i,j) }), with the formal variables y_ij in the complete Weight enumerator replaced by the entries of the Wigner function of ρ:

    ( Π̂^0_𝒮ρ^n ) = w( 𝒮^⊥; { W(ρ̂;i,j) }) = ∑_(u⃗|v⃗) ∈𝒮^⊥∏_i=1^n W(ρ̂; u_i, v_i).
  





Let the single-qudit state ρ̂ be described by Wigner function W_in(ρ̂; u,v). The n-qudit Wigner function for ρ^⊗ n is given by, 

    W^(n)_in(ρ̂^⊗ n; u⃗,v⃗) = ∏_i W_in(ρ̂; u_i,v_i).

The probability for successful projection onto the codespace is,

    ν    =     (Π̂^0_𝒮ρ_in^⊗ n) 
        =    ∑_u⃗, v⃗(∏_i=1^n W_in(u_i,v_i))  ( Π̂^0_𝒮Â(u⃗,v⃗) ).

In Appendix <ref>, Lemma <ref>,  we show that 

    (  Π̂^0_𝒮Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮^⊥
    
       1     (u⃗|v⃗) ∈𝒮^⊥.

We therefore find that,

    ν  =  ∑_(u⃗|v⃗) ∈ S^⊥∏_i W_in(ρ̂; u_i,v_i).



In magic state distillation, we first project onto the codespace of 𝒮, which we assume for simplicity to be an [[n,1]]_p code, then decode, to obtain a new state ρ' = f_MSD(ρ). We can use the above lemma to compute the Wigner function of ρ' in terms of the complete weight enumerators of 𝒮^⊥ and its cosets.

Choose any representatives of the logical Pauli operators X̅ and Z̅, to define logical Heisenberg-Weyl displacement operators D̅(x,z) and logical phase point operators A̅(x,z). The Wigner function of ρ̂' is given by,

    W(ρ̂'; x,z) = 1/ν( p^-1A̅(x,z) Π̂^0_𝒮ρ̂^⊗ n).

Let,

    p^-1A̅(0,0) Π̂^0_𝒮 =  p^-2∑_u,vD̅(u,v) Π̂^0_𝒮 =  1/p^n+1∑_(u⃗|v⃗) ∈𝒮^⊥D̂(u⃗,v⃗) ≡Π̂^0_𝒮^⊥.

Although Π̂^0_𝒮^⊥ is not a projector onto the codespace of a stabilizer code, in Appendix <ref>, we also show that,

    W(ρ̂'; 0,0) = 1/ν( Π̂^0_𝒮^⊥ρ̂^⊗ n) = 1/ν( Π̂^0_𝒮^⊥ρ̂^⊗ n) = 1/ν w( 𝒮; { W(ρ̂,i,j) }).

In general,

    W(ρ̂';u,v) = 1/ν w( 𝒮+D̅(u,v); { W(ρ̂;i,j) }).

As a cross-check, note that ∑_u,v w( 𝒮+D̅(u,v); { y_ij}) = w(𝒮^⊥; {y_ij}), so the resulting Wigner function is correctly normalized.

The problem of computing the output state of a general qudit magic state distillation routine defined by a stabilizer 𝒮 is thus reduced to computing the complete weight enumerators of 𝒮, and its cosets in S^⊥.




 §.§ The strange state and simple-weight enumerators


We now restrict our attention to the special case where ρ̂ a twirled qutrit strange state |S⟩. For this particular magic state, we will show that the complete weight enumerators in the previous subsection can be replaced by simple weight enumerators.

The qutrit strange state is given by 

    |S⟩= 1/√(2) (|1⟩- |2⟩).
  
After twirling by symplectic rotations, as described in <cit.>, all noise is converted to depolarizing noise, and noisy |S⟩ states are described by the density matrix:

    ρ̂_S(ϵ)=(1-ϵ) |S⟩⟨S| + ϵ1/3.

This corresponds to the discrete Wigner function:

    W(ρ̂_ϵ; u,v) =  x     (u,v)=(0,0) 
     y     (u,v) ≠ (0,0)  = [ y y y; y y y; x y y ],

where the parameters x and y satisfy x+8y=1, and are related to ϵ via:

    3x     =     -1+4ϵ/3
    
    6y     =     1-ϵ/3.


Applying Theorem <ref> to ρ̂_S(ϵ), we find the following lemma.

The probability for successful projection of ρ̂(ϵ)^⊗ n, ν, onto 𝒮, is given by the simple weight enumerator of 𝒮^⊥, 
    
    ( Π̂_𝒮ρ̂_S(ϵ)^n ) = w(𝒮^⊥; x(ϵ),y(ϵ)),

where x, and y are given by eqs. (<ref>) and (<ref>).



Let us assume we distilling using a stabilizer code that has a complete set of transversal Clifford gates. The output state ρ̂'=f_MSD(ρ̂) of such a distillation protocol will also be of the form in equation (<ref>), with parameter ϵ'. (Alternatively, if the stabilizer code does not have a complete set of Clifford gates, one could still twirl the output state by symplectic rotations as well to bring it into this form.) Let W(ρ̂';0,0)=x' and W(ρ̂';i,j)=y' for (i,j) ≠ (0,0). Then using equation <ref>, we find,

    x'     =    w(𝒮; x,y)/w(𝒮^⊥;x,y)
    
    y'     =    1/8w(𝒮^⊥;x,y)-w(𝒮;x,y)/w(𝒮^⊥;x,y).

The formal variables x and y used to define the simple weight enumerator in equation <ref> are now reinterpreted as entries of the discrete Wigner function in equation <ref>.

Let us rewrite this expression in terms of the noise parameter ϵ. We find, 

    ϵ' = 33A(z(ϵ))+B(z(ϵ))/4B(z(ϵ).

where, 

    z(ϵ) = 3-ϵ/8 ϵ-6,

A(z)=w(𝒮; 1,z) and B(Z)=w(𝒮^⊥; 1,z).

The error suppression of the magic state distillation routine is determined by the largest power of ϵ that divides 33A(z(ϵ))+B(z(ϵ)). By contrast, the distance of the error-correcting code is given by the largest power of z that divides B(z)-A(z).



Expressing ϵ in terms of x and y via

    ϵ/3    =     x+ 2y,

we see that 

    ϵ'/3 = 3 w(𝒮;x(ϵ),y(ϵ))+w(𝒮^⊥; x(ϵ),y(ϵ))/4w(𝒮^⊥; x(ϵ),y(ϵ)).



It is also convenient to write these results in terms of the parameter z = y/x via 

    ϵ/3  =  1+2z/1+8z,

as:

    ϵ' = 33A(z)+B(z)/4B(z).

which implies

    z'=B(z)-A(z)/8A(z).




  §.§.§ Example: the [[11,1,5]]_3 Golay code

To illustrate the above formalism, let us apply it to the 11-qutrit Golay code of <cit.>. The 11-qutrit Golay code is an [[11,1,5]]_3 CSS-code formed using two copies of the (self-dual) classical ternary Golay code.

The weight enumerator of the 11-qutrit Golay code is computed (e.g., via Magma <cit.>) to be:

    A(z) = 1+528 z^6+7920 z^8+11000 z^9+23760 z^10+15840 z^11.

Using the MacWilliams identity, we find,

    B(z) = 1+528 z^5+528 z^6+15840 z^7+40920 z^8+129800 z^9+198000 z^10+145824 z^11.
 
Substitute these results into equation (<ref>), to obtain

    ϵ'     =  3 (48336 z^11+67320 z^10+40700 z^9+16170 z^8+3960 z^7+528 z^6+132 z^5+1)/145824 z^11+198000 z^10+129800 z^9+40920 z^8+15840 z^7+528 z^6+528 z^5+1
        =  (3021 ϵ ^11-24816 ϵ ^10+92180 ϵ ^9-203280 ϵ ^8+292710 ϵ ^7-283536 ϵ ^6+181764 ϵ ^5-71280 ϵ ^4+13365 ϵ ^3)/ 
       =(990 ϵ ^11-7920 ϵ ^10+27500 ϵ ^9-50490 ϵ ^8+37620 ϵ ^7+47256 ϵ ^6-172656 ϵ ^5+243540 ϵ ^4-204930 ϵ ^3 
        +106920 ϵ ^2-32076 ϵ +4374) 
       ≈    55 ϵ^3/18 + O(ϵ^4).


The threshold of the code, ϵ_*, is the critical value of ϵ such that, ϵ<ϵ_* implies ϵ'<ϵ. Using (<ref>), we find 

    ϵ_*     =1/45(-262 √(2/405 √(109)-2981)+2^2/3√(405 √(109)-2981)+31) 
       ≈ 0.387.

Interestingly, z_*=z(ϵ_*) satisfies a simple cubic equation, 11 z_*^3+12 z_*^2+3 z_*+1=0.



 §.§ Conditions for magic state distillation


There are two conditions that a stabilizer code must satisfy for it to qualify as a magic state distillation routine for the strange state. We first require that the probability of successful projection to be nonzero in the limit ϵ→ 0, which translates into the requirement

    B( z( ϵ=0) ) =B(-1/2) ≠ 0.

We also require that the noise suppression be better than linear.

Assuming equation (<ref>) is satisfied, the noise-suppression exponent, δ, of the magic state distillation routine, ϵ' = θ(ϵ^δ), is determined by the smallest power of ϵ that divides 3A(z(ϵ))+B(z(ϵ)). Let us write

    3A(z(ϵ)) + B(z(ϵ)) = C_0 + C_1 ϵ + C_2 ϵ^2 + …
 
Generically, we expect C_0 and C_1 will be non-zero. The necessary and sufficient condition for C_0 and C_1 to vanish are,

    ( 3A(z(ϵ)) + B(z(ϵ)) )|_ϵ=0    = 0 
    d/dϵ|_ϵ=0( 3A(z(ϵ)) + B(z(ϵ)) )     = 0.

Translated into z, these conditions become

    3A(-1/2)+B(-1/2)     =     0, 
    
       3A'(-1/2) + B'(-1/2)     =     0 .

As a check, observe that, for the weight enumerators of the 11-qutrit Golay code, A(-1/2)=2187/64, B(-1/2)=6561/64, A'(-1/2)=-8019/16 and B'(-1/2)=24057/16, these conditions are satisfied. 

The MacWilliams identity simplifies at z=-1/2 to:

    B(-1/2)= 3(-1)^n A(-1/2).
 
For n odd, (<ref>) is therefore automatically satisfied. We can also ask, when do we get cubic noise suppression? If condition in equation (<ref>) is satisfied, the condition for C_2 to vanish is that 

    3 A”(-1/2)+B”(-1/2)=0.
 
Taking derivatives of the MacWilliams identity, we find that,

    B'(-1/2)     =     (-1)^n+1(3 A'(-1/2)+8n A(-1/2) ), 
    
        B”(-1/2)     =    1/3 (-1)^n (9 A”(-1/2)+48 (n-1) A'(-1/2)+64(n^2-n)A(-1/2)).

When n is odd, then the MacWilliams identities above along with (<ref>) imply that condition (<ref>) is satisfied. Thus, cubic noise suppression is guaranteed for any odd-length stabilizer code whose weight enumerator satisfies the two conditions, (<ref>) and (<ref>). 







§ SEARCH FOR DISTILLATION ROUTINES


With the above results in place, a computational search for distillation routines for the qutrit strange state is straightforward. For each stabilizer code 𝒮 in our search space, we compute the simple weight enumerator A(z), and then B(z) using the MacWilliams identity. We then check if conditions (<ref>) and (<ref>) are satisfied.



 §.§ Equivalence of distillation routines

When are two stabilizer codes equivalent for the purposes of magic state distillation? 

Conventionally, one considers two quantum error-correcting codes to be equivalent if they differ by permutations of coordinates and by local Clifford operations. However, two stabilizer codes which differ by local Clifford transformations will in general give rise to different magic state distillation protocols. 


This observation increases the size of our search space substantially – to search for all magic state distillation routines associated with a given stabilizer code 𝒮, we need to search over all orbits of 𝒮 under local Clifford transformations, and all possible eigenvalues of the stabilizers. For each [[n,1]]_3 code, this could increase the search space[Any local Clifford operator can be written as a product of V_F̂D̂(χ) where V_F̂ is a symplectic rotation. Acting on a stabilizer code with a transversal Heisenberg-Weyl displacement operator can only change the eigenvalues of the stabilizers.  Symplectic rotations leave the eigenvalues unchanged but transform the stabilizers by a product of local SL(2,ℤ_p) transformations. Therefore, in principle, a search over magic state distillation routines seems to require a search over all local SL(2,ℤ_3) orbits of each [[n,1]]_3 stabilizer codes, with all possible allowed eigenvalues for each of the stabilizers.] by a factor as large as, |SL(2,ℤ_3)|^n 3^n-1=24^n 3^n-1. 

It is possible, however, to substantially reduce this search space size, by demanding that the both the stabilizer codes we study and the magic states we wish to distill possess certain symmetries. 

Let the magic state we wish to distill be invariant under a subgroup G of the single-qudit Clifford group. It is natural to require[One, in principle, can also consider magic state distillation without twirling the input states – see <cit.>.] our noisy input states first undergo a twirling procedure (as described in detail for qutrits in <cit.>) so that they are also invariant under G. Let C be a single-qudit Clifford-unitary, if a twirled noisy magic state ρ̂_M, satisfies Cρ̂_M (C)^†=ρ̂_M, then two stabilizer codes which differ via local Clifford transformations belonging to the subgroup of the Clifford group generated by by C are equivalent for distillation of the ρ̂_M magic state. The qutrit magic state |S⟩ is a simultaneous eigenvector of all symplectic rotations, as emphasized in <cit.>. Assuming noisy input magic states are twirled by applying a random symplectic rotation, two stabilizer codes are equivalent for magic state distillation if and only if they are related to each other by local symplectic transformations. This reduction in the size of search-space is only possible for the qutrit strange state, by virtue of its exceptional symmetry – and is not expected to be true for any other qutrit or qudit magic state. 

It is also natural to require that some, or all, of the unitaries in G be transversal gates for the stabilizer code used for distillation. This is not a sufficient condition for distillation, but it is true for all magic state distillation routines studied in the literature that we are aware of, other than triorthogonal codes (which possess a transversal non-Clifford gate) <cit.>. 

In this paper, we are interested in distilling the qutrit magic state |S⟩. The symmetry group of |S⟩, is the set of all symplectic rotations. Therefore, it is natural to demand that our code possess a complete set of transversal Clifford gates. However, we can instead impose a slightly less restrictive requirement, and demand only that Ĥ^2 be a transversal gate, where Ĥ denotes the single-qutrit Hadamard gate. Notice that, acting with (Ĥ^2)^⊗ n on a stabilizer code corresponds to multiplying each row of the symplectic matrix defining the code by -1, leaving the eigenvalues unchanged. Alternatively, its action can be thought of as changing the eigenvalues ω^a of any stabilizer to ω^-a. Demanding that (H^2)^⊗ n commute with the codespace is therefore equivalent to demanding that the eigenvalues of all stabilizers in the code must be +1=ω^0. This condition allows us to restrict our search to projection onto the trivial syndrome. 

Our search space thus consists of two classes of codes: a search over all small quantum error correcting codes, and a search over CSS codes with a complete set of transversal Clifford gates. For both classes, we restrict attention to projection onto the trivial syndrome, due to our requirement that (Ĥ^2)^⊗ n be transversal.  



 §.§ Small quantum error-correcting codes


Any n-qudit stabilizer state can be thought of as a [[n,0,d]]_3 stabilizer code, and any [n-1,1]_3 stabilizer code can be obtained by shortening an [[n,0]]_3 stabilizer code.  Using the graph-state formalism, <cit.> classified all [[n,0,d]]_3 stabilizer states up to local Clifford transformations for n≤ 10, and all[Danielsen has also classified all optimal 11-qutrit stabilizer states, but we could not obtain them.] 12-qutrit stabilizer states of maximum distance[The distance of a stabilizer state is the minimum weight of any non-trivial element of its stabilizer group.] 6. 

For each [[n,0]]_3 code, we computed all inequivalent ways of shortening it to obtain an [[n-1,1]]_3 stabilizer code. For each resulting [[n-1,1]]_3 code, we computed the weight enumerator A(z) and the performance for magic state distillation, as described in the previous section. We thus searched over all [[n,1,d]]_3 codes for m ≤ 9, and all [[11,1,d]] codes that can be obtained from shortening a [[12,0,6]]_3 stabilizer state.

Remarkably the only code we found that could distill the qutrit strange state with better-than-linear noise-suppression was the 11-qutrit Golay code. There were also a few [[9,1]]_3 and [11,1]]_3 codes that distilled the strange state with linear noise suppression, (i.e., ϵ' = αϵ with α<1), all with lower thresholds than the 11-qutrit Golay code.




 §.§ CSS codes

A maximal self-orthogonal classical ternary code has parameters [n, ⌊n/2⌋]_3. For n odd, therefore, using the CSS construction <cit.>, two copies of a self-orthogonal ternary code of odd length can be combined to form a [[n, 1, d]]_3 qutrit code, whose transversal gate set is the entire single-qutrit Clifford group. Explicitly, if G_c is the generator matrix of the self-orthogonal classical ternary code, the quantum CSS code is given by symplectic matrix:

    H = ( [ G_c   0;   0 G_c ]).

Any such CSS code is a natural candidate for a magic state distillation routine for the strange state.

Classical maximal self-orthogonal ternary codes up to size n=23 have been classified in <cit.>, and are conveniently available on a website maintained by Harada and Munemasa <cit.>.

We computed the weight enumerators of all CSS-codes constructed this way from the codes given in <cit.>. We found that no indecomposable[A code is said to be indecomposable if its generator matrix cannot be written as the direct sum of two smaller generator matrices.] 13, 15, 17 or 19-qutrit CSS codes were able to distill the strange state.

Somewhat surprisingly, however, we found a total of 646 inequivalent indecomposable 23-qutrit CSS codes[The 646 inequivalent CSS codes gave rise to 263 different simple weight enumerators.] that were able to distill the strange state with cubic noise suppression. A complete list of the 646 classical ternary codes that gave rise to these codes is included in MAGMA format <cit.> as supplemental data along with this arXiv submission.

There are a total of 1928 indecomposable maximally-self-dual [23,11]_3 codes, so the probability that a randomly chosen code will give rise to a quantum CSS code that distills the strange state is 0.335, which is very close to 1/3.  The probability that a randomly chosen maximally self-orthogonal [11,5]_3 code gives rise to a CSS code that distills the strange state is also 1/3. This seems to suggest that quantum error correcting codes that distill the qutrit strange state are actually quite common.

The thresholds that arise from these codes range from 0.063 to 0.318, and are plotted in a histogram in Figure <ref>. None of these thresholds exceed that of the ternary Golay code.



The code with the highest threshold was a [[23,1,5]]_3 code. It is formed from two copies of the classical [23, 11, 6]_3 code whose generator matrix is:

    G_C = [ 1 0 0 0 0 0 0 0 0 0 0 1 2 2 0 0 2 0 2 0 0 0 0; 0 1 0 0 0 0 0 0 0 0 0 2 2 1 1 2 1 2 1 2 2 0 1; 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 2 2 0; 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 2 0 1 0 1 0; 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 1 0 2 2 2 0; 0 0 0 0 0 1 0 0 0 0 0 2 2 0 0 2 1 0 2 0 0 0 0; 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 0 1 0 0 2 0 2 0; 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0; 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 0 1 1 0 1; 0 0 0 0 0 0 0 0 0 1 0 0 0 2 1 2 0 2 1 2 2 0 1; 0 0 0 0 0 0 0 0 0 0 1 1 2 2 1 1 2 2 1 2 2 0 1 ].

The quantum CSS code has weight enumerator 

    A(z)     =  2079023616 z^23+6035662080 z^22+8258226816 z^21+7208904960 z^20+4504066560 z^19+2182781824 z^18
       + 790797312 z^17+252077184 z^16+52015680 z^15+14590080 z^14+2083104 z^13+628800 z^12+121824 z^11
        +58320 z^10+16120 z^9+4608 z^8+720 z^6+1.

It gives rise to a distillation performance 

    ϵ' ≈73 ϵ ^3/18+O(ϵ ^4),

plotted in Figure <ref>. 



The probability of successful projection onto the codespace for these codes is very low. We refer to the probability for 23 pure copies of the strange state to successfully project onto the codespace as the success probability of the code. The success probabilities range from 1/5159780352 ≈ 1.9 × 10^-10 to 1/35831808≈ 2.8 × 10^-8.  The highest probability of successful distillation is attained for the 23-qutrit code in equation (<ref>) is 1/35831808 – three other codes have the same success probability and very similar thresholds. It appears that the success probability is correlated with the threshold. A plot of success probability versus threshold for the 646 codes that distill the strange state is shown in Figure <ref>. 




§ DISCUSSION



    

Magic state distillation <cit.> is a somewhat mysterious approach to fault-tolerant quantum computing. <cit.> identified two magic states for qubits, which they denote as |H⟩ and |T⟩. For the qubit |H⟩ state of <cit.> and its qudit generalizations defined in <cit.> – magic state distillation is better understood, and systematic approaches to construct distillation routines with low overhead have been proposed in the literature via the formalism of triothogonal codes. 
However, for the |T⟩ state of <cit.> few distillation routines are known <cit.>, and the mechanism behind distillation remains unclear. The qutrit analogue of the |T⟩ state, which lies directly above a face of the stabilizer polytope, is the strange state |S⟩ <cit.> that we study here. The study of distillation of the strange state may in fact be more tractable than that of the qubit |T⟩ state, thanks to the simple relation between distillation performance and simple enumerators valid only for the qutrit strange state. We hope that, by enlarging the landscape of codes known to distill this state, will lead, in the future, to a better understanding of magic state distillation. 

In this work, we carried out a computational search and found that over 600 [[n,1,d]]_3 codes that distill the qutrit strange state, with cubic noise suppression. This seems to suggest that the ability to distill the qutrit strange state is a generic feature, at least of codes constructed to have a complete set of transversal Clifford gates. While none of these codes have thresholds that exceed that of the 11-qutrit Golay code, their existence does provide some hope that there may exist larger codes with thresholds better than the 11-qutrit Golay code. These results also do provide some support for the belief that a sequence of distillation routines of increasing size whose threshold approaches the limit set by contextuality exists, as required to prove the conjecture that contextuality is sufficient for qutrit quantum computation <cit.>.



§ ACKNOWLEDGEMENTS


SP thanks Prof. P.S. Satsangi for inspiration and guidance. We also thank Mark Howard and Amolak Kalra for comments on a draft of this manuscript. We acknowledge the support of MeitY QCAL, Amazon Braket and DST-SERB grant (CRG/2021/009137).





§ COMPLETE WEIGHT ENUMERATOR

Using the following notation

    [ y_20 y_21 y_22; y_10 y_11 y_12;    x y_01 y_02 ]

we can define the complete weight enumerator of a code as a polynomial in the above 9 variables. The simple weight enumerator is defined by seting y_ij=y.



§ NORELL STATE

Suppose we wish to distill the Norell state. The discrete Wigner function of the Norell state is of the form: 

    [ n_2 n_1 n_1; n_2 n_1 n_1;   x n_1 n_1 ]

For the mixed state

    ρ̂ = (1-ϵ_0 - ϵ_s) |N⟩⟨N|+ϵ s |s⟩ + ϵ_0 |0⟩,

we have:

    x = 1/3 (1-2 ϵ_s), n_1 = (1 - ϵ_0)/6, n_2=1/6 (3 ϵ_0+2 ϵ_s-1).


We define the Norell-weight enumerator W_N(M;x,n_1,n_2) of a code M, to be its complete weight enumerator, with the assignment y_22=y_21=y_12=y_11=y_01=y_02≡ n_1 and y_10=y_20=n_2. This weight enumerator is invariant under local Clifford transformations belonging to the Norell subgroup of the Clifford group.

The success probability for distillation by the code M is P=W_N(M^⊥; x,n_1,n_2). 

The output functions are:

    x'     =     W_N(M; x,n_1,n_2)/W_N(M^⊥; x,n_1,n_2) 
    
    n_1'     =     W_N(M+X̅;x,n_1,n_2)/W_H(M^⊥; x,n_1,n_2) 
    
    2n_2'    =     1-x'-6x_1'


For a tight distillation routine, we require that n_2'=0 when n_2=0. We can write:

    n_2'= W_N(M+Z̅;x,n_1,n_2)/W_H(M^⊥; x,n_1,n_2)

so this implies that we require

    0 = W_N(M+Z̅;x,n_1,0)

for a tight distillation scheme. This means there should be no words in M+Z̅ whose entries are only (0,0) or (1,i), or (2,i) which lead to a term in the weight enumerator proportional to  n_1 and x. 




§ HADAMARD STATES

Suppose we wish to distill the Hadamard state. The discrete Wigner function of the Hadamard state is of the form: 

    [ h_1 h_2 h_2; h_1 h_2 h_2;   x h_1 h_1 ]

These can be used to parameterize noisy input states. For the mixed state

    ρ̂ = (1-ϵ_-1 - ϵ_s) |H,1⟩⟨H,1|+ϵ_s |S⟩ + ϵ_-1|H,-1⟩,

we have:

    x     =    1/3 (1-2 ϵ_s), 
     h_1     =    1/6 (1-12 b) ϵ_-1+1/6 (1-6 b)ϵ_s+b, 
     h_2     =    (2 b-1/6) ϵ_-1+b ϵ_s-b+1/6

where: b=1+√(3)/12

We define the Hadamard-weight enumerator W_H(M; x, h_1, h_2) of a code M, to be its complete weight enumerator, with the assignment y_10=y_20=y_01=y_02≡ h_1 and y_11=y_12=y_21=y_22≡ h_2. This weight enumerator is invariant under local Clifford transformations belonging to the Hadamard subgroup of the Clifford group.

The success probability for distillation by the code M is P=W_H(M^⊥; x,h_1,h_2). 

The output functions are:

    x'     =     W_H(M; x,h_1,h_2)/W_H(M^⊥; x,h_1,h_2) 
    
    h_1'     =     W_H(M+X̅;x,h_1,h_2)/W_H(M^⊥; x,h_1,h_2) 
    
    4h_2'    =     1-x'-4h_1'


A tight distillation routine means that, h_1'=0 when h_1=0. This implies that h_2'=(1-x')/4



§ A DISTILLATION ROUTINE FOR THE HADAMARD STATE


For Hadamard states, we find the 17-qutrit CSS code, constructed from two copies of the following classical self-orthogonal ternary code:

    M_c=    (
    [ 1 0 0 0 0 0 1 2 0 1 2 0 2 1 0 2 1; 0 1 0 0 0 0 2 1 0 2 1 0 2 1 0 2 1; 0 0 2 0 1 0 2 1 0 1 2 0 0 0 0 0 0; 0 0 0 2 1 0 1 2 0 2 1 0 0 0 0 0 0; 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1;   ])

gave rise to an almost tight distillation routine for hadamard states. For this code,

    ϵ_s'     =    ϵ_s(128 ϵ_-1^2/9+O(ϵ_-1^3))+ϵ_s^2 (128 ϵ_-1^2/3+O(ϵ_-1^3))+O(ϵ_s^3) 
    ϵ_-1'     =    (ϵ_-1+O(ϵ_-1^3))+ϵ_s(ϵ_-1+O(ϵ_-1^3))+ϵ_s^2 (16/9+ϵ_-1+32 ϵ_-1^2+O(ϵ_-1^3))+O(ϵ_s^3)


We also find

    M_c' =  (
    [ 1 0 0 0 0 0 0 0 0 1 2 0 0 1 0 2 2; 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 2 2; 0 0 1 0 2 0 0 0 0 1 2 2 0 1 0 0 0; 0 0 0 1 2 0 0 0 0 2 1 1 0 2 0 0 0; 0 0 0 0 0 1 0 2 0 1 2 0 2 1 0 0 0; 0 0 0 0 0 0 1 2 0 2 1 0 1 2 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2;   ])

gave rise to an almost tight distillation routine for hadamard states. These are codes 1 and 4 in our scheme. 
For code 4, we have

    ϵ_S'     =    ϵ_S (52/289+2880 ϵ_-1^2/289+O(ϵ_-1^3))+ϵ_S^2 (12324/83521+2197440 ϵ_-1^2/83521+O(ϵ_-1^3))+O(ϵ_S^3) 
    ϵ_1'     =    (ϵ_-1+O(ϵ_-1^3))+ϵ_s(237 ϵ_-1/289+O(ϵ_-1^3))+ϵ_s^2 (300/289+56169 ϵ_-1/83521+10728 ϵ_-1^2/289+O(ϵ_-1^3))+O(ϵ_s^3)




These routines distill in the ϵ_s direction, but they do not distill in the ϵ_-1 direction. When ϵ_s=0, ϵ_-1'=ϵ_-1 exactly, and when ϵ_s ≠ 0, ϵ_-1' is slightly larger than ϵ_-1. These routines can be applied to move a state onto the line connecting |H,1⟩ to |H,-1⟩. In principle it can also be used to distill |H,-1⟩ states from anywhere on the distillable region of this line if ϵ_s ≠ 0. 

Both can be written as tensor networks, using the 4 qutrit perfect tensor, we conjecture.


We found that there are no 17-qutrit CSS codes using our construction, that distill the Norell state or the strange state. 










§ APPENDICES



§ PROJECTING PHASE POINT OPERATORS ONTO A STABILIZER CODE
 
In this appendix we prove a lemma which is needed for the proof of Theorem <ref> in the main text.


Let 𝒮 be a group of (conventionally-normalized) commuting n-qudit Heisenberg-Weyl operators.

    (  Π̂^0_𝒮Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮^⊥
    
       1     (u⃗|v⃗) ∈𝒮^⊥.
 



𝒮 can be generated by n-k independent Heisenberg-Weyl displacement operators, which we assume have been chosen.

Let 

    Π̂^0_𝒮 = 1/p^n-k∑_M ∈𝒮 M
 
denote the projector onto the ω^0 eigenspace of each M ∈𝒮. Let Π̂^s⃗_𝒮 denote the projector onto a non-trivial subspace, labeled by the vector of syndromes s⃗ of the n-k generators of 𝒮. 

Explicitly, we can write 

    Π̂^s⃗_𝒮 = 1/p^n-k∑_M ∈𝒮ω^f(s⃗,M) M,

for some phases f(s⃗,M) that depend on the syndrome s⃗. Clearly f(0,M)=0. For s⃗≠ 0, then it is easy to see that f(s⃗,M) takes on all the values j ∈ℤ_p, i.e.,  { 0, 1,  …, p-1}, and 

    |{ M | f(s⃗,M)=j }|=p^n-k-1.

(One way to prove this is to note that the operators ω^f(s⃗,M M form an abelian group, and there is a group homomorphism that maps ω^f(s⃗,M M →ω^f(s⃗,M).)

Note also that 

    Π̂^s⃗_𝒮 M = 
        0     M ∉𝒮
    
        p^k ω^f(s⃗,M^-1)    M ∈𝒮.

Therefore 

    (Π̂^s⃗_𝒮Â(0,0)^⊗ n)      =    1/p^n∑_u⃗,v⃗(Π̂^s⃗_𝒮D̂(u⃗,v⃗) ) 
        =    1/p^n∑_M ∈𝒮ω^f(s⃗,M^-1)p^k
        =    p^k/p^n· p^n-k-1( 1 + ω + ω^2 + …ω^p-1) =0    s⃗≠ 0 
    
        1    s⃗=0


Note that 

    D̂(u⃗,v⃗)^-1Π̂^0_𝒮D̂(u⃗,v⃗) = Π̂^s⃗_𝒮,

where the syndrome s⃗=0 if D̂(u⃗,v⃗) commutes with all M ∈𝒮, and s⃗≠ 0 otherwise. The condition that D̂(u⃗,v⃗) commutes with all M ∈𝒮 is equivalent to the condition (u⃗,v⃗) ∈𝒮^⊥. Therefore,

    Π̂_𝒮Â(u⃗,v⃗)
           =    ( D̂(u⃗,v⃗) Π̂_S D̂(-u⃗,-v⃗) Â(0,0)^⊗ n) 
        =     (Π̂_S^s⃗≠ 0Â(0,0)^⊗ n)     (u⃗|v⃗) ∉𝒮^⊥
     (Π̂^0_S Â(0,0)^⊗ n)     (u⃗|v⃗) ∈𝒮^⊥,

and the result follows using equation (<ref>).


Let 𝒮^⊥ be the dual of 𝒮. |𝒮^⊥|=p^n+k. Let us also point out that, if we define,

    Π̂^0_𝒮^⊥≡∑_u,vD̅(u,v) Π̂^0_𝒮 =  1/p^n+k∑_(u⃗|v⃗) ∈𝒮^⊥D̂(u⃗,v⃗),

then one can also show 

    (  Π̂^0_𝒮^⊥Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮
    
       1     (u⃗|v⃗) ∈𝒮.



§ SOME CODES THAT DO NOT DISTILL THE STRANGE STATE


It was recently <cit.> proposed that two qutrit stabilizer codes – a [[13,1]]_3 and a [[29,1]]_3 CSS code – distill the strange state, and estimated their thresholds to be of 0.425 and ≈ 0.7, respectively. We computed the distillation performance of these codes exactly, via their simple weight enumerators, and found that, contrary to the estimates in <cit.>, these two codes do not, in fact, distill the strange state. 

The [[13,1 ,4]]_3 CSS-code of <cit.> generated from two copies of the [13,6,3]_3 maximal self-orthogonal ternary code with generator matrix is, in row-reduced form,

    M_13= [ 1 0 0 0 0 0 0 0 2 0 2 0 0; 0 1 0 0 0 0 0 2 2 1 1 2 0; 0 0 1 0 0 0 1 2 0 1 0 2 2; 0 0 0 1 0 0 1 0 2 0 1 2 2; 0 0 0 0 1 0 0 0 2 2 1 1 1; 0 0 0 0 0 1 2 1 2 0 1 1 0 ].

The weight enumerator of the [[13,1 ,4]]_3 code is

    A(z) = 1+8 z^3+600 z^6+720 z^7+4320 z^8+18320 z^9+61200
       z^10+151200 z^11+178144 z^12+116928 z^13.

This gives rise to a relation 

    ϵ' = 19683-157464 ϵ+612360 ϵ^2-1516320 ϵ^3+2708235
       ϵ^4-3744144 ϵ^5+4143960 ϵ^6-3674592 ϵ^7+2543865
       ϵ^8-1312920 ϵ^9+471264 ϵ^10-104384 ϵ^11+10713
       ϵ^12/2 (6561-52488 ϵ+204120 ϵ^2-495720
       ϵ^3+844425 ϵ^4-1078704 ϵ^5+1074888 ϵ^6-849744 ϵ^7+529395
       ϵ^8-251400 ϵ^9+85152 ϵ^10-18184 ϵ^11+1827
       ϵ^12),

plotted in Figure <ref>. As ϵ→ 0, we see that ϵ' → 3/2. The reason for this is that the pure state |S⟩^⊗ 13 is orthogonal to the codespace of the stabilizer code, so the code is completely unsuitable for magic state distillation. This can be seen from the fact that B(-1/2)=0. This feature was shared by 6 out of the 7 possible 13-qutrit codes that arise from our construction using the classical ternary codes listed in <cit.>.



The [[29,1,7]]_3 CSS code has weight enumerator

    A(z) =     1+40 z^6+4280 z^9+96 z^10+2832 z^11+196584
       z^12+198768 z^13+1773408 z^14+15542368
       z^15
        +91797024 z^16  +565547232 z^17+3037545272
       z^18+13979050848 z^19+55970778960
       z^20+192507694176 z^21
        +559711606992
       z^22+1361197350960 z^23+2723501140720
       z^24+4358977591776 z^25
        +5363568387600
       z^26+4767481212256 z^27+2724627154368
       z^28+751557878400 z^29.
 (Computing the above weight enumerator on the machines we had access to took approximately one day of computation time.)

The resulting relation between ϵ' and ϵ is ϵ' = 1937 ϵ/224+O(ϵ^2), and is shown in Figure <ref> below. The threshold for distillation is zero.



Therefore, at present, the 11-qutrit Golay code has the highest threshold for distillation of the strange state, which is still quite far from the threshold set by contextuality <cit.>. We have not yet carried out a search over 29-qutrit codes for distillation, as, to our knowledge a complete list of [29, 14]_3  self-orthogonal ternary codes is not yet available. 





ssg




 Paper Title: 2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study.





romero@fisica.unam.mx
Instituto de Física, Universidad
Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, Mexico. 


§ ABSTRACT 

We derive the most general expression for the Skyrmion topological charge for a two-dimensional spin texture, valid for any type of boundary conditions or for any arbitrary spatial region within the texture. It reduces to the usual one Q = 1/4π∬f⃗·(∂_x f⃗×∂_y f⃗) for the appropriate boundary conditions, with f⃗ the spin texture. The general expression resembles the Gauss-Bonet theorem for the Euler-Poincaré characteristic of a 2D surface, but it has definite differences, responsible for the assignment of the proper signs of the Skyrmion charges. Additionally, we show that the charge of a single Skyrmion is the product of the value of the  normal component of the spin texture at the singularity times the Index or winding number of the transverse texture, a generalization of a Poincaré theorem. We illustrate our general results analyzing in detail a two-component spinor Bose-Einstein condensate in 2D in the presence of an external magnetic field, via the Gross-Pitaevskii equation. The condensate spin textures present Skyrmions singularities at the spatial locations where the transverse magnetic field vanishes. We show that the ensuing superfluid vortices and Skyrmions have the same value for their corresponding topological charges, in turn due to the structure of the magnetic field.

2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study
    S. Sánchez-Reséndiz, E. Neri, S. González-Hernández, and V. Romero-Rochín
    
=================================================================================================================================

Keywords: Skyrmion charge, Vortices, BEC, Ultracold Quantum Matter





§ INTRODUCTION


Skyrmions are topological defects or excitations that emerge in continuous fields and spin textures due to intrinsic chiral interactions  or to the presence of external inhomogeneous magnetic or gauge fields. These excitations arise in diverse materials or fields as in nuclear physics <cit.>, particle physics <cit.>, notably in magnetic systems in condensed matter, such as in 2D electron gas <cit.>, magnetic metals <cit.>, transition metal films <cit.>, frustrated magnets <cit.> and in a host of other magnetic situations <cit.>, liquid crystals <cit.> and, the context of the present paper, in Bose-Einstein condensates, both 2D <cit.> and 3D  <cit.>. Ever since Skyrme seminal work <cit.>, its analysis has been pursued in field theoretical models such as the non-linear σ-model in high energy physics <cit.>, or in terms of the Dzyaloshinskii-Moriya theory in magnetic systems <cit.>, or within the Gross-Pitaevskii (GP) equation in Bose-Einstein condensates, as in essentially all of the theoretical works already referenced. In this paper we deal with Skyrmions generated by an external gauge field in a two-component spinorial Bose-Einstein condensate in two spatial dimensions, described by two coupled GP equations. This stylized model allows for a simple and clear description of  spin textures that can show a rich variety of Skyrmions, with the allowance for the determination of all its  inherent and topological properties. In most of the experimental Skyrmion realizations, BEC studies have been performed in real alkaline weakly interacting Bose clouds with spin F=1  <cit.> and  F = 2 <cit.>, however, the topological properties of a given spin structure should be independent of their physical origin.




An outstanding feature of Skyrmions is their topological character ingrained and described by its topological charge. In most of the studies on 2D Skyrmions  the topological charge has been considered to be given by the expression <cit.>

    Q = 1/4π∬f⃗·( ∂f⃗/∂ x×∂f⃗/∂ y) dxdy ,

where f⃗(x,y) is the 3D vector spin texture or continuous field of the problem at hand, obeying f⃗·f⃗ = 1 everywhere in the XY-plane. It is expected that the charge Q is an integer. This expression, however, requires particular boundary conditions. Skyrme <cit.> and others afterwards <cit.>  considered that f⃗ = constant as |r⃗| →∞ and, in such a case, Q is a positive or negative multiple integer, with the interpretation that Q is the number of times the mapping of f⃗ covers the unit sphere. Additionally, if the spin texture lies on the plane at the boundary, then Q = ± 1/2, a defect called a half Skyrmion <cit.>. These boundary conditions, however, need not appear or exist in a given physical problem <cit.> or in a multiple Skyrmion configuration, where one may need to evaluate the local charge of the Skyrmions present. In these cases expression (<ref>) cannot account for the calculation of the whole charge that, from topological consideration, must be an integer. As an example, in figure <ref> we present a spin texture arising in the 2D spin 1/2 spinor Bose-Einstein condensate (BEC) here studied, the details of such a solution given further below. We observe a configuration of three Skyrmions, one pointing on the positive z-direction and the other two in the negative one. Each defect should have its own topological charge, properly defined, and the total one of the full configuration should be the sum of the individual ones. The main purpose of this article is to show that a topological charge Q_S of the whole or part of the spin texture configuration can be defined within an arbitrary closed curve, which may or may not be the true boundary of the spin texture. Parametrizing the curve by r⃗_c(t), with t: 0 → T and  r⃗_c(0)= r⃗_c(T), we will find that such a topological charge is given by  

    2π Q_S    =   ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy + ∮_r⃗_cf⃗·( d f⃗/dt×d^2f⃗/dt^2) /|d f⃗/dt|^2 dt  
           + ∑_i=1^s sin^-1f⃗_i ·( . d f⃗/dt|_i^ +×. d f⃗/dt|_i^ -)/| d f⃗/dt|_i^ +| d f⃗/dt|_i^ -,

with Q_S ∈ℤ, a positive or negative integer. The first term is the usual one, except for a factor of 2, with A the area enclosed  by the curve r⃗_c. The second is a boundary term with the integral along the curve r⃗_c, and the third one takes into account the possible fact that the vector t⃗ = (d f⃗/dt)/|d f⃗/dt|, tangent to the parametric 3D curve f⃗(r⃗_c(t)), 
may be discontinuous at a finite number of s points along the curve. All these aspects should become clearer in the derivation given below. The above expression indeed reduces to the usual one Q_S = 2 Q for the boundary conditions described in the previous paragraph. Figure <ref> shows the same spin texture as in figure <ref> but with several boundary curves r⃗_c. The small circular ones each enclose the local Skyrmion singularities, yielding Q_S = +1 for the  two of them pointing down and Q_S = -1 for the one pointing up, using (<ref>). The large circle and the square curve enclose the three Skyrmions and yield Q_S = +1, which is the sum of the local ones. We remark that, for the square curve, the last term of (<ref>) must be taken into account since it gives rise to discontinuities on the tangent vector mentioned above.




While we claim that the above expression is a general one for any spin texture, since the derivation that we provide makes no reference to the physical origin of the spin texture, for purposes of illustrations and for a better understanding of the Skyrmion charge, we will use the spin textures arising from  a spinorial BEC in the presence of an external magnetic field, as a case study. In addition, an important and interesting point, is that such a system shows another well known type of topological defects, namely, quantum vortices <cit.>. We will explicitly show the profound relationship between the quantum vortices and the Skyrmions, namely, that their corresponding local charges have exactly the same values. The present stylized model allows for finding that, for the present problem, the external magnetic or gauge field ultimately defines the topological structure of the vortices and the Skyrmions.


The article is organized as follows. In Sections II and III we present a detailed analysis, both in a simplified analytic and in a full numerical fashion, of stationary solutions to the Gross-Pitaevskii equations of a 2D 1/2 spinorial Bose gas in the presence of an external inhomogeneous magnetic field. We show how the quantum vortices and Skyrmions can be predicted from the knowledge of the magnetic field. In Section IV, the main contribution of this article, we provide a general derivation of the topological charge of Skyrmions present in any given spin texture, for arbitrary boundary curves. Section V is devoted to a discussion between the similarities and discrepancies between the Skyrmions topological charge and the Euler-Poincaré characteristic of a surface in terms of the Gauss-Bonet theorem <cit.>. 
In an Appendix we illustrate our main result with a variety of numerical solutions of different spin textures generated by different magnetic fields, and for different types of arbitrary boundary curves. Final remarks are given in Section VI.



§ 1/2 SPINOR BEC IN THE PRESENCE OF AN EXTERNAL MAGNETIC FIELD


We consider stationary states of a two-component spinorial BEC in 2D, with a Zeeman coupling to an external magnetic field, as described by the coupled Gross-Pitaevskii (GP) equations

    μψ_α(r⃗) = [-ħ^2/2m∇^2 + 1/2 mω^2 r⃗^ 2
    + g (|ψ_+|^2 + |ψ_-|^2) ]ψ_α(r⃗) - B⃗(r⃗)·μ⃗^αβψ_β(r⃗) ,

where r⃗ = (x,y) and α = +,- denotes the two components of the spinor. Summing over repeated indices is assumed. μ is the chemical potential and the magnetic dipole operator is given by

    μ⃗^αβ = μ_0 σ⃗^αβ,

where σ⃗= x̂σ_x + ŷσ_y + ẑσ_z  are Pauli matrices and  μ_0 > 0 the magnitude of the magnetic dipole moment. The magnetic field is 

    B⃗(r⃗) = x̂ B_x(r⃗) + ŷ B_y(r⃗) + ẑ B_z .

We denote B⃗_T(r⃗) = B_x(r⃗) x̂ + B_y(r⃗) ŷ as the transverse magnetic field and, for simplicity, we will consider B_z(r⃗) = constant . We choose  B⃗_T(r⃗)  such that it vanishes at certain points r⃗_1, r⃗_2, …, r⃗_s. We shall see that at those points there appear quantum vortices and the origins of Skyrmions <cit.>.  The normalization condition is

    ∫ d^2 r  (|ψ_1|^2 + |ψ_2|^2) = 1 .


Writing the solution as ψ_α = √(ρ_α) e^i Θ_α, the superfluid density ρ_α and  the superfluid velocity v⃗_α, in each component, are given by

    ρ_α   =    |ψ_α|^2 ,
    v⃗_α    =    ħ/m∇Θ_α.


The circulation C of the velocity field in an arbitrary closed circuit C is defined by

    C =  ∮_ Cv⃗_α· dr⃗.

As we will explicitly calculate, if the circuit does not enclose a quantum vortex the circulation vanishes, but when a vortex or vortices are enclosed, the circulation yields the vortex topological charge Q_V within the circuit <cit.>

    C =  2 πħ/m Q_V

with Q_V ∈ℤ, a positive or negative integer. 


The condensate spin texture f⃗(x,y) is an observable, independent of the spin basis, and defined at any spatial point within the condensate,

    f⃗ = 1/ρ_+ + ρ_-(ψ_+^* ψ_-^*) [x̂σ_x + ŷσ_y + ẑσ_z]
    (
    [ ψ_+; ψ_- ]) .

It can be verified that the spin texture has magnitude one f⃗·f⃗ = 1 everywhere.  


Writing f⃗ = x̂ f_x + ŷ f_y + ẑ f_z, the components explicitly are

    f_x    =   ψ_+^* ψ_- + ψ_-^* ψ_+/ρ_+ + ρ_-
       =    2 √(ρ_+ρ_-)/ρ_+ + ρ_-cos(Θ_- - Θ_+) 
    
    f_y    =   - iψ_+^* ψ_- - ψ_-^* ψ_+ /ρ_+ + ρ_-
       =    2√(ρ_+ρ_-)/ρ_+ + ρ_-sin(Θ_- - Θ_+)
    
    f_z    =   ρ_+ - ρ_- /ρ_+ + ρ_-.




§ VORTICES AND SKYRMIONS IN THE VICINITY OF POINTS OF VANISHING TRANSVERSE MAGNETIC FIELD


While we will consider a variety of magnetic fields, all have in common that they have spatial points r⃗_1, r⃗_2, …, r⃗_s where the transverse magnetic field vanishes, B⃗_T (r⃗_i) = 0. This property imprints both the appearance of vortices and of Skyrmions at those points. To understand how this occurs, with no loss of generality, we consider a transverse magnetic field that vanishes at the origin r⃗ = 0 and analyze the behavior of the spinor wavefunctions in its vicinity. For this we explicitly write both GP equations for the two spinor components, assuming that B_z = 0,

    μψ_+    =   [-ħ^2/2m∇^2 + 1/2 mω^2 r^2
    + g ρ]ψ_+ - μ_0 (B_x - i B_y) ψ_- 
    μψ_-    =   [-ħ^2/2m∇^2 + 1/2 mω^2 r^2
    + g ρ]ψ_- - μ_0 (B_x + i B_y) ψ_+ ,

where ρ = |ψ_+|^2 +  |ψ_+|^2 and r= √(x^2+y^2). In the vicinity of the origin the magnetic field generically vanishes as <cit.>

    B_x - i B_y ≈ B_0  r^n  e^i n φ

where B_0 is the field strength, not necessarily real, tanφ = y/x and n a positive integer.  We note that if B_0 is real, the resulting field obeys ∇·B⃗ = 0 and ∇×B⃗ = 0, namely, it would be a true magnetic field. However, one can consider arbitrary “artificial” gauge fields <cit.> that do not necessarily satisfy Maxwell equations but that allow for different sorts of fields and which in turn give rise to different spin textures, as we show in the rest of the paper. In this way, generically, there are four different types of fields in the vicinity of a vanishing location, with B_0 real, positive or negative,

    I)B_x - i B_y    ≈    B_0  r^n  e^i n φ II) B_x - i B_y ≈ i  B_0  r^n  e^i n φ
     III) B_x - i B_y    ≈    B_0  r^n  e^-i n φ IV) B_x - i B_y ≈ - i  B_0  r^n  e^-i n φ.

The first two obey Maxwell equations, the last two do not; as a matter of fact, case (I) with n=1 is commonly produced in a Ioffe-Pritchard trap, and n=2 is a quadrupole trap <cit.>.  Figure (<ref>) shows the structure of the magnetic fields for the case n = 1. Case (III) is the hedgehog configuration usually considered in Skyrmion discussions <cit.>. 

As we will see, the local structure of the transverse magnetic field at its vanishing points plays a fundamental role in the final signs of the topological charges of the vortices and Skyrmions. As discussed in Section V below, such a structure can be associated to a topological quantity, namely, the Index of the field at the vanishing point. <cit.>


Let us consider case (I) for the moment. We will summarize all cases below.  For this case there are two possible solutions,

    i)ψ_+(r⃗)    =    F_+(r) e^-inφ andψ_-(r⃗) = F_-(r)  
     ii)ψ_+(r⃗)    =    F_ -(r)   andψ_-(r⃗) = F_+(r) e^inφ.

Substitution of these solution into GP equations (<ref>) shows that, very near r = 0, one obtains

    F_+(r)    ≈    A  r^n 
    
    F_-(r)    ≈   μ/g,

where A is a positive constant that can be numerically determined. The important features are that F_+(r) vanishes as r^n while  F_-(r) yields a constant as r → 0. Therefore, case (i) in (<ref>) is a vortex in the ψ_+ component and a density “spike” in the ψ_- one (see below), and the opposite in (ii), namely a spike in ψ_+ and a vortex in ψ_-. As we see now, in the former case the vortex charge is Q_V = - n, while in the latter Q_V = + n. This can be concluded right away from the velocity fields,

    i)v⃗_+(r⃗)    =    - n ħ/mφ̂/r andv⃗_-(r⃗) = 0  
     ii)v⃗_+(r⃗)    =    0   andv⃗_-(r⃗) = + n ħ/mφ̂/r,

where φ̂= - x̂sinφ + ŷcosφ. Indeed, the velocity fields circulate in the clockwise (negative) direction in the ψ_+ component for case (i), and in the anticlockwise (positive) direction for the ψ_- one, in case (ii). The circulations are, using a circular contour of radius r and dr⃗ = φ̂  r dφ,

    C_±   =   ∮_ Cv⃗_±· dr⃗
        =    ∓  n ħ/m∮ dφ
        =    ∓ 2π nħ/m,

finding the values of the topological charges already quoted. Clearly, if the vortex is not enclosed by the contour, the circulation vanishes, C_± = 0. Figure <ref> shows the solution to the GP equations (<ref>), as well as the velocity fields, corresponding to figures (<ref>) and (<ref>), including the magnetic field that generates such a configuration; see the Appendix for details of the calculation. One can observe that, in the vicinity of the vortices, the top one is of type (II) while the lower ones are of type (I) in (<ref>). In panels (A) and (C) of <ref> we can also see what we mean by a “density spike”: while the vortex in one component occurs, accompanied by a density depletion, in the other component there is a rapid increase of the density at the same spatial location, namely a spike, identified in the figure by a bright spot in the density profile. This increment of the density in one spinor component almost cancels the depletion at the vortex in the other component <cit.>.  
From expressions (<ref>) of the fields, the vortex structure of case (II) is the same as case (I) just discussed, however in cases (III) and (IV) the solution is as in (I) given by (<ref>), but with the velocity circulation reversed and, therefore, the charges are now Q_V = +n for solution (i) and Q_V = -n for (ii). As we will see below, the Skyrmion charges will have exactly the same values as Q_V, with their own spin texture structure.




We now turn our attention to the spin texture and, for the sake of argument, we also study case (I). Again, in the vicinity of zero field, one finds the spin texture from (<ref>), for both solutions (i) and (ii) in (<ref>)

    f_x    =   F_+ F_-/F_+^2 + F_-^2cos n φ
    
    f_y    =    -  F_+ F_-/F_+^2 + F_-^2sin n φ
    
    f_z     =    ±F_+^2 - F_-^2/F_+^2 + F_-^2.

where the ± signs in front of f_z refer to solutions (i) and (ii), respectively, see (<ref>). First, we note that the transverse spin texture f⃗_T = f_x x̂ + f_y ŷ  is parallel and in the same direction to the transverse magnetic field B⃗_T, as physically expected, see (I) in (<ref>), thus showing that both have the same spatial pattern. This is true in general and it is exemplified in the gallery of figures in the Appendix.
Evidently, as r → 0, the components f_x and f_y vanish since either F_+ or F_- vanish as well, and the spin texture becomes parallel to the ẑ vector, pointing upwards or downwards, f⃗ = ±ẑ. This latter direction is determined by whether the solution is (i) or (ii) as above in (<ref>): for solution (i) |ψ_+| = F_+(0) = 0, |ψ_-| = F_-(0)  0 and, therefore, the spin texture points downwards, f⃗(0) = - ẑ; for solution (ii) |ψ_-| = F_+(0) = 0, |ψ_+| = F_-(0)  0 and the spin texture now points upwards, f⃗(0) = + ẑ. For the other fields (II), (III) and (IV) the same rule applies for the peaks: if the vortex appears in the ψ_+ component, the texture points downwards, but if the vortex is in ψ_- then it points upwards. The ensuing structure of the spin texture is what we call the Skyrmion and, for the sake of describing it, we shall call the location r⃗_i where f⃗(r⃗_i) = ±ẑ, as the origin of the Skyrmion. 





§ THE TOPOLOGICAL CHARGE OF THE SKYRMIONS


As mentioned in the Introduction, the topological charge of the Skyrmions is usually given by (<ref>) but, as also argued, such an expression requires particular boundary conditions on the spin texture f⃗. As also already mentioned, in general those boundary conditions need not occur, or a spin texture can have a multiplicity of coexisting local Skyrmions, whose boundaries among them can also be quite arbitrary. Moreover, in all cases one should be able to asses the individual charges of the local Skyrmions in order to validate that the charges add as any topological index. The spin textures here presented are clear examples of the just two described situations. There is another fundamental aspect to be considered as well. In descriptions of common spin textures and their charge given by (<ref>), it is affirmed that the topological charge is the number of times that the unit sphere is covered by the spin texture. This follows, on the one hand,  by assuming that f⃗(x,y), being a unit vector, is by itself a parametrization or mapping of the unit sphere and, on the other, again, by the mentioned usual boundary conditions. While such a parametrization can indeed be assumed, there is subtle aspect that we believe has not been addressed before, namely, that different spin textures give rise to mappings to the unit sphere but with different orientations. That is, the normal vector to the sphere, generated by the parametrization f⃗(x,y), can point outwards or inwards to the sphere, depending on the given spin texture. And, for configurations with several Skyrmions, this can occur within the same spin texture. This is a very interesting and important point to consider in the elucidation of a general expression for the identification of the Skyrmion topological charge. In this section we present a derivation of the Skyrmion topological charge, in an as possible straightforward fashion, by a procedure similar to the derivation of the Gauss-Bonet (GB) theorem <cit.>. We will  leave the discussion of the mentioned delicate issues to the following section, where we also argue about the similarities or, rather, dissimilarities of the Skyrmion charge with the Euler-Poincaré characteristic χ, arising from the GB theorem.


Let us assume that a spin texture f⃗(x,y) is given in all or part of the XY-plane, such that it has a single location r⃗_0 where either f⃗(r⃗_0) = +z or f⃗(r⃗_0) = -z. We call this point the origin of the Skyrmion. Consider now an area A that encloses this point. The area 
has a boundary ∂ A described by a parametrized closed curve

    r⃗_c(t) = x̂ x_c(t) + ŷ y_c(t) t : 0 → T r⃗_c(0) = r⃗_c(T) .

 Notice that although the curve is continuous, its tangent may be discontinuous at a finite number of points r⃗_c(t_1), r⃗_c(t_2), …, r⃗_c(t_s). Further, and very important,  the curve is assumed, one, to run or be oriented in the positive counterclockwise direction and, two,  the curve does not cross itself, say the red curve in figure (<ref>).
 


Evidently, since the texture f⃗(r⃗) is a unit vector, it can be considered to be the outward normal vector r̂(θ,ϕ)  to the unit sphere S^2, with (θ,ϕ) the usual azimuthal and polar sphere angles, (please note that we use ϕ for the polar angle in 3D and φ in 2D)

    cosθ   =    f_z 
    tanϕ    =    f_y/f_x.

At the same time one could assume that f⃗(r⃗) is by itself a parametrization of the unit sphere. As discussed in detail below why, we will not consider this point of view. As we show below, perhaps surprisingly at  first sight, the two mentioned assumptions are not necessarily consistent with each other. Nevertheless, any value of the spin texture f⃗(r⃗) can be identified with a single point (θ,ϕ) on the sphere, although the opposite is not true in general.  That is, depending on the order of the Skyrmion, parts of the sphere can be covered more than once. Figure (<ref>) shows a simple example of these observations, where we can see how the spin texture within and on the square are mapped to unique points on the sphere. And, with the identification of the unit normal and the spherical angles, the usual orthogonal sphere unit vectors (r̂, θ̂, ϕ̂) along the directions (r,θ,ϕ), can also be expressed in terms of the spin texture,

    r̂    =    x̂ f_x + ŷ f_y + ẑ f_z 
    θ̂    =    x̂f_z f_x/√(f_x^2+f_y^2)+ ŷf_z f_y/√(f_x^2+f_y^2) - ẑ√(f_x^2+f_y^2)
    ϕ̂    =     -x̂f_y/√(f_x^2+f_y^2)+ ŷf_x/√(f_x^2+f_y^2).

 

In an analogous way, we can observe that the curve r⃗_c(t) on the plane gets mapped to a 3D curve f⃗(r⃗_c(t)), denoted as C, that lies on the surface of the unit sphere. The arclength of such a curve can be found from

    ds = |df⃗(r⃗_c(t))/dt| dt ,

and the tangent t⃗(s) to the curve is given by, see again figure (<ref>) as an example, 

    t⃗(s) = df⃗(s)/ds.

This is a unit vector, t⃗·t⃗ = 1. Since it is also tangent to the sphere, then t⃗·r̂ = 0 and, therefore, t⃗ lies on the θ̂-ϕ̂ plane. We define  α as the positive angle that t̂ makes with θ̂ and, thus, we can write

    t⃗ = θ̂cosα + ϕ̂sinα.

α can take any positive value, see figure <ref>. Note again that for every texture vector f⃗ on the curve C there is an angle α = α(f⃗), but the inverse is not necessarily true.


With the above preliminaries we can now start the derivation of the expression for the topological charge Q_S of a single Skyrmion. Consider the following integral along the curve C,

    I = ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds) ds.

Before we proceed with the evaluation of the integral, we note two points. First, while the integrand may seem to be the geodesic curvature of C on the sphere, we will see that it is not in general, differing by a minus sign when it is not. And second, although the curve r⃗_c(t) on the plane has been considered to run in the positive counterclockwise direction in the XY-plane, the curve C on the sphere, in general, is not necessarily oriented in the same direction. These apparent discrepancies, to be discussed in the Section V, have their origin in the previous observation that f⃗ is not being formally considered to be a parametrization of the sphere.


 Substituting the tangent vector (<ref>) into the integral (<ref>) yields,

    ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds) ds = ∮_ C dα + ∮_ Ccosθ∇ϕ· ds⃗.

Let us analyze the first integral over the α angle. Here it is important to notice that, because of the assumptions on the boundary curve r⃗_c(t), the tangent t⃗ to the curve C is also discontinuous at the points (f⃗(r⃗_1), f⃗(r⃗_2), ⋯, f⃗(r⃗_s)). This, in turn, implies that the angle α is discontinuous at those  points: Let t⃗_i^ - be the incoming tangent of the curve at the point f⃗(r⃗_i) and  t⃗_i^ + the outgoing one. We call the corresponding angles α_i^- and  α_i^+. Hence, 

    ∮_ C dα   =   ∫_α(0)^α_1^- dα + ∫_α_1^+^α_2^- dα + ⋯ + ∫_α_s^+^α(T) dα
        =    α(T) - α(0) - ∑_i=1^s ε_i,

where α(0) = α(f⃗(0)) and α(T) = α(f⃗(T)) with f⃗(0) = f⃗(T). We have assumed that such a point is not on any discontinuity. ε_i = (α_i^+ - α_i^-) is the angle between the tangents t⃗_i^ + and t⃗_i^ - at the discontinuity, given by

    sinε_i = f⃗_i ·(t⃗_i^ -×t⃗_i^ +) ,

see figure <ref>. 
Below we return to the analysis of the values of α(T) - α(0). 


The second integral in (<ref>), over the angle ϕ, can be rewritten in a way that allows for the determination of another contribution of the Skyrmion charge, using Green theorem. For this, we first recall that the origin r⃗_0 of the Skyrmion occurs where the spin texture becomes f⃗(r⃗_0) = ±ẑ, that is, where the azimuthal angle equals θ = 0 or π, respectively. Since the contour C encloses either of those points, the integrand of the second integral in (<ref>) is not analytic at θ = 0 or at π, preventing the direct use of Green theorem. However, adding and subtracting an appropriate quantity one obtains 

    ∮_ Ccosθ∇ϕ· ds⃗ = - ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗±∮_ C∇ϕ· ds⃗,

where the + sign is used when the singularity is at θ = 0, namely f⃗(r⃗_0) = + ẑ, and the - sign when θ = π or f⃗(r⃗_0) = - ẑ. In this way, the integrand of the first integral on the right side of (<ref>) is analytic at the respective points and the integral can be performed using Green Theorem. This yields, up to a sign, the solid angle subtended by the contour C around the North or the South Pole,

    ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗  = ±∬_ C dΩ.

The sign on the right side depends on the positive or negative orientation of the contour C, which in turn depends on the structure of the Skyrmion in the vicinity of the singularity. The above term can also be seen as a Berry phase type contribution <cit.>. However, the previous integral can be written in a manner that is more appropriate for our purposes, namely, by rewriting it in terms of the spin texture itself:

    ∮_ C(± 1 - cosθ) ∇ϕ· ds⃗    =    ∮_r⃗_c(± 1 - f_z )  ∇(arctanf_y/f_x) · d r⃗
        =    ∬_A{∂/∂ x[ f_z ∂arctanf_y/fx/∂ y] -
    ∂/∂ y[ f_z ∂arctanf_y/fx/∂ x] } dxdy 
        =    ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy .

The first equality follows by writing the angles θ and ϕ in terms of the f⃗ components, see (<ref>), hence allowing for expressing the line integral along the original contour r⃗_c(t). The second equality is obtained by means of Green Theorem, the surface integral being performed over the whole area A enclosed by the curve r⃗_c(t). The derivation of the last equality is a laborious exercise that requires the use of f⃗·f⃗ = 1. Note that in the expression of the last line one does not need to worry about the sign of the integral, as this will emerge naturally from the current spin texture.  The important issue of these signs will be further addressed below. 


Collecting the previous results (<ref>), (<ref>) and (<ref>) into (<ref>), we partially obtain

    α(T) - α(0) ±∮_ C∇ϕ· ds⃗   =   ∬_A f⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy + ∮_ Cf⃗(s) ·(t⃗(s) ×dt⃗(s)/ds)ds 
           + ∑_i=1^N arcsinf⃗_i ·(t⃗_i^ +×t⃗_i^ -)  .

First of all, it is simple to conclude that the left-hand-side of this expression must be 

    2π Q_S = α(T) - α(0) +  ∮_ C∇ϕ· ds⃗

 with Q_S ∈ℤ. This integer will be carefully  identified below as the topological charge of the Skyrmion enclosed by the curve r⃗_c. Quite generally, it must be an integer since, first, f⃗(0) = f⃗(T), then sinα(f⃗(0)) = sinα(f⃗(T)) and, hence, α(T) - α(0) = 2 π N with N an integer. Similarly, since

    ±∮_ C∇ϕ· ds⃗   =   ±∮_ C d ϕ
        =     2 π M,

with M a positive or negative integer, since the contour C encloses either the North or the South Pole of the unit sphere. Therefore, Q_S =  N +  M is an integer. However, it remains to understand how this integer number relates to the given spin texture f⃗(x,y) and to its behavior in the neighborhood of the singularity f⃗(r⃗_0) = ±ẑ. 



Let us address the role of the angles α and ϕ in the determination of the integer value of the Skyrmion charge Q_S in (<ref>). First, one can conclude that α(0) = α(T), namely N = 0. 
This follows from expression (<ref>) for α and from the geometrical facts, one, that 
θ̂ always points in the direction of increasing θ, two, that the tangent vector t⃗ has a definite orientation, counter or clockwise for any given curve C and, three, that either the North or South Pole are enclosed by C.
These three facts imply that the angle α retraces itself around C without completing a single turn. Thus, this leaves the value of Q_S to depend solely on the integral of ϕ in (<ref>) and (<ref>). This has an intrinsic topological solution. For this, we write the integral over ϕ in terms of an integral of the spin texture over the original contour r⃗_c(t),

    ±∮_ C_i dϕ = ±∫_r⃗_c^(i)∇(arctanf_y/f_x) · d r⃗_c^(i).

The interesting and relevant observation if that the integral on the right-hand-side is precisely the Index I(f⃗_T(r⃗_0)) of the transverse spin texture field f⃗_T(r⃗), times 2π, around r⃗_0 <cit.>. Such an Index is I = ± n with n a positive integer that counts the number of turns the angle ϕ does around the corresponding Pole. 
Based on our findings that the  transverse spin texture f⃗_T(r⃗) follows the transverse magnetic field B⃗_T(r⃗), we can extrapolate and assert that any spin texture, in the vicinity of the singularities f⃗_T(r⃗_0) = 0, behaves just as the magnetic field given in (<ref>). Hence, for cases (I) and (II) the index is I = -n while for (III) and (IV) one obtains I = +n. Therefore, in a very general way, the topological charge of a single Skyrmion can be expressed as,

    Q_S = f_z(r⃗_0)  I(f⃗_T(r⃗_0))  ,

demonstrating that the topological charge is a positive or a negative integer, its sign being determined by both, the value of f_z and the Index of f⃗_T, at the singularity. We believe that this last identification of the Skyrmion charge has not been pointed out before. Figure <ref> shows two cases of  Skyrmions with singularities at f⃗ = +ẑ and f⃗ = +ẑ, produced by the same transverse magnetic field but with opposite B_z component; all the thermodynamic properties are the same, except that the Skymion charges have opposite values.




Returning to the identification of  Q_S in (<ref>), and rewriting the integrals in terms of the spin texture f⃗ and the original boundary curve r⃗_c, one arrives at the sought for expression for the topological Skyrmion charge given in (<ref>) in the Introduction, but for a single Skyrmion. The extension to the total topological charge of a spin texture that holds an arbitrary number N of Skyrmions, whose origins are at r⃗_1, r⃗_2, …, r⃗_N, is straightforward. This is achieved with the 
reasonable assumption that, while in the vicinity of the origins the spin texture behaves as in the case of a single Skyrmion, the “local” textures can be  joined smoothly in a continuous and differentiable fashion. This assumption is verified by the numerical examples shown in the Appendix.
Very importantly, the boundaries between those local cases can be quite arbitrary and complicated, such as the boundary of our example of the three Skyrmions in figure (<ref>). This corroborates the importance of having obtained the local charge of a Skyrmion for an arbitrary boundary curve r⃗_c. Figure <ref> shows a 
sketch of the union of local boundary curves r⃗_c^(i) of several Skyrmions that yields a total curve r⃗_c, inside of which all Skyrmions are enclosed. Writing the different contributions of a local Skyrmion charge as Q_S^(i) = Q_2^(i) + Q_1^(i) +Q_0^(i), where

    Q_2^(i)   =   ∬_A_if⃗·(∂f⃗/∂ x×∂f⃗/∂ x) dx dy 
    
    Q_1^(i)    =    ∮_r⃗_c^(i)f⃗·( d f⃗/dt×d^2f⃗/dt^2) /|d f⃗/dt|^2 dt  
    
    Q_0^(i)    =    ∑_j=1^s^(i)sin^-1f⃗_j ·( . d f⃗/dt|_j^ -×. d f⃗/dt|_j^ +)/| d f⃗/dt|_j^ -| d f⃗/dt|_j^ +,

it is evident, since all boundary curves r⃗_c^(i) run with same positive orientation, that 

    Q_S = ∑_i=1^N Q_S^(i)

with an analogous expression for all contributions Q_2, Q_1 and Q_0. The obtained expression for the total Skyrmion charge Q_S = Q_2 + Q_1 + Q_0 is given by (<ref>) of the Introduction, with r⃗_c = ⋃_i=1^N r⃗_c^(i) the external boundary curve, A =  ⋃_i=1^N A_i, the total area, and the sum over the angles is only on those on the external boundary r⃗_c since all the internal angles cancel out, see figure <ref>. It is also of interest to conclude that the total Skyrmion charge can be written as,

    Q_S = ∑_i=1^N  f_z(r⃗_i)  I(f⃗_T(r⃗_i))  .

As we will argue in the following section, this last expression appears as a generalization of Poincaré theorem <cit.>. 



In the Appendix we illustrate the results of this section with explicit numerical calculations of several spin textures obtained by solving GP equations (<ref>) for our case study of a spinor BEC in the presence of an external field. 




§ DISSIMILARITIES BETWEEN THE SKYRMION CHARGE Q_S AND THE EULER-POINCARÉ CHARACTERISTIC Χ



The expression (<ref>) for the Skyrmion topological charge Q_S has a deceivingly similar appearance to the Euler-Poincaré characteristic χ of the Gauss-Bonet theorem of a given 2D surface embedded in 3D <cit.>. As a matter of fact, the derivation here presented for Q_S is similar, but not equal, to the derivation of the Gauss-Bonet theorem presented by do Carmo <cit.>. However, as we have been alluding to in the text, there are notable differences between them: A technical difference being that the spin texture vector f⃗ cannot always be both the normal vector and the parametrization to the sphere at the same time. A fundamental difference being that in the Gauss-Bonet theorem the main role is played by the surface itself, while in the Skyrmion case the protagonist is the spin texture f⃗. 


To be precise, given an orientable surface S, the GB theorem states that the Euler-Poincaré characteristic χ of a region R of the surface S, with a given closed boundary C and with M possible exterior angles θ_i is, <cit.>

    2πχ = ∬_R K  dA + ∮_C k_g ds + ∑_i=1^M θ_i

where K is the gaussian curvature at a point on the surface, k_g the geodesic curvature at a point of the curve C, the boundary of R. The curve C must be oriented in the positive direction. These are intrinsic properties of the surface and of curves drawn on it. If the region R is a simple region of the surface S, then χ = +1 always.  If the surface is compact with no boundaries, then χ = 2 - 2g where g is the genus or number of handles of the surface. And there are surfaces with χ = -1 such as the pseudo sphere and the so-called pair of pants. The above expression for χ is written in an invariant form, independent of any parametrization of the surface. For sure, one can write it in terms of a parametrization, but this one must be consistent with an assumed orientation of the surface and with the positive orientation of the curve C. To see this, let X⃗(σ_1,σ_2) be such a parametrization, then, the normal to the surface is

    n̂ = ∂_1 X⃗×∂_2 X⃗/|∂_1 X⃗×∂_2 X⃗|,

where ∂_α is the derivative with respect to σ_α. This normal must have the same orientation on the whole surface. For instance, if the surface is a sphere, then n̂ must point outwards or inwards over the whole sphere. The gaussian curvature can be expressed as,

    K = n̂·(∂_1 n̂×∂_2 n̂) .

Along the same lines, the tangent to the curve C is

    t⃗ = ∂X⃗(s)/∂ s,

and, hence, the geodesic curvature of C is

    k_g = n̂·(t⃗×dt⃗/ds) .

The exterior angles θ_i are, in an obvious notation,

    sinθ_i = n̂·( t⃗_i^-×t⃗_i^+) .


Therefore, if we compare χ, from (<ref>)-(<ref>), with Q_S from (<ref>), they look quite similar if we identify the parametrization X⃗ with f⃗, the surface thus being a sphere, and the normal n̂ with f⃗. However, these two identifications are incompatible in general. This can be readily seen if we calculate the normal to the sphere from the parametrization X⃗ = f⃗, 

    n̂ = ∂_x f⃗×∂_y f⃗/|∂_x f⃗×∂_y f⃗|.

One finds that, depending on the spin texture, n̂ = f⃗ or n̂ = -f⃗. And this can occur within the same spin texture, such as in the example of figure (<ref>). Therefore, if f⃗ is a parametrization, then f⃗ cannot be the normal always, and viceversa. This would result in severe problems since the normal would discontinuously change on the sphere. Hence, we have opted to consider 
that f⃗ is the outward normal to the sphere, at the expense of abandoning f⃗  as a parametrization. It is then not a surprise that we do not obtain GB theorem and, therefore, Q_S is not χ, since the sign of the former depends on the spin texture f⃗ and the latter is a topological intrinsic property of the given surface. Indeed, the fact that we can find positive or negative Skyrmion charges, simply by inverting the value of the origin of the Skyrmion f⃗(r⃗_0) is a consequence of not having considered f⃗ as a parametrization of the sphere. In this way, the integrands of Q_2 and Q_1 are not necessarily the gaussian or the geodesic curvatures of the sphere or of the curves drawn on it.
It is also of interest to point out that there is a theorem, due to Poincaré <cit.>, that states that χ = ∑_i I_i with I_i the Indices of a differentiable vector field on a compact surface with isolated singular points. That is, given the surface, the singularities must exist and their Indices add to χ. In the present case, see (<ref>), the singularities can be quite arbitrarily given by the the spin texture and, as a consequence, their topological charges depend on an additional sign given by f_z(r⃗_0) at the singularity. 


The previous observations are not definite, however, as we are aware that topology can yield results that may be difficult to visualize geometrically. That is, although we have pointed out evident differences between the Skyrmion topological charge and the Euler-Poincaré characteristic, we also concede that their respective expressions appear strikingly similar. Hence, we leave open the question of whether they may truly be the same topological object or not, perhaps for not so obvious or evident surfaces such as the unit sphere. This should be the subject of further scrutiny. Along the same lines, an approach in terms of homotopy theory <cit.> should complement our approach, basically rooted in differential geometry.




§ FINAL REMARKS


In this article we have provided a derivation of the Skyrmion topological charge Q_S of an arbitrary spatial region A ⊂ℝ^2 of a spin texture f⃗(r⃗), given by (<ref>). If the spin texture is continuous and differentiable everywhere, except perhaps at the origins of the Skyrmions, where f⃗ = ±ẑ, one can find the local topological charges of the present Skyrmions and, simply by joining the local regions, one can calculate the topological charge of several Skyrmion regions and, eventually, of the whole area where the spin texture is defined. While we can calculate the topological charge as the sum of the three main contributions Q_S = Q_2 + Q_1 + Q_0, one can also demonstrate that the local Skyrmion charge is the product Q_S = f_z(r⃗_0) I(f⃗_T(r⃗_0)), with I the Index of the transverse spin texture at the singularity r⃗_0. The total Skyrmion charge of a region, or of the whole spin texture, is the addition of the local Skyrmion charges. It may be of formal interest to highlight that the 2D contribution Q_2, the 1D Q_1 and the 0D Q_0 are all expressed in terms of a triple scalar product involving the spin texture f⃗ times the relevant cross product of the contribution at hand. We also find that, depending on the value of the spin texture at the boundary curve or curves, one recovers the usual expression for the Skyrmion charge, given solely in terms of the contribution Q_2. For instance, if  the texture completely covers the sphere, f⃗ = ±ẑ, or f_z = 0 at the boundary, the last two terms in (<ref>) or (<ref>) vanish. Notice also that we have defined the Skyrmion charge with a factor 1/2π in (<ref>), instead of 1/4π as in (<ref>), because of the similarities  with the Gauss-Bonet representation of the Euler-Poincare characteristic. Thus, while usually the Skyrmion charge for a closed sphere is Q = ± 1, here it is Q_S = ± 2, and as the charge of a half Skyrmion is considered Q = ± 1/2, here it is a Q_S = ± 1. As we indicated in the Introduction, the derived expression is quite general and independent of the origin of the spin texture.


In order to produce spin texture within a physically sensible system, we have studied a 1/2 spinor BE condensate in the presence of an external magnetic field. This system, in addition to generate interesting and non trivial spin textures, with which we can make a detailed and systematic verification of our main result, presents quantum vortices, defects of topological nature as well. As we have indicated, one can trace the appearance of the vortices and of the Skyrmions to the behavior of the external magnetic field in the vicinity of the locations r⃗_0 where the transverse magnetic field vanishes. Indeed, by looking a the emergence of quantum vortices, say, in (<ref>), one can conclude that the vortex local topological charge is Q_V = ± I(B⃗_T(r⃗_0)), where the sign ± is if the vortex appears in the ψ_+ or ψ_- spinor component, respectively, and I(B⃗_T(r⃗_0)) is the Index of the transverse magnetic field in the vicinity of its vanishing location r⃗_0. However, since the transverse spin texture f⃗_T follows exactly the same pattern as the transverse magnetic field B⃗_T, see Appendix, then the local quantum vortex and Skyrmion topological charges are exactly the same. That is, for the present case study, the external magnetic field determines the topology of all ensuing structures. These results can definitely be tested with the currently achieved experimental capabilities in the study of spinor Bose-Einstein condensates. For other physical spin textures one can perhaps always find the underlying field or interaction responsible for the emergence of the corresponding topological structures. 


While we have left out several additional observations of this problem, related to its intrinsic differential geometry and topology, we mention one that should be of importance for further analysis. This regards the issue of what we call a “Skyrmion” structure in a spin texture. That is, the emphasis is always in the role of the location where one finds the origin of the Skyrmion f_z(r⃗_0) = ±ẑ and, therefore, one may believe that the determination of the topological charge needs of a boundary curve that encloses such a point or points. However, this is not completely true or necessary. For this we point out expression (<ref>) where we identified the Skyrmion charge in terms of the angle difference α(T)-α(0) and the contour integral over ϕ. We recall that we derived our results by assuming a contour that enclosed the singularity yielding  α(T)-α(0) = 0 and the topological charge was contained in the ϕ contour integral. However, if we choose a contour that does not enclose the singularity, then the Index of the transverse spin texture vanishes, but now α(T)-α(0)  0. And, for sure, the topological charge is now accounted for by the angle α, namely 2π Q_S = α(T)-α(0). This indicates that if we take a fully arbitrary boundary curve r⃗_c, yet closed and positively oriented, that does not necessarily enclose singularities, one still finds the topological charge contained within the encircled spin texture. In other words, the topological content of the Skyrmion is within the whole spin texture, as Topology may have indicated us in advance.


Acknowledegment. This work was partially funded by grant PAPIIT-UNAM IN117623. VRR acknowledges fruitful discussions with Leonardo Patiño at the early stages of this work.




§ APPENDIX: A GALLERY OF VORTICES AND SKYRMIONS, SOLUTIONS TO GP EQUATIONS (<REF>).


The figures throughout the text and in this Appendix are found from numerical solutions of the two coupled Gross-Pitaevskii equations (<ref>). We use a standard method described in Refs. <cit.>, in a GPU processor, within a 256 × 256 grid. We consider arbitrary dimensionless units such that the solutions are numerically stationary and the desired configuration is clearly observed. We use ħ = m = 1, the interatomic coupling g = 15000 and, in all cases, the computational box is a square of side L_0 = 56.72. The values of the isotropic frequency trap ω, the magnetic fields and the corresponding chemical potential μ are given in the figures captions. 



 Recalling that the full magnetic field is B⃗ = B⃗_T + B_z ẑ, an interesting point to highlight is that both solutions (i) and (ii) in (<ref>) are possible in each position where the transverse magnetic field vanishes. However, if B_z = 0, we find numerically that these solutions appear at random, each 50% of the time, because our numerical methods requires a random guess of the initial wavefunctions <cit.>. However, if we turn on the constant B_z field, we find, as expected, that the sign of this field component forces one of the solutions, namely, if B_z > 0, then solution (i) occurs, while (ii) ensues for B_z < 0. 



The solutions of figures <ref> and <ref>, as well as those below <ref> and <ref>, have been obtained with transverse magnetic fields that become zero at appropriate locations on the XY-plane within the BE condensate. These fields are in turn tailored by superposing magnetic fields of a set of long (infinite) wires at appropriate locations, carrying also an appropriate current I. We recall that the field of a long straight wire in the ẑ-direction, located at  (x_0,y_0), with current I, is

    B⃗_ wire(r⃗) = 2I/c-(y-y_0) x̂ + (x-x_0) ŷ/(x-x_0)^2 + (y-y_0)^2.

In the figures captions we indicate the values of the currents and the locations of the wires. The wires are placed outside of the numerical calculation box.


In the following gallery of figures we show the condensate superfluid density ρ_α and velocity v⃗_α, for component α = +, panels (A) and (B); and for component α = -, panels (C) and (D), respectively. In panel (E), we show the Skyrmion spin texture and in panel (F) the transverse spin texture field lines f⃗_T = f_x x̂ + f_yŷ. In panel (G) we show the transverse magnetic field lines B⃗_T =  B_x x̂ +  B_y x̂ that produces the corresponding condensate. Note that the transverse spin texture streamlines follow those of the transverse magnetic field. Details of the corresponding cases are given in the figures captions. 













§ REFERENCES





 Paper Title: Entropy production due to spacetime fluctuations.









thiagohenriquemoreira@ufg.br
QPequi Group, Institute of Physics, Federal University of Goiás, Goiânia, Goiás, 74.690-900, Brazil


lucas@qpequi.com
QPequi Group, Institute of Physics, Federal University of Goiás, Goiânia, Goiás, 74.690-900, Brazil

§ ABSTRACT 
Understanding the quantum nature of the gravitational field is undoubtedly one of the greatest challenges in theoretical physics. Despite significant progress, a complete and consistent theory remains elusive. However, in the weak field approximation —where curvature effects are small— we can explore some expected properties of such a theory. Particularly relevant to this study is the quantum nature of gravitational waves, which are represented as small perturbations in flat spacetime. In this framework, a quantum description of these perturbations, as a quantum field, is feasible, leading to the emergence of the graviton. Here we consider a non-relativistic quantum system interacting with such a field. We employ the consistent histories approach to quantum mechanics, which allows us to frame classical questions in a quantum context, to define a fluctuation relation for this system. As a result, thermodynamic entropy must be produced in the system due to its unavoidable interaction with the quantum fluctuations of spacetime. 


Entropy production due to spacetime fluctuations
    Lucas C. Célerihttps://orcid.org/0000-0001-5120-8176
    < g r a p h i c s >

    
=================================================================================














§ INTRODUCTION


Fluctuation theorems are strong results in nonequilibrium statistical mechanics, with applications spanning a wide variaty of fields. These range from human learning <cit.> and cosmology <cit.> to biology <cit.>, chemistry <cit.>, and quantum field theory <cit.>, to name just a few. Essentially, these theorems illustrate a strong connection between equilibrium quantities such as Helmholtz free energy and non-equilibrium ones, like energy and work. While various versions of these theorems exist, the most renowned are those formulated by Jarzynski <cit.> and Crooks <cit.>. For a historical overview of this topic, see Ref. <cit.>, and for technical details on the general theorems, consult Refs. <cit.>.

In the development of quantum field theory, one key implication is the inherent fluctuation of the quantum field energy. These fluctuations underpin various physical phenomena, such as the Casimir effect <cit.> and the Lamb shift <cit.>. When a system interacts with such fields and attempts to alter these fluctuations, it induces frictional forces, leading to dissipation and entropy production <cit.>.

In this context, a particularly intriguing aspect is the quantum nature of the gravitational field. Although a complete theory of quantum gravity remains elusive, it is possible to construct a consistent quantum field theory in the linear regime of Einstein’s theory, where curvature is small. This approach, known as the weak field approximation, introduces gravitons as the particles associated with the gravitational field <cit.>. The primary aim of this work is to examine entropy production in a system interacting with a weak quantum gravitational field. Specifically, by modeling the system as an open quantum system interacting with a bath of gravitons, we derive a fluctuation theorem for the system, the first moment of which is the entropy production. It is interesting to observe here that, contrary to other fields where energy fluctuates on a well defined spacetime, the quantum nature of the gravitational waves introduces fluctuations of the spacetime itself.

One major conceptual challenge in formulating a quantum fluctuation theorem is defining work. In classical systems, work is a well-defined quantity dependent on the trajectory and does not fluctuate, unless the system is immerse in a fluctuating environment, like a heat bath. In such cases, stochastic thermodynamics is required to describe the system and physical quantities emerge as averages over the fluctuations. In quantum systems, fluctuations are unavoidable, even in isolated systems. Moreover, the lack of the notion of a trajectory for a quantum system makes unambiguous the extension of the notion of thermodynamic quantities, like work.

For closed quantum systems, the definition of work using a two-time measurement scheme <cit.> is well-established. However, defining work for open quantum systems remains challenging. In this paper, we adopt the approach proposed in Ref. <cit.>, which defines work in a way analogous to classical systems by interpreting trajectories within quantum mechanics. To achieve this, we employ the decoherent histories formalism, primarily developed by Griffiths, Omnès, Gell-Mann, and Hartle <cit.>, with additional insights from Ref. <cit.>. This formalism was designed to describe quantum systems without relying on external observers. It employs Schrödinger's equation to help assigning probabilities to different histories, allowing us to consistently address classical questions within a quantum framework. One such question is the definition of work, which we approach by considering individual quantum histories, keeping in mind the classical thermodynamic idea of trajectory dependent quantity.

The remaining of the paper is structured as follows. First, we define our system of interest in the next section, Sec. <ref>, where the notation is also established. The decoherent histories approach to quantum mechanics is reviewed in Sec. <ref>. The main result, the fluctuation theorem, is presented in Sec. <ref>, where entropy production is directly related to quantum fluctuations of the spacetime. Finally, Sec. <ref> presents a physical discussion and our concluding remarks. We employ natural units in which ħ=c=k_B=G=1 and the metric signature (-,+,+,+).






§ LOCALIZED QUANTUM SYSTEM IN LINEARIZED QUANTUM GRAVITY



Our system of interest consists of two massive particles interacting with a gravitational wave within a flat spacetime background. Upon quantization, the fluctuations of the gravitational waves introduce a stochastic dynamic to the particles. In this section, we provide a brief overview of the approach presented in Refs. <cit.>, which derives a Langevin-like equation governing the geodesic separation between the two masses in the presence of the weak quantum gravitational field. For additional details, the reader is also referred to Ref. <cit.>.

Let us start by writing the classical action for a weak gravitational field interacting with a pair of massive particles. The total action is given by S = S_ m + S_ EH, where S_ m represents the action of the particles and their interaction with the gravitational field, and S_ EH is the Einstein–Hilbert action that describes the gravitational field itself. Denoting the coordinates of the particles as ζ^μ and ξ^μ, the action for the particles is expressed as

    S_ m   =-M∫ t √(-g_μνζ̇^μζ̇^ν)
       -m∫ t √(-g_μνξ̇^μξ̇^ν)-V(ξ).

We consider that the first particle (M) is free falling, while the second one (m) suffers the action of a potential V. Moreover, we take M≫ m such that the heavier particle is on shell with worldline ζ_0^μ(t). In this way, its dynamics becomes negligible. Furthermore, we take this particle to be at rest at the origin of our coordinate system, ζ_0^μ(t)=tδ_0^μ. Therefore, we can interpret the coordinate time t as its proper time. Since our interest lies on the lighter particle, from here on we simply refer to it as the system.

Under these assumptions it becomes appropriate to think of (t,ξ^i) as the Fermi normal coordinates defined with respect to the worldline of the heavier particle <cit.>. In these coordinates, we find <cit.>

    √(-g_μν(ξ)ξ̇^μξ̇^ν)≃1-1/2δ_ijξ̇^iξ̇^j+1/2R_i0j0(t,0)ξ^iξ^j

in the nonrelativistic limit. Here, R_μνρσ is the Riemann curvature tensor.

Now we write the spacetime metric as g_μν=η_μν+h_μν, with h_μν≪ 1, where η_μν=diag(-1,1,1,1) is the flat Minkowski metric while h_μν is a small perturbation. This is the weak field regime for the gravitational field, where the spacetime curvature is small compared with the relevant scales of the problem. Under this condition, the Einstein field equation reduces to a linear wave equation for h_μν. By taking advantage of the gauge invariance of linearized gravity, we choose to work in the transverse-traceless gauge such that the metric perturbation obeys h_0ν=0, η^μνh_μν=0 and ∂^μh_μν=0, with h_μν = h_μν - h η_μν/2 and h = h^μ_μ. In this gauge, the non-vanishing components of the Riemann curvature tensor are given by R_0i0j(t,0)=-ḧ_ij(t,0)/2, thus implying that the matter action takes the form

    S_ m=∫ t [1/2m δ_ijξ̇^iξ̇^j-V(ξ)+1/4m ḧ_ij(t,0)ξ^iξ^j],

where we have dropped non-dynamical terms.

It can be shown that the coupling with the weak gravity field leads to the total action given by <cit.>

    S   =π^2/8∫ t∫^3k ∑_s[q̇^2_s(t,k)-k^2q_s^2(t,k)] 
       +∫ t [1/2m δ_ijξ̇^iξ̇^j-V(ξ)] 
       +m/4∫ t ∫^3k ∑_sq_s(t,k) X^s(t,k),

where we have defined

    X^s(t,k)≡[2]t[ϵ_ij^s(k)ξ^iξ^j].

The gravitational field is expressed in terms of its two physical degrees of freedom, corresponding to the two polarization components by means of a Fourier transform

    h_ij(t,x)=∫^3k ∑_sϵ_ij^s(k)q_s(t,k)e^ik·x,

with ϵ_ij^s denoting the polarization tensor that satisfies the normalization ϵ_ij^s(k)ϵ^ij_s'(k)=2δ^s_s', the transversality k^iϵ_ij^s(k)=0 and the traceless δ^ijϵ_ij^s(k)=0 conditions.

Now, we assume that the initial state of the entire system is a product state, implying that initially there are no correlations between any of the components of the system —the two particles and the field. Using the Feynman-Vernon influence functional approach to open quantum systems <cit.>, we can express all (weak) quantum gravitational influences on the system through the influence functional

    e^iS_ IF[x,x'] = ∫𝒟𝒩P[𝒩]e^i∫ t𝒩^ij(t)[x_ij(t)-x_ij'(t)]
       × e^i/2∫ t t' [x_ij(t)-x_ij'(t)]D^ijkl(t,t')[x_kl(t')+x_kl'(t')],

where x_ij(t)=ξ_i(t)ξ_j(t). Here, 𝒩^ij(t) is a stochastic variable with Gaussian probability density P[𝒩] such that 𝒩^ij(t)=0 and 𝒩^ij(t)𝒩^kl(t')=N^ijkl(t,t').
The quantities D^ijkl(t,t') and N^ijkl(t,t') are called the dissipation and noise kernels, respectively.

In summary, the system is found to be described by a stochastic effective action given by <cit.>

    S_ SEA=S_ p[ξ]-S_ p[ξ']+S_ IF,

with S_ p[ξ]=∫ t [1/2m δ_ijξ̇^iξ̇^j-V(ξ)] and S_ IF being the influence action defined in Eq. (<ref>). The equation of motion is obtained by setting δ S_ SEA=0 <cit.>, yielding

    mξ_i(t)-2∫ t' D_ijkl(t,t')ξ^j(t)ξ^k(t')ξ^l(t') 
    
        =-Vξ^i+2𝒩_ij(t)ξ^j(t).

Note that the second term on the left-hand side introduces a non-linearity, making this integral differential equation very difficult to solve. It is for this reason that we choose to work on a perturbative regime, thus dropping the term involving the dissipation kernel as it was done in Ref. <cit.>. In this regime, we obtain

    mξ_i(t)=f_i(t)+2𝒩_ij(t)ξ_0^j(t),

where f_i is defined such that V(ξ)=-f_i(t)ξ^i(t) and ξ_0^i is the solution to mξ_0^i=0.

The above equation is a Langevin-like equation involving the stochastic force term 2𝒩_ij(t)ξ_0^j(t). The well-known solution for the initial conditions ξ^i(0)=ξ̇^i(0)=0 is given by

    ξ_i(t)   =1/m∫_0^t t' (t-t')f_i(t') 
       +2/m∫_0^t  t' (t-t')𝒩_ij(t')ξ_0^j(t').

This can be viewed as the classical limit for the geodesic separation between the two masses, where the stochastic behavior arises from the quantum fluctuations of the gravitational field. The key observation here is that, under certain conditions, we can employ the solutions to the Langevin equation to define work for the quantum system similarly to how it is defined for classical systems. This approach relies significantly on the concept of trajectories in space. To explore how this is feasible, we will work within the consistent (or decoherent) histories approach to quantum mechanics. This is a crucial step that allows us to properly build a fluctuation theorem.






§ DECOHERENT HISTORIES



Shortly, a quantum mechanical history is a sequence of events at successive instants of time, being characterized by a sequence of successive projection operators. Let us begin by considering a closed quantum system initially described by a density matrix ρ_0≡ρ(t_0) and we wish to consider histories consisting of n projections at times t_1<t_2<…<t_n, with t_1>t_0. In order to establish whether or not we can assign probabilities to histories (and how we do this), we need to look at the so-called decoherence functional. For a pair of histories, [α] and [β], where [α] denotes the sequence of events {α_i}_i=1^n at times { t_i}_i=1^n, and analogously for [β], the decoherence functional is defined as

    𝒟[α,β]=Tr[P_α_n… P_α_1ρ_0P_β_1… P_β_n],

Here, P_α_k is some set of projection operators compatible with event α_k (at time t_k), P_α_k=U^†(t_k,t_0)P_α_0U(t_k,t_0), where U(t_k,t_0) is the unitary time evolution operator for the system. Note that we can have different types of projections at different moments of time (although in this work we are mainly concerned with projections in position) <cit.>. The decoherence functional can be written in a more compact form as <cit.>

    𝒟[α,β]=TrT[∏_j=1^nP_β_j]T[∏_i=1^nP_α_i]ρ_0,

where T (T) stands for time (anti-time) ordering. It is straightforward to show that the decoherence functional satisfies the following properties

    
  * 𝒟[α,β]=𝒟^*[β,α],
    
  * ∑_[α],[β]𝒟[α,β]=1,
    
  * 𝒟[α,α]≥0,
    
  * ∑_[α]𝒟[α,α]=1,

where we are employing the shorthand notation ∑_[α] ≡ ∑_α_1∑_α_2…∑_α_n.

The last two properties suggest that we identify the diagonal elements 𝒟[α,α] with the probability for the history (ρ_0,t_0)→(α_1,t_1)→…→(α_n,t_n), namely p(α)=𝒟[α,α] <cit.>. However, in order for such elements to represent a genuine probability, Kolmogorov's third axiom of probability theory —the probability sum rule— must be satisfied. It is not difficult to show that

    p(α∨β)   =𝒟[α,α]+𝒟[β,β]+2Re𝒟[α,β] 
       =p(α)+p(β)+2Re𝒟[α,β].

Clearly, the sum rule is not obeyed in general due to the last term on the right-hand side. This term characterizes the quantum mechanical interference between distinct histories and it is in this sense that interference effects prevents us to give a meaningful notion of trajectory in quantum mechanics. However, this notion acquires meaning when there is strong decoherence, case in which 𝒟[α,β] vanishes for distinct trajectories. One may then write the fundamental formula for the quantum mechanics of histories as

    Re𝒟[α,β]=p(α)δ_α_1,β_1…δ_α_n,β_n.

This equation provides the necessary and sufficient condition for assigning probabilities to individual histories, as well as specifies what those probabilities are. However, for most applications of the formalism, it is often sufficient to assert that a pair of mutually exclusive histories is consistent when Re𝒟[α,β]<<𝒟[α,α], 𝒟[β,β] for [α]≠ [β] <cit.>.

From now on, let us consider projections in position basis, which is a kind of history implemented naturally in the path integral formalism. The projectors are represented by window functions P_α_k→ w_α[x(t_k)], which are equal to one if the configuration at t_k satisfies the requirement of history [α] and zero otherwise <cit.>. Equation (<ref>) can then be written as

    𝒟[α,β]=   ∫𝒟y𝒟y' e^i(S[y]-S[y'])ρ(y(0),y'(0),0) 
       ×∏_iw_α[y(t_i)]∏_jw_β[y'(t_j)].

Here S is the action for the total system described by some generic coordinate y.

Let us apply this to the problem at hand. By factoring the total action as shown in Eq. (<ref>), the integration over the gravitational variables leads to the influence functional given in Eq. (<ref>). We can then consider [α] and [β] as histories represented by two non-overlapping paths ξ^(1)(t) and ξ^(2)(t), in which case the window functions would be simply delta functions at each time instant. For this scenario, the decoherence functional was analyzed in Ref. <cit.>, considering four possible types of initial graviton states and estimating the decoherence time as a function of the gravitational energy scale (either a cutoff Λ_ g∼ξ^-1 <cit.> or the temperature β^-1 of a thermal state of gravitons)[The analysis done in Ref. <cit.> considered that the particle interacting with the gravitons was described by additional dynamical internal degrees of freedom that also coupled with the gravitational field. This coupling turned out to increase even more the decoherence rate. Although this is a feature to be taken advantage of, the internal degrees of freedom noise kernel also introduces a non-trivial modification to the equation of motion.]. Decoherence induced by gravitons was also analyzed in Ref. <cit.>, where it was established that decoherence occurs for particles with mass much greater than the Planck mass M_ pl∼10^-8 kg. For such particles, this allows us to give meaning to the concept of trajectory for the quantum system since the interference terms may be neglected due to decoherence. Within this scenario we derive a quantum fluctuation theorem for our system. 






§ FLUCTUATION THEOREM



In order to discuss the validity of the fluctuation theorem, we proceed as in Ref. <cit.>. By fluctuation theorem we mean Jarzynski-like relation, which is the integral form of Crook's fluctuation theorem <cit.>. For closed quantum systems, the derivation of the theorem relies on the hypothesis of initial thermal state of the entire system. However, our analysis of a quantum particle interacting with the gravitons was built under the assumption of initial product state, which is not a thermal state. 

To circumvent this issue, we take the initial time to be t_0=-∞. At this time instant, we assume that the total state of the system is described by a tensor product, ρ(-∞)=ρ_ sys(-∞)⊗ρ_ grav(-∞), where ρ_ grav is a thermal state of gravitons. We then allow the system to evolve according to the total action with f^i(t)=f^i(0) for t<0, so that we obtain a total thermal state at t=0. At t=0, the driving force starts changing according to some arbitrary protocol until a given time instant t=τ. Assuming the decoherence conditions discussed in the previous section hold, the work performed on the system in the interval [0,τ] is defined as

    W≡-∫_0^τ t ḟ_i(t)ξ^i(t),

where ξ^i(t) is the solution to the Langevin-like equation given in Eq. (<ref>). Physically, this is the work that must be performed on the system in order to move it through the bath of gravitons. By using the explicit forms of the solutions we can write W = W_f + W_n, with

    W_f=-1/m∫_0^τ t  t' ḟ^i(t)g(t-t')f_i(t'),

denoting the work associated with the driven protocol, while 

    W_n = -2/m∫_0^τ t t' ḟ^i(t)g(t-t')𝒩_ij(t')ξ^j_0(t'),

contains the quantum fluctuations coming from the noise. In these equations, g(t-t')=(t-t')θ(t-t'), with θ(x) being the step function. 

While W_f is entirely determined by the driven protocol, which is arbitrary, W_n encompasses both the driven contribution, as it includes f, and the noise term resulting from the interaction with the gravitons. Since gravity cannot be shielded and its quantum nature inherently involves fluctuations, this latter contribution, unlike the one arising from the driven protocol, is unavoidable.

We note that the work performed on the system is linear in 𝒩_ij(t), which is a Gaussian random process. Consequently, W itself must also be a Gaussian random variable. Its statistics are characterized by the first two moments W and σ_W^2=W^2-W^2. Therefore, we can write

    P(W)=1/√(2πσ_W^2)e^-(W-W)^2/2σ_W^2.

The brackets denote stochastic average with a Gaussian probability density.

Before we proceed, it is important to emphasize that the average with P(W) is equivalent to the stochastic average with P(𝒩) due to the linear dependence of W on 𝒩_ij(t). In conventional formulations of the classical fluctuation theorem, such stochastic forces are not typically considered, and the probabilistic nature of the work done on the system arises from the assumption of an initial thermal state. Each sampling from the initial state produces a trajectory, and the average is taken over an ensemble of such realizations. For the quantum fluctuation theorem, the inherent quantum uncertainty of the quantum state introduces an additional probabilistic aspect. However, we have opted to use an equivalent initial state preparation method based on a product initial state for the total system. This choice replaces the system's dependence on the initial state with the properties of noise statistics, thereby leaving only one probabilistic element instead of two. For further discussion on this topic, see Ref. <cit.>.

The moments of the work distribution can be directly computed as follows. First, since 𝒩^ij(t)=0, we have

    W=-1/m∫_0^τ dtdt' ḟ^i(t)g(t-t')f_i(t') = W_f.

Additionally, by using 𝒩^ij(t)𝒩^kl(t')=N^ijkl(t,t'), which is the noise kernel of gravitons, we also find

    σ_W^2=∫_0^τ dtdt' ḟ^i(t)σ_ij(t,t')ḟ^j(t'),

where

    σ_ij(t,t')   =   4/m∫_0^τ dt_1dt_2 g(t-t_1)g(t'-t_2) 
       ×    N_ijkl(t_1,t_2)ξ^k_0(t_1)ξ^l_0(t_2).

Note that, while W is independent of the noise kernel of gravitons, the second moment σ_W^2 is not. The noise kernel for an initial thermal distribution of gravitons was computed explicitly in Refs. <cit.>.

With the expressions for the first and second moments of work, the probability distribution is completely specified. One can now compute

    e^-β W=∫ dW P(W)e^-β W,

where β=T^-1 is the inverse temperature associated with the gravitons. Since Jarzynski equality holds for the entire system, we must have e^-β W=e^-βΔ F, where Δ F=F(τ)-F(0) is the total free energy difference. F(t) is defined with respect to the Hamiltonian of the system at time t, under the assumption that the temperature of the gravitons does not change. 

A direct calculation leads to

    e^-β W=e^-β(W_f-βσ_W^2/2),

allowing us to identify the free energy difference of the total system as

    Δ F=W_f-β/2σ_W^2,

which is a sum of a term that depends exclusively on the force protocol f [see Eq. (<ref>)] and another that also contains the contribution of gravitons via the noise kernel [see Eqs. (<ref>) and (<ref>)].

From this we can directly determine the entropy production Σ = β( W-Δ F) = β( W_f-Δ F), resulting in

    Σ = β^2/2σ_W^2.

This result intuitive says that spacetime fluctuations determine the entropy production in the system. There are two interesting limits to discuss here. First, when the gravitational field is turned off, σ_W^2 = 0, thus implying that there is no entropy production in the system. This is expected since we are dealing with a quantum system that is driven to follow a trajectory in spacetime. Thus, the absence of gravitational waves must imply that there will be no entropy being produced in the system. Secondly, by turning off the driven force, f=0, we also have Σ=0. This can be understood because, without the driving force, the system is in equilibrium with the fluctuating environment, and no net work or entropy production should be observed. 






§ DISCUSSIONS



Here, we analyze a driven quantum system interacting with a (weak) quantized gravitational field and derive an expression for the entropy production due to gravity by using an integral fluctuation theorem within the framework of decoherent histories. In the regime of low energy scales, perturbations of spacetime are treated quantum mechanically as a quantum field theory, leading to quantum fluctuations of spacetime. These fluctuations are unavoidable, as no system can be shield from gravity —there is no equivalent of a Faraday cage for gravitational effects. Thus, every system in nature experiences entropy production not only due to processes that potentially drive the system out of equilibrium but also because of the fundamental fluctuations of the spacetime in which the system live. The main result of this work is to characterize these fluctuations.

It is important to note that the key aspect of decoherent histories is the need for strong decoherence to eliminate interference among histories. Although crucial for the definition of work, this requirement is not a severe limitation in our case, as decoherence induced by quantum fluctuations of the gravitational field is indeed strong <cit.>. Furthermore, quantum particles typically possess additional internal degrees of freedom, which further increases the decoherence rate <cit.>.

An interesting question for future investigation is the impact of considering a curved spacetime instead of flat spacetime, as was done here. While this introduces many complexities, it should be possible to uncover some fundamental features of entropy production <cit.>.

The result that entropy is produced in quantum systems due to their interaction with gravitational waves could imply decoherence of spatial superpositions of massive particles. This raises the question of how long such superpositions persist. As shown in Ref. <cit.>, these effects are significant and should be considered when discussing the gravitational field generated by such states <cit.>.

Finally, we observed that no entropy is produced when the system is in free fall, in the absence of an external agent. This observation aligns with the equivalence principle. However, this does not imply that quantum spacetime fluctuations are undetected in such a frame, and further investigation is necessary. For example, instead of focusing on the work done by an external agent, one could analyze the work done by the stochastic force resulting from gravitational fluctuations. This distribution of work is not expected to be Gaussian, making the analysis more complex.






§ ACKNOWLEDGEMENTS

This work was supported by the National Institute for the Science and Technology of Quantum Information (INCT-IQ), Grant No. 465469/2014-0, by the National Council for Scientific and Technological Development (CNPq), Grants No 308065/2022-0, and by Coordination of Superior Level Staff Improvement (CAPES).





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 Paper Title: Dephasing-assisted transport in a tight-binding chain with a linear potential.


1

Dephasing-assisted transport with a linear potential]Dephasing-assisted transport in a tight-binding chain with a linear potential 

] 
 
 





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    Received ..... .., ....; accepted ......, ....
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§ ABSTRACT 


§ 


An environment interacting with a quantum system can enhance transport through the suppression of quantum effects responsible for localization. In this paper, we study the interplay between bulk dephasing and a linear potential in a boundary-driven tight-binding chain. A linear potential induces Wannier-Stark localization in the absence of noise, while dephasing induces diffusive transport in the absence of a tilt. We derive an approximate expression for the steady-state current as a function of both dephasing and tilt which closely matches the exact solution for a wide range of parameters. From it, we find that the maximum current occurs for a dephasing rate equal to the period of Bloch oscillations in the Wannier-Stark localized system. We also find that the current displays a maximum as a function of the system size, provided that the total potential tilt across the chain remains constant. Our results can be verified in current experimental platforms and represents a step forward in analytical studies of environment-assisted transport.

 

§ KEYWORDS:
 Dephasing-assisted transport, Wannier-Stark localization





§ INTRODUCTION


The quantum features of an open system interacting with a macroscopic environment are inevitably destroyed in a process known as decoherence <cit.>. However, the notion that the environment is always detrimental for quantum processes – such as information processing and transport – has been challenged for more than a decade, prompted by investigations of quantum effects in biological systems <cit.>. It is now understood that the environment can assist energy transport in non-interacting quantum systems <cit.>, an effect which has been experimentally verified in quantum networks of photons <cit.>, trapped ions <cit.> and superconducting circuits <cit.>. Although several mechanisms for environment-assisted quantum transport have been proposed and debated <cit.>, a clear mechanism is at play in localized quantum systems – when destructive interference responsible for localization and transport suppression is destroyed by the environment, quantum transport is enhanced. This is indeed the expected impact of environmental coupling on Anderson and Wannier-Stark localization. In Anderson localized systems, quantum transport is suppressed as a consequence of lattice disorder <cit.>. On the other hand, Wannier-Stark localization occurs in the presence of a linear lattice potential such as an electric field <cit.>; in this case, coherent (Bloch) oscillations take place within the region of localization <cit.>. Despite the fact that Wannier-Stark localized systems subject to noise have been investigated <cit.>, the literature on this subject is still scarce.

Environment-assisted transport can also affect the non-equilibrium transport properties of many-body quantum systems. A common approach to evidence this effect is to drive the quantum system to a non-equilibrium steady state by high temperature reservoirs located at the boundaries <cit.>. The dephasing effect of the environment renders the transport diffusive, which has been shown to enhance the steady-state current in boundary-driven systems with disorder <cit.> or quasi-periodicity <cit.> and to modify the transport properties of systems with long-range hopping <cit.>. Moreover, Markovian boundary-driven non-interacting systems are amenable to analytical treatment even in the presence of dephasing, with exact and approximate solutions of steady-state quantities available for tight-binding chains <cit.>.

In this paper, we study dephasing-assisted transport in a boundary-driven, tight-binding chain with a linear potential. We make use of a steady-state ansatz elaborated in Refs. <cit.> to numerically access the relevant steady-state quantities for up to one thousand chain sites. By performing a minimal approximation, we derive an analytical expression for the steady-state current which matches the exact dynamics for a vast range of parameters. From it, we find that the current is maximized at a dephasing rate equal to the period of Bloch oscillations in a Wannier-Stark localized system. We also find that the current displays a maximum as a function of the system size, provided that the total potential tilt across the chain is kept constant. Our results present a significant contribution in the analytics of environment-assisted transport and can be experimentally verified in several platforms.

The paper is organized as follows. In Sec. <ref> we introduce the model and the steady-state ansatz used in our study. In Sec. <ref> we present and discuss our results for dephasing-assisted transport. In Sec. <ref> we present the conclusion and outlook of our work. Sec. <ref> is the supplementary material, containing technical details referenced throughout the main text. 



§ SETUP





 §.§ The model


We study a non-interacting, one-dimensional lattice of L sites with Hamiltonian given by the XX model

    H = J ∑_j=1^L-1  (σ_j^x σ^x_j+1 + σ_j^y σ^y_j+1) + ∑_j=1^Lϵ_j σ_j^z   .

Here, {σ^x_j, σ^y_j, σ^z_j }_j=1^L are Pauli matrices, J is the hopping term and ϵ_j is a local field which will be specified shortly. In order to induce a non-equilibrium state in the system, we couple the boundary sites to high-temperature reservoirs with
different chemical potentials; in addition, each site is also exposed to its own local reservoir which induces dephasing. All the reservoirs are assumed to be ideal and couple weakly to the system. The dynamics of the system's density operator ρ(t) at time t is given by the Lindblad master equation <cit.>

    d ρ(t)/dt = -i/ħ[H,ρ(t)] + ∑_α = l,r,dℒ^α[ρ(t)]   .

Here the superoperator ℒ^α describes dissipation induced by the reservoirs α = l,r,d corresponding to left boundary, right boundary and dephasing, respectively. It is a sum of local Lindblad jump operators of the form

    ℒ^α  [·] = ∑_j,β L^αβ_j · L_j^αβ† - 1/2{ L_j^αβ† L^αβ_j, ·}  ,

where j labels the site acted on by the reservoir α and β labels different jump operators on that same site. The effect of each boundary reservoir (α = l,r) is represented by two jump operators

    L^l ±_1 = √(Γ(1 ± f)/2)σ_1^± ,  L^r±_L = √(Γ(1 ∓ f)/2)σ_L^±  ,

where Γ is the coupling rate at the boundaries, f is the chemical potential bias and σ_j^± = (σ_j^x ± i σ_j^y)/2. For forward bias (0 ≤ f ≤ 1) excitations are mostly created on the first site and annihilated on the last site; the opposite reasoning applies for reverse bias (-1 ≤ f ≤ 0). In order to describe the effects of dephasing, we consider L jump operators acting on individual sites as if each of them were coupled to its own reservoir

    L_j^d = √(γ/2)σ^z_j   ,  j = 1, 2, ..., L   ,

where γ is the dephasing rate. 

The key observables in the system are the magnetization and its associated current. We denote by ⟨ A ⟩(t) ≡Tr[A ρ(t)] the expectation value of an observable A at time t. Differentiating with respect to time and using Eq. (<ref>) yields a continuity equation (see Sec. <ref> for more details)

    d ⟨σ^z_j ⟩(t)/dt    = ⟨ I_j-1⟩(t) - ⟨ I_j⟩(t) + δ_j,1Γ [f - ⟨σ^z_j ⟩(t)] - δ_j,LΓ [f + ⟨σ^z_j ⟩(t)]    ,

involving  the expectation value of the magnetization current ⟨ I_j ⟩(t)  flowing from j to j+1, where

    I_j = 2J/ħ (σ_j^xσ_j+1^y - σ_j^yσ_j+1^x)

is the corresponding current operator. Note that the dephasing reservoir does not change the average magnetization and therefore does not contribute explicitly to Eq. (<ref>), although it still affects the average magnetization and currents implicitly. Moreover, the boundary reservoirs do not contribute to the expression of the current in the bulk of the chain. 



When the steady state is reached, the left-hand side of Eqs. (<ref>) and (<ref>) vanishes. We denote by ⟨ A ⟩_∞≡Tr[A ρ(∞)] the expectation value of A at the steady state ρ(∞). From Eq. (<ref>) we obtain the following equations for the steady-state current

    ⟨ I_1 ⟩_∞    = Γ (f - ⟨σ^z_1 ⟩_∞)   , 
    ⟨ I_j-1⟩_∞    = ⟨ I_j⟩_∞  ,  j = 2, 3, …, L-1 
    ⟨ I_L-1⟩_∞    = -Γ (f + ⟨σ^z_L ⟩_∞)      .

This current is uniform across the chain, with its value dictated by the average magnetization at the boundaries; from now on, we remove the subscript and denote it simply by ⟨ I ⟩_∞. In the absence of any bias (f=0), the boundary reservoirs induce an infinite temperature (maximally-mixed) steady state ρ(∞) = 𝕀/2^L, in which the values of magnetization and current all vanish.



 §.§ Steady-state ansatz


In order to study the transport properties of the model at the steady state, both numerically and analytically, we make use of a perturbative ansatz characterized by an expansion in terms of potential bias f as <cit.>

    ρ    = 1/2^L[𝕀 + f (H + B) + 𝒪(f^2) ]   , 
    
        H     = ∑_r=1^L∑_j=1^L+1-r h^(r)_j H^(r)_j   , 
     
        B     = ∑_r=2^L∑_j=1^L+1-r b^(r)_j B^(r)_j   ,

where h_j^(r) and b_j^(r) are expansion coefficients, and the corresponding operators are given by

    H^(1)_j    = - σ^z_j , 
    
        H^(r)_j    = σ^x_j Z^(r-2)_j+1σ^x_j+r-1 + σ^y_j Z^(r-2)_j+1σ^y_j+r-1  ,  r ≥ 2 
    
        B^(r)_j     = σ^x_j Z^(r-2)_j+1σ^y_j+r-1 - σ^y_j Z^(r-2)_j+1σ^x_j+r-1  ,  r ≥ 2   .

These operators are thus strings of Pauli matrices starting at site j and having length r, where Z^(r-2)_k+1 = σ^z_k+1σ^z_k+2 ... σ^z_k+r-3σ^z_k+r-2 are strings of Pauli z matrices of length r-2. The defining characteristic of this ansatz is that all operators in the expansion are orthogonal according to the Hilbert-Schmidt inner product ⟨ A, B⟩≡Tr[A^†B]. This implies that the expectation values of H^(r)_j and B^(r)_j are exactly determined within first order in f. In particular, H_j^(1) and B_j^(2) are respectively proportional to the magnetization and current

    ⟨σ^z_j⟩_∞ = -f h^(1)_j  , ⟨ I_j ⟩_∞ = f 4J b^(2)_j/ħ  .

Once we determine the first order expansion coefficients h^(r)_j and b^(r)_j, we have access to the exact current and magnetization profile <cit.>. In order to obtain them, we insert Eq. (<ref>) into Eq. (<ref>) and set the left-hand side to zero as demanded at stationarity (see Refs. <cit.> and Sec. <ref> for more details). This gives rise to a set of equations for the coefficients

    ħΓ(1+h^(1)_1) - 4J b_1^(2) = 0   , 
       ħΓ(1-h^(1)_L) - 4J b_L-1^(2) = 0   , 
        b_j^(2) - b_j-1^(2) = 0   ,  j = 2,...,L-1   ,

for r = 1. Note that Eqs. (<ref>)–(<ref>) just express the fact that the current is uniform in the steady state, with its value dictated by the boundary reservoirs as discussed before. For r ≥ 2 we have

    J [ h_j^(r-1) - h_j+1^(r-1) + h_j^(r+1) - h_j-1^(r+1)] +  (ϵ_j - ϵ_j+r-1)h_j^(r) + ħ[ γ + Γ/4(δ_j,1 + δ_j+r-1,L) ] b^(r)_j = 0   , 
        J [b_j^(r-1) - b_j+1^(r-1)+ b_j^(r+1) - b_j-1^(r+1)] + (ϵ_j - ϵ_j+r-1)b_j^(r) - ħ[ γ + Γ/4(δ_j,1 + δ_j+r-1,L) ] h^(r)_j = 0   .

In total, Eqs. (<ref>)–(<ref>) are a closed set of L^2 coupled equations and will be the starting point of our study. This set of equations is equivalent to a Lyapunov-type equation for the correlation matrix in the presence of dephasing <cit.>. Instead of the exponential scaling with the system size, the quadratic scaling makes the problem computationally tractable. Moreover, as we show below, it also allow us to find analytical expressions for the current under some mild assumptions.



§ DEPHASING-ASSISTED TRANSPORT



We now discuss separately the effect of the tilt and dephasing on the transport properties of the model, while their interplay is addressed in the next section. The bias f of each reservoir at the boundaries induces transport along the chain. Intuitively, a similar effect is expected in the presence of an electric field which increases/decreases at each site at a rate U, or tilt between neighbour sites. If the tilt decreases from left to right, the local potentials are

    ϵ_j = - Uj   .

Setting the bias to f=1 drives excitations in the same direction (see Fig. <ref>). Note that changing the sign of f inverts the sign of the current and magnetization, while changing the sign of U affects neither <cit.>. 

Since the tilt U represents the energy difference between nearest neighbours in the chain, the total tilt V=U(L-1) represents the difference in energy across the whole chain. This quantity diverges in the thermodynamic limit, so that a physically meaningful analysis requires fixing the total tilt V and allowing U to decrease with the system size L. As discussed in Secs. <ref> and <ref>, this procedure introduces a non-trivial dependence of the current with system size in the presence of dephasing: while for fixed U the current is expected to decrease with the system size L, this will no longer be the case for fixed V.



Such a linear potential is known to induce Wannier-Stark localization in the absence of dephasing, with U/ħ being the frequency of Bloch oscillations <cit.>.
In the absence of dephasing γ = 0 and for non-vanishing tilt V>0, the steady state magnetization profile shows the formation of domain walls <cit.> – regions of constant magnetization ⟨σ^z_j ⟩_∞≃± f starting at the boundaries and extending inwards as shown in Fig. <ref>(a). According to Eqs. (<ref>) or (<ref>), this implies a vanishing current, which characterizes the localized system. Indeed we observe in Fig. <ref>(b) that, as V increases, the current displays an exponential decay with the system size ⟨ I ⟩_∞∼ e^-L/L_0 with L_0 being the localization length. Note that L_0 here is different from the localization region of a single particle in a Wannier-Stark localized system given by J/U <cit.>. We did not find any direct relationship between the two.



The results in the absence of tilt U = V = 0 are shown in Figs. <ref> and <ref>(a), respectively for the magnetization and the current. We distinguish two regimes: γ = 0 and γ > 0. The first case corresponds to ballistic transport, characterized by a small (but non-zero) magnetization at the boundaries and zero bulk magnetization (inset of Fig. <ref>), together with a size-independent current (see later Fig. <ref>(a)). The second case corresponds to diffusive transport, characterized by an emerging linear magnetization profile as the dephasing rate increases, accompanied by a decreasing current in accordance with Eqs. (<ref>) or (<ref>). For very large dephasing or system sizes the current scales as ⟨ I ⟩_∞∼ 1/(γ L) signaling diffusive transport. The exact expression for the current in the absence of tilt is given by Eq. (<ref>) (with U=V=0) as we discuss in Sec. <ref>.



 §.§ Exact current for small system sizes


Having shown the degrading effect of the tilt and dephasing separately on the spin transport across the system, we now study their interplay which gives rise to dephasing-enhanced transport. In this section, we focus on transport for small system sizes L=2,3. An exact and compact solution for the current can be found from Eqs. (<ref>)–(<ref>) in these cases. For L = 2 we obtain

    ⟨ I ⟩_∞ = f 16 J^2 Γ (2 γ + Γ)/16 J^2 (2 γ + Γ) + 4 Γ U^2 + ħ^2Γ(2 γ + Γ)^2  .

Dephasing-assisted transport can be characterised through the maximum of Eq. (<ref>) with respect to the dephasing rate γ, yielding an optimal dephasing rate

    γ_max = U/ħ - Γ/2≥ 0   .

The behaviour of Eq. (<ref>) is shown in Fig. <ref>. Note that since the dephasing is always positive, the maximum only occurs if U/ħ≥Γ/2.

For L=3 we have a similar expression, namely

    ⟨ I ⟩_∞ = f 16 J^2 Γ (4 γ + Γ)/16 J^2 (4 γ + Γ) + 16 Γ U^2 + ħ^2Γ(4 γ + Γ)^2   ,

where the maximum occurs at

    γ_max = U/ħ - Γ/4≥ 0   .

The complexity of these expressions increases rapidly for L > 3, so the exact evaluation of the current and its maximum is too involved to be displayed here for larger systems. In the following section, we resort to an approximate solution for the current.



 §.§ Tridiagonal approximation 


In order to find an exact solution for systems of arbitrary size, we perform the tridiagonal approximation which consists of retaining strings of Pauli matrices up to r=2, that is, ignoring higher order coefficients in Eqs. (<ref>)–(<ref>) so that h^(r)_j,b^(r)_j = 0 for r ≥ 3. In the correlation matrix approach, this is equivalent to retaining only the diagonal and first off-diagonal elements <cit.>. Physically, this implies that spin and current correlations can be ignored within first order in f, which allows the derivation of an exact expression for the current as a function of all parameters as we now show. We only have to consider Eqs. (<ref>) and (<ref>) for r=2. Summing Eqs. (<ref>) from j=1 to j=L-1 we obtain

    J(h^(1)_1 - h^(1)_L)     + ∑_j=1^L-1 (ϵ_j - ϵ_j+1) h^(2)_j +  ħ b [ γ(L-1) + Γ/4] = 0   ,

where b ≡ b^(2)_j is the coefficient associated with the current, which is independent of j as a consequence of Eq. (<ref>). First, we can write h^(1)_1 - h^(1)_L as a function of b by using Eqs. (<ref>) and (<ref>), which yields

    h^(1)_1 - h^(1)_L = 8 J b/ħΓ - 2   .

Second, we can also express h^(2)_j as a function of b by using Eq. (<ref>), which leads to

    h^(2)_j =  (ϵ_j - ϵ_j+1) b/ħ[γ + (δ_j,1 + δ_j,L-1)Γ/4]  .

We can now substitute the previous two equations into Eq. (<ref>) and obtain an expression for b, namely

    b = 4J/2 ħγ(L-1) + ħΓ + 16J^2/ħΓ  + 2/ħ∑_j=1^L-1(ϵ_j - ϵ_j+1)^2/γ +(δ_j,1 + δ_j,L-1)Γ/4  .

The current is now obtained from Eq. (<ref>). For the local tilted potential in Eq. (<ref>) it becomes

    ⟨ I ⟩_∞ = f 16J^2 Γ/16J^2 + ħ^2 Γ[Γ + 2 γ(L-1)] + 4 Γ U^2 
        
          1/(2γ + Γ)     L = 2 
    
          (L-3)/(2γ) + 4/(4γ + Γ)     L ≥ 3
          .

We note that this expression is exact for L=2 and L=3, in which case we recover Eqs. (<ref>) and (<ref>), respectively. It is also exact in the absence of tilt U=V=0 <cit.>, which is verified in Fig. <ref>(a). In last three panels of Fig. <ref> we plot the exact solution and the tridiagonal approximation for non-zero values of total tilt V > 0. We observe that the approximation works remarkably well, deviating from the exact solution only for very small dephasing rates and system sizes; when γ = 0, the approximation fails dramatically and only the exact solution is shown. The latter case corresponds to a Wannier-Stark localized system which we already discussed. Moreover, we see that for fixed γ the current can now increase with the system size up to a critical size L_max where it is maximal, and only then decreases. In the same way, we see that for fixed L there is a critical dephasing rate γ_max at which the current is maximal. 





 §.§ Interplay between dephasing and tilt 


Since the validity of Eq. (<ref>) is established, we now use it to determine both L_max and γ_max. The maxima of Eq. (<ref>) as a function of L (for fixed V and γ) and γ (for fixed L and V) can be determined exactly. However, as before, the expressions are too lengthy to be displayed. More physically intuitive expressions are obtained by assuming that L ≫ 1 and L ≫ 8 γ / (4γ + Γ) in the denominator of Eq. (<ref>) which leads to the approximate maxima

    L_max   ≃V/ħγ  , 
    γ_max   ≃V/ħ L = U/ħ  .

These maxima characterize dephasing-assisted transport as we show in Fig. <ref>(a). 

Below the critical length (L_max≫ L ≫ 1), dephasing is not yet effective and an increase in L decreases the local tilt U = V/(L-1) and thus localization, which increases the current according to Eq. (<ref>). Above this length scale (L ≫ L_max≫ 1) dephasing becomes effective and diffusive transport clearly emerges ⟨ I ⟩_∞∼ 1/(γ L). Analogously, below the critical dephasing rate (γ_max≫γ) dephasing jump events happen at a much lower frequency than Bloch oscillations, so the current flowing remains hampered by localization. Increasing the rate of dephasing destroys localization and increases the current. When the dephasing rate passes this threshold (γ≫γ_max)  diffusive behaviour emerges. We illustrate this in Fig. <ref>(b) where we plot the (finite size) diffusion coefficient defined by ⟨ I ⟩_∞≡ f D/L as a function of dephasing rate for the largest system size we reached L = 1000.

The imprint of dephasing-enhanced transport is also evident in the magnetization profile, as shown in Fig. <ref>. Here, varying the dephasing strength towards its critical value (<ref>) decreases the magnetization gradient, rendering the magnetization profile almost uniform. This is in contrast with the results obtained in the absence of tilt, where an increasing dephasing rate tends to increase the magnetization gradient and decrease uniformity as shown in Fig. <ref>(a). Uniformity as a feature of environment-assisted transport has been pointed out for quantum systems subject to on-site disorder <cit.>; our results confirm that the same phenomenon appears in the presence of a linear potential inducing Wannier-Stark localization.





§ CONCLUSION AND OUTLOOK


In this paper, we have exploited an exact ansatz to obtain the steady state of a non-interacting spin chain subject to bulk dephasing, a linear potential and boundary-driving. This description has allowed us to unravel the delicate interplay between Wannier-Stark localization and dephasing-induced diffusive transport. We applied the tridiagonal approximation, where only on-site and nearest-neighbour correlation matrix elements are retained, to derive an approximate expression for the steady-state current as a function of both dephasing and tilt. This expression is found to closely match the exact solution for a wide range of parameters, and reveals that the maximum current occurs for a dephasing rate equal to the period of Bloch oscillations in the Wannier-Stark localized system. Fixing the total tilt across the system then revealed a maximum in the current as a function of system size. This evidences a critical system size beyond which Bloch oscillations are suppressed by dephasing.

Our work motivates future analytical and numerical analysis on the interplay between dephasing and a tilted potential in more complex and richer systems, e.g. those including interactions between neighboring sites <cit.>, time-periodic driving <cit.>, or more realistic (non-Markovian) boundary-driving schemes <cit.>. Furthermore, our results can be readily verified in current photonic <cit.>, ion-trap <cit.> and cold-atom <cit.> experimental platforms.



§ FUNDING


The authors declare financial support was received for the research, authorship, and/or publication of this article. S.L.J. acknowledges the financial support from a Marie Skłodowska-Curie Fellowship (Grant No. 101103884). S.R.C. gratefully acknowledge financial support from UK's Engineering and Physical Sciences Research Council (EPSRC) under grant EP/T028424/1 which is part of an EPSRC-SFI joint project QuamNESS funded also by the SFI under the Frontier For the Future Program. J.G. is supported by a SFI Royal Society University Research Fellowship.



§ ACKNOWLEDGEMENTS


The authors want to thank Gabriel T. Landi for useful discussions.



§ SUPPLEMENTARY MATERIAL





 §.§ Heisenberg equation



For completeness, we show here how to obtain Eq. (<ref>). Differentiating ⟨σ^z_j ⟩(t) ≡Tr[σ_j ρ(t)] with respect to time and using Eq. (<ref>) yields

    d ⟨σ^z_j ⟩(t)/dt = i/ħ Tr[ [H,σ^z_j] ρ(t) ] + ∑_α = l,r,dTr[ σ^z_j ℒ^α[ρ(t)] ]   .

Throughout this section, we make use of several relations for the Pauli matrices, which can all be deduced from their commutation and anti-commutation relations [σ^α,σ^β] = 2i ε_αβγσ^γ and {σ^α,σ^β} = 2 δ_αβ𝕀_2, respectively. Here α, β, γ∈{ x, y, z }, 𝕀_2 is the identity operator for a two-level system and ϵ_αβγ is the Levi-Civita symbol with the Einstein summation notation being implicit. The commutator in Eq. (<ref>) can then be easily computed

    i/ħ[H,σ^z_j]     = iJ/ħ∑_k=1^L-1[ σ_k^x σ^x_k+1 + σ_k^y σ^y_k+1,σ^z_j] 
        =  2J/ħ (σ_j-1^xσ_j^y - σ_j-1^yσ_j^x) - 2J/ħ (σ_j^xσ_j+1^y - σ_j^yσ_j+1^x) 
        = I_j-1 - I_j  ,

In the first line only the kinetic energy part of the Hamiltonian is non-vanishing. In the second line, only the pairs (k,k+1) which overlap with j survive. In the last line, we identified the current operator in Eq. (<ref>). Regarding the last term in Eq. (<ref>), we can make progress by considering the dual superoperator acting on σ^z_j defined by Tr[ σ^z_j ℒ^α[ρ(t)] ] = Tr[ ℒ^α†[σ^z_j] ρ(t) ] with

    ℒ^α†[·] = ∑_j,β L^αβ†_j · L_j^αβ - 1/2{ L_j^αβ† L^αβ_j, ·}  .

The dephasing superoperator is equal to its dual and reads explicitly

    ℒ^d[·] = γ/2∑_j=1^L ( σ_j^z ·σ_j^z - · ) = ℒ^d †[·]   .

It is straightforward to see that its action vanishes when applied to any Pauli z-matrix. Therefore dephasing does not contribute to Eq. (<ref>). For the reservoir at the left boundary, we have the dual superoperator

    ℒ^l †[·]     = Γ(1+f)/2 O_1^l+ †[·] + Γ(1-f)/2 O_1^l- †[·] 
    
        O_1^l±†[·]    ≡σ_1^∓·σ_1^± - 1/2{σ_1^∓σ_1^±,·}  ,

and a similar expression holds for the right reservoir in Eq. (<ref>). Applying these superoperators to the Pauli z-matrix yields after some manipulation

    ℒ^l †[σ^z_j]     = δ_j,1Γ(f - σ^z_1)   , 
    ℒ^r †[σ^z_j]     = - δ_j,LΓ(f + σ^z_L)   .

Using the last two expressions and Eq. (<ref>) in Eq. (<ref>) immediately leads to Eq. (<ref>).



 §.§ Steady-state ansatz



In order to derive Eqs. (<ref>)–(<ref>), we insert Eq. (<ref>) into Eq. (<ref>) and set the left-hand side to zero as demanded at steady state. This gives

    0     = 1/2^L∑_α = l,r,d,0[ ℒ^α[𝕀] + f ( ℒ^α[H] + ℒ^α[B])]   , 
    ℒ^α[H]     = ∑_r=1^L∑_j=1^L+1-r h^(r)_j ℒ^α[H^(r)_j]   , 
    ℒ^α[B]     = ∑_r=2^L∑_j=1^L+1-r b^(r)_j ℒ^α[B^(r)_j]   .

In the last expression we ignore all terms beyond first order in f and defined the superoperator for free evolution

    ℒ^0 [·] ≡ -i/ħ[H, · ]   .




  §.§.§ Action of the dephasing reservoir


We start by computing the action of the dephasing superoperator which reads explicitly

    ℒ^d[·] = γ/2∑_j=1^L ( σ_j^z ·σ_j^z - · ) ≡γ/2∑_j O^d_j[·]   .

It is straightforward to see that the action of this superoperator vanishes when applied to 𝕀 and σ^z. For r ≥ 2, we have to compute terms of the following form

    ℒ^d[σ^α_j Z^(r-2)_j+1σ^β_j+r-1]     = γ/2[O^d_j[σ^α_j] Z^(r-2)_j+1σ_j+r-1^β + σ^α_j Z^(r-2)_j+1 O^d_j+r-1 [σ_j+r-1^β] ]   ,

Note that the right-hand side has only two non-vanishing terms on the boundaries since the action on σ^z vanishes. We then have

    O^d_j[σ^α_j] = σ_j^z σ_j^ασ_j^z - σ_j^α = - 2 σ_j^α  (δ_α,y + δ_α,x)   .

Using this last expression and carrying through simple algebraic manipulation, the action of the dephasing is written as

    ℒ^d[𝕀]     = ℒ^d[H_j^(1)] = 0   , 
    ℒ^d[H_j^(r)]     = -2 γ H_j^(r) ,  r ≥ 2 
    ℒ^d[B_j^(r)]     = -2 γ B_j^(r) ,  r ≥ 2   .




  §.§.§ Action of the boundary reservoirs


We first illustrate the action for the left boundary reservoir. The superoperator is given by

    ℒ^l[·]     = Γ(1+f)/2 O_1^l+[·] + Γ(1-f)/2 O_1^l-[·]   , 
    
        O_1^l±[·]    ≡σ_1^±·σ_1^∓ - 1/2{σ_1^∓σ_1^±,·}  .

In opposition to dephasing, this superoperator does not preserve the identity

    O_1^l±[𝕀] = ±σ_1^z = ∓ H^(1)_1   .

As a consequence, its action on the Pauli matrices σ_j^z depends on j. Namely, we obtain

    O_1^l±[σ_j^z] = 
        
          - σ^z_1   ,     j = 1 
    ±σ_1^zσ^z_j   ,     j = 2,3...L   .

Inserting these results into Eq. (<ref>) we obtain the actions

    ℒ^l[𝕀]     = -Γ f H^(1)_1   , 
    ℒ^l[H^(1)_j]     = 
        
            - Γ H_1^(1)    j = 1 
    
            f Γ H_1^(1) H_j^(1)    j = 2,3,…,L   .

Inspecting the last expression together with Eq. (<ref>), we see that for j=2,3,..L the reservoir creates terms in the expansion which are proportional to f^2 instead of f – they represent higher order spin correlations which do not affect the steady state values of magnetization and current. We henceforth ignore these second order terms by considering the action of the reservoir on Pauli matrices located at the reservoir site. Proceeding with the calculations for r ≥ 2, we have to compute terms of the form

    ℒ^l[σ^α_j Z^(r-2)_j+1   σ^β_j+r-1] = δ_j,1{Γ(1+f)/2 O^l+_1[σ^α_1] + Γ(1-f)/2 O^l-_1[σ^α_1] } Z^(r-2)_j+1σ^β_j+r-1  .

It is easy to verify that

    O^l ±_1[σ^α_1] = -σ_1^α/2 , α = x,y   .

Using the last two expressions and carrying out the algebra, the results for the left reservoir to first order in f are

    ℒ^l[𝕀]     = - Γ f H^(1)_1   , 
    ℒ^l[H^(1)_j]     = -δ_j,1Γ H^(1)_j   , 
    ℒ^l[H^(r)_j]     = - δ_j,1Γ/2 H_j^(r) ,  r ≥ 2   , 
    ℒ^l[B^(r)_j]     = - δ_j,1Γ/2 B_j^(r) ,  r ≥ 2   .

Proceeding the same way for the right boundary reservoir we obtain the result

    ℒ^r[𝕀]     = Γ f H^(1)_L   ,
    ℒ^r[H^(1)_j]     = -δ_j,LΓ H^(1)_j   , 
    ℒ^r[H^(r)_j]     = - δ_j+r-1,LΓ/2 H_j^(r) ,  r ≥ 2   , 
    ℒ^r[B^(r)_j]     = - δ_j+r-1,LΓ/2 B_j^(r) ,  r ≥ 2   .




  §.§.§ Action of Hamiltonian


Regarding the Hamiltonian superoperator in Eq. (<ref>), it is easy to see that it preserves the identity ℒ^0[𝕀] = 0 while ℒ^0[H_j^(1)] yields the well-known result

    ℒ^0[H_j^(1)] = - 2J/ħ(B^(2)_j - B^(2)_j-1)   .

This result was already implicitly used in Eq. (<ref>) and simply expresses that the change in magnetization at a given site is due to differences between incoming and outgoing currents at that site. For r ≥ 2, it is useful to write the superoperator  
as a sum of two contributions ℒ^0≡ℒ^0_1 + ℒ^0_2, where

    ℒ^0_1·   ≡ -i J/ħ∑_j=1^L-1 { [σ_j^x σ_j+1^x,·] + [σ_j^y σ_j+1^y,·] }  , 
    ℒ^0_2·   ≡ -i/ħ∑_j=1^L ϵ_j [σ_j^z,·]   .

We compute only the action of ℒ^0_1 and the one for ℒ^0_2 is obtained in a similar manner. We first evaluate operators for r = 2 which contain no Pauli z-strings (see Eq. (<ref>) for the exact expressions). It involves computing terms of the form

    ∑_j=1^L-1 [σ_j^γσ_j+1^γ, 
     σ_k^ασ_k+1^β]     = σ^γ_k-1[σ_k^γ,σ_k^α]σ_k+1^β+ [σ_k^γσ_k+1^γ,σ_k^ασ_k+1^β] + σ^α_k[σ_k+1^γ,σ_k+1^β]σ_k+2^γ
        = 2i [ε_γαωσ_k-1^γσ_k^ωσ_k+1^β + ε_αβω(δ_γασ_k+1^ω - δ_γβσ_k^ω) + ε_γβωσ_k^ασ_k+1^ωσ_k+2^γ]   .

In the first line, the only non-vanishing terms are for those pairs (j,j+1) that partially or completely overlap with (k,k+1). In the second line, we used the standard properties of the Pauli matrices. The last expression is enough to compute the action of ℒ^0_1 on H^(2)_k and B^(2)_k which reads

    ℒ^0_1[H^(2)_j]     = -2J/ħ(B_j^(3) - B_j-1^(3))   , 
    ℒ^0_1[B^(2)_j]     = -4J/ħ(H_j+1^(1) - H_j^(1))-2J/ħ(H_j^(3) - H_j-1^(3))   .

For r ≥ 3, the operators in Eq. (<ref>) have Pauli-z strings. We need to compute terms of the form

    ∑_j=1^L-1   [σ_j^γσ_j+1^γ,σ_k^α Z^(r-2)_k+1σ_k+r-1^β] = σ_k-1^γ [σ_k^γ,σ_k^α] Z^(r-2)_k+1σ_k+r-1^β + [σ_k^γσ_k+1^γ,σ_k^ασ_k+1^z] Z^(r-2)_k+2σ_k+r-1^β
        + σ_k^α Z^(r-2)_k+1[σ^γ_k+r-1,σ^β_k+r-1]σ^γ_k+r + σ_k^α Z^(r-3)_k+1 [σ_k+r-2^γσ_k+r-1^γ,σ_k+r-2^zσ_k+r-1^β]   , 
        = 2i [ ε_γαωσ_k-1^γσ_k^ω Z^(r-2)_k+1σ_k+r-1^β + ε_γβωσ_k^α Z^(r-2)_k+1σ_k+r-1^ωσ_k+r^γ
        + (ε_γαωδ_γ zσ_k^ω + ε_γ z ωδ_γασ_k+1^ω) Z^(r-3)_k+2σ_k+r-1^β + σ_k^α Z^(r-3)_k+1 (ε_γ z ωδ_γβσ_k+r-2^ω + ε_γβ zδ_γ zσ_k+r-1^ω)]   .

Again, the only non-vanishing terms are for those pairs (j,j+1) that partially or completely overlap with the string (k,k+1,...k+r-1). No commutator appears containing pairs of σ_z since these also vanish. The last expression suffices to compute the action of ℒ^0_1 on H^(r)_k and B^(r)_k for r ≥ 3 and a similar reasoning holds for ℒ^0_2. We summarize the results below:

    ℒ^0[𝕀]     = 0   , 
    ℒ^0[H_j^(1)]     = - 2J/ħ(B^(2)_j - B^(2)_j-1)   , 
    ℒ^0[B_j^(2)]     = 4J/ħ(H_j^(1) - H_j+1^(1))+2J/ħ(H_j^(3) - H_j-1^(3)) + 2(ϵ_j - ϵ_j+1)/ħH_j^(2)  , 
    ℒ^0[H_j^(r)]     = - 2J/ħ[ (B_j^(r-1) - B_j+1^(r-1)) + (B_j^(r+1) - B_j-1^(r+1)) ] - 2(ϵ_j - ϵ_j+r-1)/ħB_j^(r) ,  r ≥ 3   , 
    ℒ^0[B_j^(r)]     = 2J/ħ[ (H_j^(r-1) - H_j+1^(r-1)) + (H_j^(r+1) - H_j-1^(r+1)) ] + 2(ϵ_j - ϵ_j+r-1)/ħH_j^(r) ,  r ≥ 3   .

With the action of all superoperators computed and using Eq. (<ref>) we obtain the set of Eqs. (<ref>)–(<ref>) for the coefficients used in the main text.

Frontiers-Harvard




 Paper Title: Enhancing Quantum Entanglement Through Parametric Control of Atom-Cavity States.






[]arthur.vesperini@unisi.it
DSFTA, University of Siena, Via Roma 56, 53100 Siena, Italy
INFN Sezione di Perugia, I-06123 Perugia, Italy


[]roberto.franzosi@unisi.it
DSFTA, University of Siena, Via Roma 56, 53100 Siena, Italy
QSTAR & CNR - Istituto Nazionale di Ottica,    Largo Enrico Fermi 2, I-50125 Firenze, Italy
INFN Sezione di Perugia, I-06123 Perugia, Italy


 

§ ABSTRACT 
Dicke states form a class of entangled states that has attracted much attention for their applications in various quantum algorithms. They emerge as eigenstates of the Tavis-Cummings Hamiltonian, a simplification of the Dicke model, which describes an assembly of two-level atoms trapped in an electromagnetic cavity. In this letter, we show that in the regime where the field energy is large with respect to the atomic energy splitting, precise control of the ground state can be implemented. Specifically, pure Dicke states can be selected and produced by appropriate tuning of the parameters. This result may have important applications in quantum engineering and quantum information theory.





Enhancing Quantum Entanglement Through Parametric Control of Atom-Cavity States
    Roberto Franzosi
    August 5, 2024
===============================================================================



Introduction. – 
The ability to manipulate quantum states with the aim of enhancing their entanglement degree constitutes one of the main steps to be achieved in the development of quantum technologies for the second quantum revolution. In fact, entanglement is considered an essential resource in quantum information, quantum cryptography, teleportation, and quantum computation.
Recently, quantum entanglement has also been proposed as a key resource for the realization of high-efficiency quantum batteries <cit.>. Indeed, long-distance entanglement allows for a charging speedup, resulting in a super-extensive power of quantum batteries. In this context, some theoretical schemes for describing quantum batteries are inspired by the Tavis-Cummings or Dicke models <cit.>.
In the latter, the fast charging effect is associated with the superradiant phase transition occurring in such systems. 
Also in the context of quantum technologies and quantum computation, Dicke states constitute a valuable resource (see <cit.> and references therein).



In the present work, we address the study of the off-resonance Tavis-Cummings model, demonstrating that it indeed undergoes a superradiant quantum phase transition (QPT) at a finite system size, persisting as the system size approaches the thermodynamic limit. Moreover, we identify the Hamiltonian parameters that, at the quantum phase transition, drive the system's atomic ground state towards any chosen Dicke state. Among these, the most interesting one, in terms of its applications, is the maximally entangled Dicke state. This result can therefore be applied as an experimental scheme for preparing maximally entangled states.

Model. –
The Jaynes–Cummings (JC) model was initially proposed in 1963 <cit.> to describe the interaction of a two-level atom with an electromagnetic field. Today, it is considered an intriguing model due to the various physical effects it exhibits, such as Rabi oscillations, collapses and revivals of Rabi oscillations, and the superradiant phase transition <cit.>.
The Tavis–Cummings (TC) model is an interesting generalization of the JC model, considering M two-level atoms (qubits) interacting with a single mode of a quantized electromagnetic field <cit.>. The TC model provides the opportunity to observe additional non-classical effects, including state squeezing and quantum state entanglement. These phenomena, in particular, are considered of primary importance, for instance, in enhancing the performance of optical atomic clocks towards the Heisenberg limit <cit.>.
The TC model Hamiltonian is given by

    H = ω_c a^† a + ω_z S_3 - λ/√(M)(a^† S_- + a S_+),

where a^† and a are the creation and annihilation operators of photons in the cavity, satisfying [a, a^†] = 1. The total spin operator components S_j, j=1,2,3, satisfy the usual commutations [S_i, S_j] = i ϵ_ijk S_k, and S_± = S_1 ± i S_2. Here and in the following, we consider units in which ħ=1. In terms of the Pauli matrices σ^α_j (j=1,2,3), where α=0,…,M-1 runs over the index of the distinguishable atoms, we have S_i = ∑^M-1_α=0σ^α_i/2.
The model is characterized by three tuning parameters: the photon frequency, denoted as ω_c, the atomic energy splitting, denoted as ω_z, and the photon–atom coupling strength, denoted as λ. The TC model possesses desirable properties as it is fully solvable <cit.>.
In fact, the calculation of the eigenstates can be simplified by expressing the Hamiltonian of the full system as a sum of two commuting terms, namely H = H^I + H^II and, where
H^I=ω_c (a^† a+ S_3), H^II= ω_c (η-1) S_3 - ω_c g√(η / M) (a^† 
S_- + a S_+), where η = ω_z /ω_c and g=λ/√(ω_c ω_z). 

For the sake of simplicity, from now on we will set ω_c=1 or, in other words, express energies in units of ω_c. The model can thus be studied by solely considering the two parameters η and g, rewriting the Hamiltonian as

    H = a^† a + η S_3 - g√(η)/√(M)(a^† S_- + a S_+)   .


Quantum phase transition. –
The TC model undergoes a quantum phase transition, the nature of which can be understood by studying the system's ground state as a function of the Hamiltonian parameters.
While the TC model has an infinite-dimensional Hilbert space, since [H^I, H^II]=0, the energy spectrum can be organized into multiplets of simultaneous eigenstates for H, H^I, and H^II <cit.>. Each multiplet is an eigenspace of H^I that is identified by an index k∈ℕ. 
In particular, the eigenvalues of H^I and their corresponding multiplicities are E^I_k = k-M/2 and d^I_k = min(k+1,M+1) for k∈ℕ, respectively. 
The energy spectrum of the full Hamiltonian is therefore derived by diagonalizing H^II in each eigenspace H^I_k (of dimension d^I_k), for k ∈ℕ. We denote by |n, M_3⟩ the tensor product of an n-photons Fock state and a normalized eigenstate of S_3 with eigenvalue M_3, for -M/2 ≤ M_3≤ M/2. The k-th eigenspace H^I_k is spanned by the vector basis |k,-M/2⟩, …, |0,k-M/2 ⟩, for k≤ M, and |k,-M/2⟩, …, |k-M,M/2 ⟩, for k>M. Note
that in each eigenspace H^I_k we have k=n_φ + M_3 + M/2, where n_φ≥ 0 is the number of photons in the cavity. The general form of the eigenstates of H in H^I_k is

    |Ψ⟩ = ∑_n=0^min{k,M}a_n|k-n, n - M/2⟩, with  k∈ℕ  .

Let us denote by E_k(η, g) the lowest eigenvalue of H in each eigenspace H^I_k, depending on the Hamiltonian parameters. The corresponding ground state for H is

    |GS (η,g) ⟩  =  |E_k^*(η,g) ⟩

where k^*∈ℕ is such that

    E_k^* (η,g)= min_k 
    { E_k (η,g) }  ,

thus k^* = k^* (η,g) is named excitation number of ground state of the full Hamiltonian.

For g<1, |GS (η,g)⟩=|0,-M/2⟩ and k^*=0.
However, at g=g_1:=1, the energy level E_0(η,g) crosses the level E_1(η,g). Corresponding to this level crossing k^* jumps from 0 to 1. This first level crossing is followed by further crossings between the minimum energy levels of successive multiplets as the magnitude of g increases. 
If, by increasing the magnitude of g, E_k(η,g), the minimum energy level of H, crosses the minimum energy level of H E_k+1(η,g) at g=g_k+1(η), then at this value of g, the value of k^* changes from k to k+1.
Since n_φ = k - M_3  -M/2≥ k - M, the greater g is, the greater k^* becomes, and consequently, the greater n_φ becomes.
Remarkably, in the limit of strong spin energy separation, i.e., as η approaches infinity, all the crossing-level points converge at the quantum phase transition point g=1. We expect that, in the opposite limit, as η approaches zero, the differences between successive critical values g_k increase. We will show below that it is the case, provided that g remains small enough.

Maximally entangled Dicke states. – 
The eigenstates |S=M/2,M_3=n-M/2⟩=|D^M_n⟩ of S_3 are in fact Dicke states, namely degenerate states of n excited qubits. They are defined as

    |D^M_n⟩ =Mn^-1/2∑_j P_j{|e⟩^⊗ n⊗ |g⟩^⊗ (M-n)}  ,

where we denote with ∑_j P_j the sum over all the possible permutations and, for μ=0,…,M-1, |e⟩^μ and |g⟩^μ are the eigenstates of σ^μ_3 with eigenvalues +1 and -1, respectively. 

Yet, the Dicke states are known to possess a number of interesting properties. In particular, some of them are known to exhibit genuine multipartite entanglement. It is thus of great interest to devise ways to generate such target states in a controlled fashion. 

Hereafter, we use the following definition of pure state entanglement per qubit, derived by the Entanglement Distance (ED) <cit.>,

    E(| ψ⟩):=1 - 1/M∑^M-1_μ=0∑^3_j=1|⟨ψ|σ_j^μ| ψ⟩|^2,

where the first sum runs over all the qubits.

For even M, the state |D^M_M/2⟩ maximizes the ED E, since it is E(|D^M_M/2⟩)=1. 
On the other hand, for odd M, the most entangled Dicke states are |D^M_M/2±1⟩, with E(|D^M_M/2±1⟩)=1-4/M^2.

Parametric state control. – 
The form of Hamiltonian (<ref>) suggests that η is a key parameter governing the probability amplitudes a_n. In fact, the smaller η is, the greater the number of excited atoms, since a smaller η corresponds to a smaller energy cost of the atomic excitation compared to that of photons.
Guided by this observation, we expect that straightforward control of the ground state can be implemented through fine-tuning the parameters (g,η),  given the dependence k^*=k^*(g,η) and a_n=a_n(g,η). 

In particular, the atomic part of the ground state writes

    ρ_s(g,η)   =_φ[|Ψ⟩⟨Ψ|]
       = ∑_n=0^min{k,M}w_n(g,η)|D^M_n⟩⟨ D^M_n|,

where _φ[·] denotes the partial trace over the photonic part of the system, and w_n=a_n^2. Thus, controlling the weights w_n(g,η), one may be able to engineer mixtures of Dicke states. 

Obviously, for all practical purposes, the most interesting targets are pure Dicke states and, even more, the maximally entangled ones.

Perturbative development. – 
Restricting ourselves to an eigenspace H^I_k, we can rewrite the Hamiltonian as H_k = H^I_k + H^II_k, where H^I_k= E^I_k𝕀, and

    H^II_k = H_k^(0) + √(η)H_k^(1) + η H_k^(2)  ,

with H_k^(0)=-S_3, H_k^(1)=-g/√(M) (a^† S_- + a S_+) and H_k^(2)=S_3. We set aside the constant term H^I_k for now.

Considering √(η)≪ 1, we can apply perturbation theory to H^II_k. Namely, we assume that its ground state and ground state energy can be perturbatively expanded in the parameter √(η), that is | E_k⟩=∑_j η^j/2| E_k^(j)⟩ and E_k^II =∑_j η^j/2E_k^(j), where j=0,1,2,⋯

From the eigenvalue equation H^II_k | E_k⟩ = E_k| E_k⟩, we straightforwardly obtain, after some computations, solutions up to second order in √(η). 
At zeroth order, we have E_k^(0) = -(k - M/2) and |E_k^(0)⟩=|0, k-M/2⟩ for k≤ M (|E_k^(0)⟩=|k-M, M/2⟩ for k> M).
At first order, we have E_k^(1) =0 and | E_k^(1)⟩ = g√(k(1 - k-1/M)) | 1, k-1 -M/2 ⟩ for k≤ M (| E_k^(1)⟩ = g | k-M + 1, M/2 -1 ⟩ for k> M).
Finally, the second order energy correction reads E_k^(2) = - g^2k(1-k-1/M) + (k- M/2) for k≤ M (E_k^(2) = - g^2 + M/2 for k> M).
It results the total ground state energy E_k = E_k^I + E_k^II in the k-th eigenspace

    E_k    = η( k - M2 - g^2k(1-k-1/M)) + o(η^3/2)   ,  k≤ M 
    
        E_k    = k - M + η(M2 - g^2 ) + o(η^3/2)  ,  k> M   .

Solving E_k+1-E_k=0, we retrieve the position of the level crossings.
In the case k> M there are no solutions, in the case k ≤ M we get

    g_k≈√(M/M - 2k + 2)  ,

and, therefore, for the total excitation number of the ground state of the full Hamiltonian as a function of g we have

    k^*≈⌈M/2( 1 -  1/g^2)⌉  .


Clearly, in the perturbative regime, the targeted value k^*=M/2 (k^*=(M+1)/2) is reached for high enough values of g, namely when g>√(M/2) (g>√(M)) for M even (odd). No further level crossing is predicted by our perturbative analysis; however, for any finite η, this analysis ceases to be valid for large g. This is because the term √(η)H_k^(1) scales as g√(η). An appropriate trade-off should thus be found, with high enough g and low enough g√(η).
In Figures <ref> and <ref>, we see that the agreement with numerical results is quite good in the regime where η≪ 1 and g√(η)≪ 1. 

Furthermore, the perturbative expansion predicts that the atomic part of the ground state is, up to a correction of order o(η g^2 k), in the pure state |D^M_k⟩⟨ D^M_k|.

We numerically computed w_M/2(g,η) (w_M/2+1(g,η)) for a wide range of values of these parameters, and different even (odd) M, as shown in Fig. <ref>. 

As expected, for any finite M, there exists a parametric region for which w_M/2≈1, and 
    ∀ M<∞, ∃ g such that  w_M/2(g=g,η)η→ 0⟶ 1.
 

It results that the fine-tuning of the parameters of the TC model allows for the generation of an almost pure highly entangled state of M qubits. 

Evidently, as M increases, the parametric region of interest shrinks (as can be seen in Fig. <ref>); this implies that such fine-tuning cannot be achieved for arbitrary system sizes.


It is worth noting that a mere projective measurement on the photonic degrees of freedom of the ground state suffices to obtain a pure Dicke state.
To generate the state |D^M_k⟩, we simply place ourselves in the bulk of the parametric region where k^*=k, cool the system down to its ground state, and perform a measurement of the photon number. The parameters can be fine-tuned so that the latter has a high probability to be null, namely w_k≈ 1, as can be seen in Figure <ref> in the specific case k=M/2; in this case, the atomic part of the system is in the desired pure state. It has on the other hand a probability of order o(η g^2 k) to be non-zero; in this case, we reinitialize the process. 

We showed in this letter that a chosen pure Dicke state may be obtained as the atomic part of the ground state of the Tavis-Cummings model, upon appropriate tuning of the parameters. Furthermore, the maximally entangled state |D_M/2^M⟩, most useful in algorithmic applications, is shown to be particularly easy to target, as it corresponds to a wide parametric region. 
Follow-ups of this work include the search of possible physical implementations, and to take into account the noise occurring at finite temperature and in opened systems. One could further study to which extent these results are applicable to other models, such as the Dicke model.

This framework could be of great use in quantum technologies.

Acknowledgments. –
We acknowledge support from the RESEARCH SUPPORT PLAN 2022 - Call for applications for funding allocation to research projects curiosity-driven (F CUR) - Project ”Entanglement Protection of Qubits’ Dynamics in a Cavity”– EPQDC and from INFN-Pisa.
Furthermore, 
we acknowledge support from the Italian National Group of Mathematical Physics (GNFM-INdAM). 






*



 Paper Title: An open-source framework for quantum hardware control.













CERN-TH-2024-126










^1Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE
^2TIF Lab, Dipartimento di Fisica, Università degli Studi di Milano
^3Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
^4Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore
^5European Organization for Nuclear Research (CERN), Geneva 1211, Switzerland
^6Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore

§ ABSTRACT 

    The development of quantum computers needs reliable quantum hardware 
    and tailored software for controlling electronics specific to various quantum platforms. 
    Middleware is a type of computer software program that aims to provide
    standardized software tools across the entire pipeline, 
    from high-level execution of quantum computing algorithms to low-level driver
    instructions tailored to specific experimental setups, including instruments.
    This paper presents updates to , a software library that leverages 
    's capabilities to execute quantum algorithms on self-hosted quantum 
    hardware platforms. 
    offers an application programming interface (API) for instrument control
    through arbitrary pulses and driver operations including sweepers.
    This paper offers an overview of the new features implemented in since Ref. <cit.>,
    including the redefined boundaries between platform and channel classes, the integration of an emulator 
    for simulating quantum hardware behaviour, and it shows updated execution times benchmarks
    for superconducting single qubit calibration routines.






§ INTRODUCTION


Nowadays, quantum computing research and development needs both reliable quantum hardware and classical hardware. 
The latter is what is usually called the control electronics and takes care 
of synthesizing the simultaneous synchronized control waveforms, specific to each different quantum
platform technology, required to operate quantum hardware.
The objective of middleware is to provide standardized software tools for the whole pipeline,
e.g., from the high-level execution of quantum computing algorithms based on the quantum circuit
paradigm, to operate the low-level control instrument instructions tailored to a specific experimental setup.


The commitment to build a software with a platform-agnostic interface helps the transition from theory
to experiments by reducing the effort and expertise required to operate a quantum platform
and develop novel quantum algorithms.
Additionally, it ensures reusability and this allows the creation of a community of quantum 
laboratories that can share experiments and data significantly reducing 
the burden coming from the different setups. 

Our commitment to open-source and community-driven software started with
 <cit.>, a framework for gate-based and adiabatic 
quantum computing with hardware acceleration.


In this paper, we present updates to  <cit.>, a software library 
that leverages ’s potential to execute quantum algorithms on self-hosted
quantum hardware platforms.


takes care of all the necessary operations to prepare the execution of quantum circuits
on a fully characterized device.

is running on a host computer, which communicates, typically via a
network protocol, with the control electronics used for generation of synchronized signal sequences. 
These electronics are connected to the quantum processing unit (QPU) via different channels. 

In the case of operating a superconducting quantum device: the readout and
feedback channels in a closed loop for measuring the qubit, the drive channels
for applying gates and, for flux-tunable qubits, the flux channels for tuning
their frequency.
Moreover, some architectures there may have coupler qubits between some 
computational qubits that need flux pulses to control computational qubit interactions.


The library offers a dedicated application programming interface (API) not only for quantum circuit design,
but also qubit calibration, instrument control through arbitrary pulses, driver operations 
as sweepers.

Thanks to , we can operate the control electronics required to fully calibrate a quantum device performing
specific experiments. In order to make this process as smooth as possible, we have also
developed on top of the  <cit.> library collecting some calibration
protocols for superconducting qubits with a user-friendly interface.


Successful implementation of the whole ecosystem will provide the research community with a prototype
of an extensible, quantum hardware-agnostic, open-source hybrid quantum operating system, 
fully tested and benchmarked on superconducting platforms.

In the following sections, we first give an overview of the project in Sec. <ref>,
highlighting the differences with the old software version described in Ref. <cit.>, 
then we provide the updated time benchmarks in Sec. <ref>, at the end we
provide some details about the emulator in Sec. <ref>.



§ PROJECT OVERVIEW



provides four main interface objects:
the  object, which is used to define arbitrary pulses played on the quantum device,
a series of pulses can be collected in  and executed on 
a specific QPU through the .


Pulses constitute the basic building blocks of programs executed on quantum hardware.
They represent a physical pulse, i.e., they are used to read 
the state of a qubit, drive it to change its state,
or flux bias a qubit to change its resonant frequency to probe two-qubit interactions.


provides pulse objects for each one of these operations as its
 object holds the required information about amplitude,
duration and phase of the pulse for the generation of physical pulses.
Differently from the previous version, the  object now further 
facilitates the integration with different control electronics.


Abstract pulse sequences defined using the  API can be deployed on
hardware using a .  is the  core object and it
orchestrates the different instruments for qubit control.
Each  instance corresponds to a specific quantum device controlled
by a specific set of instruments. It allows users to execute a single sequence,
a batch of sequences, or perform a sweep, in which one or more pulse parameters
are updated in real-time, within the control instrument without external communication. 
This new version allows the  to execute all of them
within a single instrument connection step, increasing the uptime of the QPU.



The  object also contains single qubit, two qubit interactions 
and coupler qubits information and eventually any other quantum components like 
parametric amplifiers that may be present in the experimental setup.
Since, computational qubits and coupling qubits are similar objects from the 
instrument point of view, as elements you send pulses to, so in the software level 
they can be defined by the same class we called .



To define two qubit interactions,  objects 
contain information about coupled qubits pairs in a given device for a given 
configuration and their corresponding two-qubit native gates.



Finally,  represents the connection between the quantum device 
inputs and outputs to the proper instrument port.This connection is essential 
for the instrument our interface to target the desired .
It also provides a -centric interface instrument parameter setting,
which is useful in calibration routines.



 are also responsible of signal generation. 
Differently from the previous version, now the readout is decoupled
into probe and acquisition channels, thus giving more freedom to the end user to decide
which part of the readout pulse to acquire or extend the acquisition beyond the 
time interval defined by the readout pulse.


With the new refactoring, the boundaries between the  and 
classes are redefined: there is no more distinction between readout 
and drive pulses. Since, they are just defined by the physical channel where
the pulses are sent, the frequency is now a  property, 
so this class is closer to a logical definition of channel.


Another significant change is  which got moved
to .  This choice is driven by a rethinking of the respective roles
of the ecosystem libraries and transparency. 
The transpilation procedure consists of processing the sequence of gates 
required by an algorithm and integrate SWAP gates, if necessary, so that the connectivity
of the chip is respected.
In practice, given the topology of the device, this is a task completely solvable within itself.
Once a connectivity-compatible sequence of gates is defined, provides a compiler,
that is, the translation of abstract gates into native gates, and thus into pulses.






§ CROSS-PLATFORM BENCHMARK



In this section, we show the updated benchmarks already presented in the  paper.
They are related to the execution times of a set of calibration experiments performed
on a single qubit by different control electronics (see Tab.<ref>) currently supported by  out of the box.


In contrast to the previous benchmark version,  side, we have implemented the possibility to unroll a list of
sequences to a single pulse sequence that contains multiple measurements. This approach achieves much faster execution 
compared to executing the sequences one by one in a software loop. On the side of the control electronics, we have updated the 
Zurich Instrument software to  <cit.>.


The experiments chosen for this benchmark represent the minimal set of routines 
required for superconducting single qubit calibration. 

They also offer a view of the different execution modes supported by : in particular 
the Single shot classification experiment executes fixed pulses sequences, 
while the Spectroscopies perform different sweeps over pulse parameters.

All the results are summarized in Fig. <ref>, for each calibration protocol 
we provide also the theoretical execution time T_ideal, 
defined as the total duration of the pulse sequences executed
during the acquisition, whose formula <cit.> is

    T_ideal =n_shots∑_i (T_sequence, i + T_relaxation).

where T_sequence, i is the duration of the whole pulse sequence
in the i-th point of the sweep, T_relaxation the time we wait
for the qubit to relax to its ground state between experiments,
n_shots the number of shots in each experiment and
the sum runs over all points in the sweep.
The ideal time denotes how long the qubit is used during
an experiment and provides the baseline for our benchmark.

As already pointed out in Ref. <cit.>, the comparison between
the ideal and real execution times show that the first one is always less than the second one, 
because of different overheads, indeed we can express the real execution time T_real as



    T_real = T_qibo + T_inst + T_ideal,

 
where T_qibo is the overhead coming from  backend and T_inst the instruments one.
We found that the overhead coming from the , T_qibo, 
is negligible compared to that of the control instruments, i.e.,


    T_real≃ T_inst + T_ideal.








§ EMULATOR



The new version supports emulators to simulate quantum hardware. It is a crucial tool especially for , as it
could be used both for testing the software itself (especially when access to the real device is limited or unavailable)
and serve as a digital twin to a specific quantum hardware to assist with a range 
of functionalities including predicting output of calibration experiments, 
coarse-grain calibrations, pulse-shaping and test-bedding, and more.
For the latter, the emulator requires the device parameters that characterize 
the quantum hardware of interest as inputs to build an accurate model of the hardware.


To integrate the emulator into the  ecosystem, we deployed an ad-hoc controller class called .
The  is exclusively called by the emulator and it serves
primarily as a middleman to translate and communicate objects between  and 
the selected quantum dynamics simulation library (hereafter referred to as the
simulation engine) used to numerically solve the underlying quantum dynamics of
the device model in the presence of time-dependent control pulse sequences. 
Specifically, it initializes the simulation engine with a device model specified
by the device parameters and simulation settings in the runcard, extracts 
modulated signal waveforms from pulse sequences and sends them to 
the simulation engine which in turn performs the dynamics simulation, and at
the end of the simulation, translates the results generated from the simulation
engine back to  (or ) result objects. 


From the user's point of view therefore, the emulator platform behaves like 
a quantum hardware platform and both are equivalent in terms of attributes and 
functions. Consequently, it requires a platform folder and it is initialized 
as any other device platform. In addition, the emulator returns simulated 
results – a time series of statevectors (or density matrices when including dissipation)
of the underlying quantum system that was generated sequentially by the simulation
engine as it solves the dynamics described by the Lindblad master equation,


    ρ̇ (t) = -i [ H (t), ρ (t)] + ∑ _kγ_k/2[ 2 A_k ρ(t) A_k^† - ρ (t) A_k^† A_k - A_k^† A_k ρ (t) ].


In the above, ρ (t) is the density matrix of the system's quantum state at
time t, H(t) is its time-dependent Hamiltonian, and A_k are the operators 
through which the environment couples to the system with rate γ _k.
As currently only supports superconducting-qubits-based hardware, 
the device model used by the emulator is based on N capacitively coupled 
transmon qubits modelled as Duffing oscillators <cit.> 
with number of energy levels predefined by the user in the  settings:


    H (t)    =    H_ sys + H_ drive (t), 
    
      H_ sys   =   ∑ _i=1^N ( ω _i b_i ^† b_i + α_i/2 b_i ^† b_i (b_i ^† b_i - 1) ) + ∑ _i ≠ j g_ij (b_i ^† b_j + b_i ^† b_j ),
    
      H_ drive (t)    =   ∑ _i=1^N [Ω_X,i (t) cos (ω _ drive, i t) + Ω_Y,i (t) sin (ω _ drive, i t) ] (b_i ^† + b_i).


In the above, ω _i, ω _ drive, i and α _i denote the 
resonant frequency, drive frequency and anharmonicity respectively for transmon
i, b_i (b_i^†) its annihilation(creation) operator, and
Ω_X,i(t), Ω_Y,i(t) the drive amplitudes on its quadratures, 
while g_ij denotes the coupling strength between transmon i and j. 
Decoherence is incorporated using the Bloch-Redfield model <cit.>,
whereby each transmon is characterized by its longitudinal and transverse 
relaxation times T_i,1 and T_i,2 respectively,


    A_i,1 = 1/2(σ _i,X + i σ_i,Y),    A_i,2 = σ _i,Z,   γ _i,1(2) = 2π/T_i,1(2),


with σ _i,μ = X,Y,Z the Pauli matrices corresponding to transmon i.
As an example, we show in Fig. <ref> the state overlap between the
transmon qubit, modelled as a three-level system, with each of its energy modes
as it interacts with the pulse for the X gate followed by the Hadamard gate.
This can be easily extended to include other quantum computing technologies when
they are available on .
The first supported simulation engine is based on  <cit.>,
with plans to incorporate  <cit.> support for GPU acceleration, as well as
tensor-network based quantum dynamics simulation libraries to speed up the
simulation of larger system sizes with a lower memory footprint but with a small accuracy cost.



For the initial release, resonators are not included in the simulation model and
therefore the only acquisition modes supported are the discrimination and integration, where the 
latter is currently implemented as a projection onto the in-phase pulse component. 
Despite this limitation, the available emulator can already execute most protocols to simulate calibration and device characterization.
This part of the library is still a work in progress and we plan to expand it
with new features.  Increasing the complexity of the simulated quantum system with
the inclusion of flux tunable qubits, resonators and couplers is part of our roadmap.




§ CONCLUSIONS


In this proceedings, we have presented significant updates to , a software library
designed to harness the full potential of  for executing quantum algorithms
on self-hosted quantum hardware platforms. The enhancements introduced in include a more platform-agnostic approach to pulse definitions, streamlined 
orchestration of quantum hardware control through the  object,
and the integration of an emulator for simulating quantum hardware behaviour.


Our work goal is a platform-agnostic interface to facilitate
the transition from theoretical quantum computing models to practical experiments.
This approach not only reduces the expertise required to operate diverse quantum
platforms but it also ensures tools reusability as well as experiment and data 
sharing across a community of quantum laboratories. 
This collaborative environment significantly lower the barriers associated with
different experimental setups.


Benchmark results demonstrate that improvements lead to more
efficient execution times for qubit calibration routines, highlighting the 
library's capability to optimize the performance of various quantum control electronics.
The refined boundaries between the  and  classes
and the relocation of the  to  further streamline the 
software architecture, ensuring a clearer separation of roles and greater transparency.


The addition of the emulator component represents a crucial tool for both software 
testing and predictive analysis of calibration experiments. While currently, it is 
still a work in progress, the emulator sets the stage for future expansions,
including more complex quantum systems and advanced acquisition modes.

In conclusion, the advancements in  not only enhance its functionality 
and efficiency but also contribute to the broader goal of creating an extensible,
quantum hardware-agnostic, open-source hybrid quantum operating system. This system,
fully tested and benchmarked on superconducting platforms, offers a robust foundation
for ongoing research and development in the field of quantum computing.
By fostering a collaborative community and providing powerful, flexible tools,
 and  collectively advance the frontier of quantum technology, 
supporting both current and future innovations in quantum research and applications.



§ ACKNOWLEDGMENTS

This project is supported by ’s Quantum Research Center. 
This work was further supported by the National Research Foundation Singapore, under its Quantum Engineering Programme 2.0 (National Quantum Computing Hub, NRF2021-QEP2-02-P01).
The authors thank all contributors for helpful discussion and Dr. Ye Jun. 
M.R. is supported by ’s Quantum Technology Initiative () through the Doctoral Student Program.
J.Y.K acknowledges funding support from A*STAR C230917003.

unsrtnat



 Paper Title: Beyond full statevector simulation with Qibo.



 TIF-UNIMI-2024-7







^1Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE
^2TIF Lab, Dipartimento di Fisica, Università degli Studi di Milano
^3School of Computing, Australian National University, Canberra, ACT, Australia
^4Instituto de Física, Universidade Federal do Rio de Janeiro, P.O. Box 68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil

^5European Organization for Nuclear Research (CERN), Geneva 1211, Switzerland


andrea.pasquale@unimi.it

§ ABSTRACT 
    In this proceedings, we present two new quantum circuit simulation protocols recently added as optional backends to , an open-source
    framework for quantum simulation, hardware control and calibration.
    We describe the current status of the framework as for version .
    In detail, the two new backends for Clifford and tensor networks simulation are presented and benchmarked against the state-of-the-art.




§ INTRODUCTION

With the significant achievements reached by quantum technologies <cit.>,
the interest towards quantum computing is consistently growing. However, the noise
affecting these early devices is preventing large-scale applications of quantum algorithms.
Waiting for quantum computers to reach a level of reliability that allows for
full-scale execution of quantum computing routines without limitations on the problem size,
it is necessary to continuously improve the simulation tools we have at our disposal.
In particular, the development of classical simulation techniques is crucial to
overcome the intrinsic representation limit in full statevector simulation, where
the memory required to fully represent a qubit system explodes exponentially as
the number of qubits increases.

Several open-source software packages for classical simulation of quantum systems
are available. While some of them, including  <cit.> and
 <cit.>,  <cit.>,
 <cit.> and  <cit.>, provide general tools for quantum
simulations, other libraries such as  <cit.>
mostly focus on specific applications like quantum machine learning.

In this context, for the last 4 years we have been developing  <cit.>,
an open-source software
framework for quantum computing. At its early stages, was mainly dedicated
to full statevector (including density matrix) simulation <cit.>,
and managed to achieve performance competitive with state-of-the-art simulators,
thanks, also, to a Just-In-Time compilation approach <cit.>.
The current layout of the framework, as of version , is shown in
Fig. <ref>. provides a language Application Programming Interface
(API) to deploy quantum algorithms using either a circuit-based or a quantum
annealing-based approach. Moreover, we provide general-purpose tools that are useful in quantum information theory,
including calculation of distances among quantum states and quantum channels.

's modular structure enables the deployment of any component of the language
API on different software and hardware platforms, which we refer to
as backends.
We are now going to briefly present all the backends available in version .
The package <cit.> upon installation is equipped with a backend based on
 <cit.>, which provides adequate performances for simulating circuits with
a relatively low number of qubits n (i.e. n ≤ 20). We also provide a more efficient
general purpose simulator, called ,
which supports hardware acceleration. More specifically, allows multi-threaded CPU execution
using  <cit.>, while GPU support is enabled both through  <cit.>
and through compatibility with NVIDIA's  <cit.> library.
To improve the performance, defines custom operators that
exploit the sparsity of the matrix representation of several quantum
gates that are often used in quantum computing.
Such improvements are also reflected in the quantum annealing approach when the Trotter decomposition is used.
We also provide backends supporting automatic differentiation specifically designed for quantum machine learning
applications, including one based on  <cit.>
primitives, and a recently added one based on the popular machine learning framework  <cit.>.
The possibility of integrating automatic differentiation tools within
such a modular environment can be exploited to develop and test both pure quantum
and hybrid classical-quantum machine learning algorithms <cit.>.

As a middleware software package, also provides a backend for quantum hardware execution.
This backend is called , which defines a dedicated API to perform instrument
control, driver operations, as well as compilation
of circuits into customizable native gates sets.
Alongside , we provide a dedicated package to characterize and calibrate
self-hosted quantum devices:  <cit.>.
Besides simulation and hardware backends, there is an ongoing effort to provide cloud access, giving
users the possibility to manage their self-hosted quantum hardware by using 's hardware control tools,
or to deploy quantum circuits on well-known quantum cloud providers, including IBM Quantum <cit.> and IonQ <cit.>.

This was a brief overview to showcase the wide modularity and diversity that provides. As a demonstration of the
ongoing effort to continuously support new platforms and extend 's compatibility with state-of-the-art quantum
computing software stacks we recently added a backend that interfaces with  <cit.>.

Below, we focus on two simulation backends recently added to the framework, which are dedicated to tensor network simulation of quantum circuits and fast simulation of Clifford circuits.



§ SIMULATION OF CLIFFORD CIRCUITS

With full statevector simulation, it is only possible to simulate circuits up to
a limited number of qubits: John Preskill describes this limitation in Ref. <cit.>
introducing the approximate yet explanatory term of 50-qubit barrier.
The main limitations are imposed by  huge memory requirements and very long computation times <cit.>.
Over the last decades, researchers have studied extensively Clifford circuits, a class of circuits that can be simulated in polynomial times <cit.>.
The quantum states generated by Clifford circuits are called stabilizers states <cit.>,
and have many applications in quantum information theory <cit.>.
As current quantum hardware gets closer to the requirements needed for quantum error correction<cit.>,
such circuits have started to receive more attention, leading to the development of techniques to classically simulate them.



Based on this, a new backend has been developed in which focuses on fast and efficient simulation of Clifford circuits.
The implementation in is based on the phase-space formalism introduced in Ref. <cit.>.
From the perspective of code design, the implementation makes full use of 's modularity, allowing for alternative backends to be easily plugged in.
The basic implementation proposed is based on  primitives. This enables the single-threaded simulation of Clifford circuits on CPUs.
By taking advantage of , we also provide implementations based on  and .
The integration with the  backend for multi-threaded computations on CPUs is done via custom kernels that are compiled just-in-time.
For GPU integration, we make use of custom CUDA C kernels through .

To evaluate our implementation, we compare our Clifford simulator with Clifford simulators available in  and .
The results are shown in Fig. <ref>.
We tested all the simulators on the same dataset consisting of 100 random Clifford circuits for each system syze n.
These circuits were sampled uniformly following Ref. <cit.>.
We ran the CPU benchmarks on an AMD EPYC 7773X processor and the GPU benchmarks on a NVIDIA RTX Quadro a6000.
Our  and  backends asymptotically approach performance similar to ,
whereas our -based implementation displays an advantage over the whole range of qubits considered.
For instance, for n = 1000, our  backend is up to one order of magnitude faster than , and almost two order of magnitudes faster than .

Here, we make a remark about our GPU implementation.
As expected, the overhead of copying the data from the host to the device is dominating the results for a small number of qubits.
However, an improvement due to the huge parallelization capabilities of GPUs is expected to appear at n ≳ 2^11.
Since we benchmarked average runtimes up to 1000 qubits, this crossing in performance was not observed yet, making further investigation of bigger systems a necessity.




§ TENSORNETWORK SIMULATION USING 


After showing that for specific type of quantum circuits we can reduce the computational time,
we now introduce a second popular approach to simulate large quantum circuits:
classical approximation methods.
A popular method for approximating quantum circuits are tensor networks (TN) <cit.>,
which represents states or operators as network of smaller tensors reducing both memory and computational requirements.
They are successfully used, for instance, to solve one-dimensional strongly-correlated quantum systems <cit.>.
On the other hand, the effectiveness of representing large-scale systems
comes at the cost of introducing truncation errors, which make these techniques less
effective when there is the need to know the quantum state more exhaustively.

Within the framework, we have recently developed , a subpackage which enables to execute quantum circuits using tensor network like computations,
allowing to support large-scale simulation of quantum circuits in .
interfaces with state-of-the-art quantum TN simulation libraries such
as  <cit.> from NVIDIA and  <cit.>.
Both Matrix Product States and generic TN are supported. is designed to support
High Performance Computing configurations including single node GPUs, as well as multi-node multi-GPU configuration using
Message Passing Interface or the NVIDIA Collective Communication Library (NCCL) from .

To showcase the capabilities of the library we perform comparison between and performances in Fig.<ref>.
We execute a variational quantum circuit on a  A100 GPU for several number of qubits.
As expected, using full statevector simulation with it possible to run only up to 40,
while using we show how the curve flattens and we observe that we are able to simulate a variational circuit with up to 400 qubits.
Moreover, given the slow rise of the curve we expect to increase the number of qubits with appropriate memory requirements.



§ OUTLOOK


In this proceedings, we have described the latest updates available in .
After a brief overview on all modules currently available in the framework, we have put the focus
on two new simulation methodologies: Clifford simulation and Tensor Networks (TN). We have shown that our Clifford simulator
is competitive with state-of-the-art libraries. We further demonstrated that, despite its early development stage,
our TN implementation is able to simulate efficiently circuits with up to 400 qubits.
Future developments of the framework include having a dedicated module to perform QML algorithms,
which we refer to as , and we are looking forward to expand to support also quantum chemistry, as well as Quantum optimization problems.
Finally, although has been developed in , we are looking to separate core elements to take advantage of the better performance offered by other languages,
e.g.  and .



§ ACKNOWLEDGEMENTS

This project is supported by ’s Quantum Research Center.
The authors thank all contributors for helpful discussion and Liwei Yang and Andy Tan Kai Yong
for their support in developing .
M.R. is supported by ’s Quantum Technology Initiative () through the Doctoral Student Program.

unsrtnat





 Paper Title: Scaling and assigning resources on ion trap QCCD architectures.



































Scaling and assigning resources on 
 ion trap QCCD  architectures
    Anabel Ovide, Daniele Cuomo and Carmen G. Almudever
 
 
Computer Engineering Department, Universitat Politècnica de València, Valencia, Spain

Email: aovigon@upv.es,  cuomo.daniele@outlook.com, cargara2@disca.upv.es
    
===========================================================================================================================================================================================================================





§ ABSTRACT 
Ion trap technologies have earned significant attention as potential candidates for quantum information processing due to their long decoherence times and precise manipulation of individual qubits, distinguishing them from other candidates in the field of quantum technologies. However, scalability remains a challenge, as introducing additional qubits into a trap increases noise and heating effects, consequently decreasing operational fidelity. Trapped-ion Quantum Charge-Coupled Device (QCCD) architectures have addressed this limitation by interconnecting multiple traps and employing ion shuttling mechanisms to transfer ions among traps. 
This new architectural design requires the development of novel compilation techniques for quantum algorithms, which efficiently allocate and route qubits, and schedule operations. The aim of a compiler is to minimize ion movements and, therefore, reduce the execution time of the circuit to achieve a higher fidelity.

In this paper, we propose a novel approach for initial qubit placement, demonstrating enhancements of up to 50% compared to prior methods. Furthermore, we conduct a scalability analysis on two distinct QCCD topologies: a 1D-linear array and a ring structure. Additionally, we evaluate the impact of the excess capacity – i.e. the number of free spaces within a trap – on the algorithm performance. 


scalability quantum computing systems, modular quantum computers, mapping of quantum algorithms, ion trap QCCD. 



§ INTRODUCTION


It is widely known that, in theory, quantum computers have the potential to solve intractable problems for classical computers due to quantum mechanics phenomena such as superposition, entanglement and interference. Nowadays, there are various quantum processors based on different qubit implementations, all facing issues such as high error rates, reduced qubit counts, and relatively short coherence times, which limit their computational power. These technologies, known as Noise-Intermediate-Scale Quantum (NISQ) devices, include superconducting circuits <cit.>, 

quantum dots <cit.>, neutral atoms <cit.>, photonic qubits <cit.>, and ion traps <cit.>, among others.  

Although currently there is not an outperforming quantum hardware, ion trap platforms offer several advantages such as long coherence times, identical qubits, all-to-all qubit connectivity (in a single trap), and precise qubit control, being a promising candidate not only for quantum computation but also for quantum networking <cit.>. However, one of the main challenges, which is shared with all other quantum technologies, is scalability. The number of qubits needs to be substantially increased to build fault-tolerant systems and achieve the full computational power that quantum computers can offer. 

Ion traps utilize the energy states of atomic ions such as Ca+ or Yb+ to encode qubits, which are then manipulated through quantum gates facilitated by lasers. These qubits are confined or trapped, typically in a linear arrangement, within specific regions called traps via electromagnetic fields <cit.>. Current ion trap chips integrate tens of qubits on a single-core architecture, such as the IonQ Forte quantum processor consisting of 36 qubits <cit.>. However, to develop larger devices, one cannot keep on increasing the qubit counts within a single trap as the addition of qubits highly amplifies the ion chain's vibrational mode, leading to increased heating and diminished overall fidelity during execution. 





Given the constraints in confining more and more qubits within a single trap, Quantum Charge-Couple Devices (QCCD) <cit.> are a promising approach to scale up ion trap platforms. QCCD interconnects multiple traps using electric fields to shuttle ions between distinct trapping regions. This architectural modular approach allows an increase in the total number of qubits in the device by expanding the number of traps and still maintaining a suitable number of ions per trap. Practical implementations of trapped-ion QCCD-based processors already exist <cit.>, with notable advancements demonstrated by Quantinuum in enhancing scalability<cit.>.


The emergence of this new architectural design highlights the need to develop novel quantum circuit mapping techniques. Note that they play a crucial role in optimizing the overall circuit execution. More precisely, they increase the circuit fidelity by reducing the total number of operations and parallelizing them while minimizing the frequency of ion SWAPs (relocation of qubits within a trap) and shuttles (movement of qubits between traps). Notably, mapping methods used in single ion trap devices may no longer be applicable to this multi-trap architecture due to the inherent all-to-all connectivity and lack of shuttle requirements. A few circuit mapping techniques for this new quantum technology have been proposed <cit.>, which involves algorithms for qubit allocation, routing of qubits and operation scheduling.

In this work, we focus on the qubit assignment problem for QCCD architectures and present a novel qubit allocation algorithm, which enhances previous state-of-the-art approaches <cit.> by showing an improvement of up to 50% in the total circuit execution time. This advancement is achieved by considering not only the overall interaction of qubits in the circuit but also the temporal distribution of these interactions. To further explore the potential of this scalable design, we conduct a scalability analysis on two distinct QCCD topologies: a 1D-linear array and a ring structure (Figure <ref>). Additionally, we evaluate the algorithm performance based on the impact of the excess capacity, namely the number of free spaces within a trap, set at the beginning of the execution. 

The paper is structured as follows. Section II provides an overview of single and QCCD ion trap devices. In Section III, several mapping techniques for quantum algorithms are presented, with a focus on ion trap processors and highlighting the differences between single and QCCD approaches. Next, in Section IV, our qubit allocation strategy is introduced. In Section V, the selected benchmarks and experimental framework are described. The different experiments and results are presented in Section VI, including a comparison of our qubit allocation method with previously proposed strategies, and its evaluation across different processor topologies. This is followed by a scalability analysis and a study of the excess capacity. The paper ends with the conclusions and future work.


§ TRAPPED-ION QUANTUM PROCESSORS

Nowadays, numerous quantum technologies are competing to demonstrate the full computational capabilities inherent in quantum information processing. Among these quantum computing platforms, ion trap devices emerge as a promising approach due to their long decoherence times and precise qubit manipulation. In ion trap processors, ions are retained within traps, which are specific regions of the quantum chips designed to confine them via electric and magnetic fields (Penning traps) or an oscillating electric field (Paul traps) <cit.>. Typically, ions within traps are arranged linearly. To manipulate the states of qubits and perform operations, including single and two-qubit gates, various techniques can be employed depending on the qubit type, such as optical methods (laser manipulation) or microwave-based approaches <cit.>.

The prolonged coherence times offered by ion traps make this technology a promising candidate not only for quantum computation <cit.> but also for quantum network applications <cit.> including distributed quantum computing  <cit.>. They entail the execution of quantum programs or algorithms across multiple quantum devices interconnected via quantum and classical links. These executions need constant communications, both classical and quantum, among different processors, which may incur prolonged waiting times as devices must await responses from others to proceed with program execution. These waiting times can potentially lead to qubit state degradation highlighting the suitability of ion trap devices for quantum networking <cit.>. 

Multiple trapped-ion processors have been developed, integrating tens of qubits within a single trap <cit.>. However, scalability poses a significant challenge for this technology. As previously mentioned, qubits are confined via electric and magnetic fields, and increasing their counts makes the confinement process more complex. Moreover, the precision required to perform operations becomes more intricate with the increase in the ions' vibrational states <cit.>, which also leads to a growth in the system heating. Consequently, as the number of qubits per trap gets larger and hardware complexity rises, the fidelity of executions drastically reduces.

An alternative and promising approach to address this problem is the modular trapped-ion Quantum Charge-Coupled Device (QCCD) architecture consisting of several traps<cit.>. QCCD employs shuttles to connect different traps, enabling ions to be physically transported from one trap to another with high fidelity <cit.>. By interconnecting traps in this manner, the total number of qubits can be increased without a significant decrease in fidelity, as a reduced number of ions are confined in each trap. These devices support a variety of potential topologies, including a 1D linear array, ring devices, grid structures <cit.>, or X-junction configurations <cit.>. Note that this approach is not exclusive to ion trap processors. Next generations of superconducting quantum devices are expected to be multi-core in which multiple quantum chips are interconnected via quantum and classical links <cit.>.

QCCD ion traps introduce a set of new challenges, encompassing both hardware and quantum circuit mapping methods. Hardware-related difficulties arise from the complexity of the structures and connections between traps. Furthermore, existing quantum algorithm mapping techniques applicable for single-ion trap devices are no longer compatible with this new approach, which will be discussed in the subsequent section.


§ MAPPING OF QUANTUM CIRCUITS ON NISQ DEVICES

Hardware-agnostic quantum algorithms or programs cannot be directly executed on quantum processors as they do not consider hardware restrictions. The main limitations arise from gates, which necessitate decomposition into the ones supported by the quantum chip, and from the restricted qubit connectivity, which limits possible interactions requiring qubit relocation for executing two-qubit gates. To overcome these constraints, quantum circuit mapping techniques are employed to translate and accommodate the quantum circuit to the designated quantum hardware. They usually consider multiple quantum processor characteristics,  including constraints related to qubit connectivity, gate timing, parallel execution of operations, idle time minimization, resource availability, gate fidelity, and qubit coherence times. 







Note that the quantum circuit mapping process is crucial for maximizing the algorithm performance, given the impairments of current quantum devices.




 §.§ The quantum circuit mapping process


The mapping of quantum circuits or algorithms has three fundamental steps across all quantum technologies: qubit allocation or initial placement, qubit routing, and operations scheduling.

Allocation of qubits or initial qubit placement. In this step, logical qubits (i.e. qubits in the quantum circuit) are strategically assigned to physical qubits in order to reduce and facilitate subsequent movements during routing. This paper will focus on this stage and introduce a novel initial qubit placement algorithm that will be explained in subsequent sections. 

Routing of qubits. Given the qubit connectivity constraint, qubits may need to be relocated within the quantum chip to perform two-qubit gates. For instance, in superconducting devices, qubits must be adjacent to perform two-qubit gates,  whereas in trapped-ion QCCD architectures, qubits involved in a two-qubit operation must be placed within the same trap. In this process, additional gates for `moving' qubits around are introduced to relocate ions. The number of movements should be minimal, as they result in a longer execution time and an increase in total operations, consequently decreasing the overall execution fidelity. 

Scheduling of the operations. This step aims at minimizing the execution time of the quantum circuit by maximally parallelizing quantum operations. It typically involves the utilization of a quantum operation dependency gate graph (nodes represent gates and edges dependencies between them). 

Considering that different quantum technologies are subject to different constraints — for instance, the requirement in superconducting devices for qubit adjacency when executing two-qubit gates and the insertion of the corresponding SWAPs or the necessity of shuttling ions in  QCCD architectures — the quantum circuit mapping procedure should be adapted accordingly.



 §.§ From single-trap to  multi-trap architectures


An important feature of ion trap technology is the presence of all-to-all connectivity within each trap. This property simplifies the mapping process in single ion trap devices, as logical qubits can be randomly assigned to physical qubits (qubits remain stationary during operations). Furthermore, qubit routing is unnecessary, as operations can be executed without the need for ion rearrangement. Therefore, the scheduling process becomes the primary focus during mapping, as it involves the strategic prioritization of available gates for execution. 



The complexity of the mapping procedure notably scales when considering ion trap QCCD architectures. In the initial qubit placement stage, qubits have to be optimally placed to minimize or even eliminate the need for future shuttling between traps. Moving qubits from trap to trap will require not only adding shuttling operations but also qubit relocation within the trap itself, as the ions to be shuttled need to be positioned in the trap extreme for being transferred. The routing of qubits also becomes complex,  as it must account for both shuttles required to transport ions throughout the device and for the optimal placement of qubits within the trap. In addition, the scheduler must now contemplate the additional gates introduced to facilitate ion reallocation within a trap and the newly added shuttling movements. Note that in current QCCD architectures, parallel operations within a single trap are not allowed. However, parallel execution of non-dependent operations is possible when these are performed in qubits allocated in different traps. It is worth noting that the maximum number of parallel operations corresponds to the number of traps the device has. An example of mapping a quantum circuit on a QCCD architecture is shown in Figure <ref>.

Currently, a few mapping methodologies have been proposed for efficiently executing quantum circuits on QCCD  architectures. For instance, Saki et al. <cit.> proposed a scheduling and qubit routing approach, which are based on the number
of gates a qubit has in different traps, aiming to identify the most efficient ion movement between traps; it also considers full traps when moving ions, trying to avoid or decongest them. Upadhyay et al. <cit.> introduced an initial qubit placement that considers future operations when placing the qubits. Additionally, in <cit.>, the authors propose a circuit mapping procedure for a four-trap architecture interconnected by an X junction. Furthermore, a routing algorithm formulated as a Boolean satisfiability (SAT) problem was proposed by Daniel et al. in <cit.>, improving their method in subsequent work <cit.>.

Further research of the mapping procedure for scalable trapped-ion architectures is needed not only to efficiently execute quantum algorithms but also, and more importantly, to investigate the potential of new modular architectural designs and guide future developments. In this paper, we introduce a novel initial qubit placement that improves previous qubit allocation strategies by considering the number of qubits interacting with each qubit, the frequency of these interactions, and when those interactions take place (time domain), as will be explained in the next section. In addition, a scalability analysis is conducted in which weak and strong scaling experiments are performed. Furthermore, the initial optimal excess capacity is studied.


§ QUBIT ALLOCATION FOR QCCD ARCHITECTURES

State-of-the-art initial qubit placement techniques for ion trap QCCD processors have been proposed by Murali et al. <cit.> and Upadhyay et al. <cit.>. These methodologies rely on a qubit interaction graph (see Figure <ref>b), where nodes denote qubits and edges represent interactions between them, describing two-qubit gates. The edges are weighted, indicating the number of two-qubit gates between each qubit pair. In both works, logical qubits are assigned to physical traps based on the edge weight, placing edges with higher weights first. 



The main distinction between these two approaches resides in the use of gate prioritization. More precisely, Upadhyay's method prioritizes gates occurring at the initial stages of the program over those appearing later, updating the weight of the edges based on this information. This is achieved by utilizing a decaying function for any re-occurrence of gates, assigning a negative value after a certain number of operations. However, they both overlook the consideration of the total number of qubits interacting with each qubit, which serves as an indicator of the extent to which a qubit will need to be relocated. Specifically, for a given qubit, a higher and more diverse number of operations with other qubits will potentially result in an increase in movements to reach the traps where other qubits are placed, compared to qubits with fewer interactions. In addition, Upadhyay's method, while considering the timing of operations (i.e. when the qubit interactions take place), is suboptimal for lengthy algorithms due to the introduction of a negative weight for late occurrences. Moreover, they do not incorporate ion repositioning within traps based on the operations they are involved in and the temporal distribution of those operations, thereby minimizing ion movements within traps for potential shuttling operations.

In this work, we introduce a novel initial qubit placement strategy, the “Spatio-Temporal Aware Qubit Allocation Algorithm" (STA). STA enhances previous qubit allocation approaches by incorporating not only the number of gates between qubit pairs and the temporal aspect of interactions but also the ratio of qubit-to-qubit interactions. Additionally, it includes a reordering of qubits within traps considering the time operations are executed.



 §.§ The Spatio-Temporal Aware Qubit Allocation Algorithm

The Spatio-Temporal Aware Qubit Allocation Algorithm (STA) contemplates: (i) the involvement of qubit pairs in two-qubit gates while (ii) considering the time in the circuit in which the operations take place, and (iii) the ratio of qubit interactions for each qubit in the circuit. It also considers the initial excess capacity of each trap, which is defined as the number of free spaces within a trap <cit.>, at the beginning of the execution (i.e. in the routing of the qubits, those free spaces can be used). The pseudocode of the STA algorithm is shown in Algorithms <ref> (main routine), <ref>, and <ref>.

The algorithm works based on two data structures: the qubit interaction ratio (R), consisting of N elements (being N the number of logical qubits) and the temporal weight (T), whose values are calculated for each interacting pair of qubits considering the gate execution time. The two lists are derived as follows. After obtaining the operations per time slice (Algorithm <ref> line 1, “S"), which are the operations that can be performed at a specific time in the circuit execution, the ratio of qubit interactions R(q_i) for each qubit and the temporal weight T(q_i,q_j) for each interacting pair of qubits can be computed. The qubit interaction ratio (Algorithm <ref> line 2, "R") is calculated as:

    R(q_i)=r_i/N

where r_i represents the number of different qubits q_i interacts with, and N the number of logical qubits involved in the program. This calculation will give information about how much each qubit interacts with others, a crucial consideration when positioning qubits within traps. Qubits with higher interaction ratios will necessitate more movements, thus optimizing their placement becomes crucial.

Subsequently, a value for each interacting pair of qubits (Algorithm <ref> line 3,  “T") is calculated. This value is based on the number of two-qubit gates involving a pair of qubits within the respective time slice. The temporal aspect is determined by the following formula, which is inspired by the work of Baker et al. <cit.>:


    T(q_i,q_j)=∑^sI(s,q_i,q_j)×2^-s


where q_i,q_j denote the interacting pair of qubits, s represents the current time slice (starting from 0), and the function I(s,q_i,q_j) is equal to zero if the specified qubits do not interact during the given time slice; otherwise, it is equal to one. 2^-s is the lookahead component (considers time). A duplicate of this list is generated for the purpose of relocating qubits within traps at the end of the algorithm (T_copy).












 An illustrative example of how STA allocates qubits on a 2 trap device is shown in Figure <ref> and Figure <ref>. Given a quantum circuit (described as a list of gates), the structures R and T are derived (Figure <ref>(a)). Observe that qubit 2, in R list, demonstrates the highest ratio (4/5) by computing the different qubit interactions (4) divided by the number of qubits (5).In list  T, the pair (0,2) is positioned at the beginning of the list, as those qubits are involved in the first gate to be executed and exhibit the highest number of two-qubit gate operations between them.

Once the two data structures are calculated, the algorithm will run until there are no elements remaining in the qubit ratio list (Algorithm <ref> line 5-7). It operates recursively (Algorithm <ref>), prioritizing the placement of qubits with higher interaction ratios. Therefore, the first input will be the qubit with the highest ratio (first element in R); qubit 2 in the provided example. Followed by subsequent qubits with progressively lower ratios.

 Each time the function map_qubit is called, a second qubit forming a pair with the input qubit is identified from the T structure (Algorithm <ref> line 3, q_2), being qubit 0 the corresponding pair of qubit 2 following the example. Subsequently, it is verified whether this identified qubit has no interactions closer in time or a higher number of operations with another qubit; if it does, the algorithm is recursively called with the qubit that interacts nearest in temporal proximity (algorithm <ref> line 7-9). Qubit 0 does not appear earlier in T list, suggesting that it has not previously engaged in any interaction and/or has participated in a significant number of operations with qubit 2, being both scenarios in this case.

Following the confirmation that the pair of qubits is the closest in time or has the higher number of operations compared to other potential candidates regarding qubit interaction, both qubits are placed within the same trap if none of them have been previously allocated and removed from the lists. Suppose only one of the qubits has already been placed. In that case, the other one is allocated by considering the shortest path between them using the Dijkstra algorithm and afterward removed from both lists (algorithm <ref> line 10-14). In the provided example, no qubits have been placed yet. Thus, both qubits are assigned to the same trap (Figure <ref> (b)), and their respective entries are removed from the lists. These steps will be repeated until all qubits are allocated. 



Continuing with the example, the subsequent input for the function is qubit 4 in the R list. Similar to previous steps, the pair for this qubit is searched in T, as performed before, resulting in qubit 1. Afterward, it is verified whether qubit 1 has interacted previously with another qubit, resulting in qubit 1 having a higher value with another qubit (i.e. (1,3)), so the function is called recursively with qubit 1. 
Following this recursive process, it is determined that qubit 1 interacts with qubit 3, which either has no previous interactions or has participated in a significantly limited number of operations with other qubits. Consequently, both qubits, 3 and 1, are placed in the same trap. Since the first trap does not have enough space for both qubits (considering the excess capacity), they are instead placed in the second trap and removed from the lists.
Upon completing the recursion, the focus of the function shifts to qubit 4. With the remaining qubits already placed, qubit 4 is allocated to the second trap, thereby concluding the recursion process. The main algorithm proceeds by invoking the function responsible for reordering the qubits within traps.

Once all qubits are placed (i.e., when there are no more qubits in the "R" structure), they will be reordered within each trap (algorithm <ref> line 8, algorithm <ref>). This arrangement is accomplished by iterating through the copy of the interaction qubit pairs structure created earlier in a reverse order (algorithm <ref> line 4, T_copy). Each qubit pair ordered from the lowest value to the highest one will be checked whether they are placed within the same trap (algorithm <ref> lines 2-4). If that is not the case, qubits will be relocated to the extremes of the trap closest to the trap the other qubit is (algorithm <ref> lines 5-6). After the relocation process for all qubits is completed, the STA algorithm will be finalized.



As illustrated in Figure <ref> (b), by iterating through T structure, the first pair of qubits is selected (2,3), and it is checked whether they are situated within the same trap. Since this is not the case, both qubits are re-positioned at the end of the trap closer to the other qubit, resulting in the shortest path between them.
Subsequently, the next pair of qubits, (4,2), is considered. Qubit 2 does not require relocation, whereas qubit 4 is now positioned at the end of the trap as done previously with qubit 3. Notably, qubit 3 becomes adjacent to qubit 4, meaning that qubits with closer execution times or more operations are prioritized, followed by subsequent ones. The process continues until all qubits are relocated, if necessary.


The STA algorithm, though presented initially for linear and ring topologies, can be applied to a variety of structures such as those that have X/T junctions. Furthermore, while it was initially designed for ion trap technology, its applicability is not limited solely to these quantum processors. The algorithm can also be employed in other modular quantum computing architectures, such as multicore (multinode) superconducting quantum devices <cit.>, where the initial placement provided by STA serves as an initial strategy for allocating qubits across the different cores.






 






























§ SIMULATION FRAMEWORK AND BENCHMARKS

The experiments of this work were conducted using the QCCDSim framework provided by  <cit.>, a simulator for ion-trapped QCCD architectures, offering comprehensive features, including information about heating considerations, and execution times and fidelities, among others. This simulator also allows the implementation of custom mapping techniques, as the initial qubit placement proposed in this paper. Moreover, it gives the flexibility to utilize various trapped-ion QCCD topologies and performance metrics.

To perform experiments with the STA algorithm, an already implemented scheduling and qubit routing method, presented in  <cit.>, has been used. The routing is based on the number of gates a qubit has in different traps. This analysis aims to identify the most efficient movement, determining which ion should be transferred to another trap. 





The scheduler works in an earliest-ready gate-first approach as previously done in <cit.>, where the order is determined by the gate dependency graph. 

A set of diverse algorithms has been chosen to benchmark STA for different scalable QCCD architectures. It includes the Cuccaro Adder (CA), the Draper Adder (DA), the Quantum Approximate Optimization Algorithm (QAOA), the Quantum Fourier Transform (QFT), Quantum Volume (QV), and Randomly generated Circuits (RC) in which experiments were conducted 20 times. The selection of these benchmarks was driven by their structural diversity, as shown in Table <ref>. Algorithms like the CA, DA, QAOA, and QFT are more structured. The CA is characterized as the lightest due to its notably fewer two-qubit gates compared to other algorithms, conversely to the QFT and QAOA algorithms, which also exhibit a high average of two-qubit gates per slice. Additionally, their degree of parallelization varies; QAOA and QFT exhibit the highest level of gate parallelization (average number of gates per slice, last column in Table I), whereas CA is a more sequential algorithm.

Conversely, unstructured algorithms, such as RC and QV, incorporate randomness. Among these benchmarks, Quantum Volume stands out as the heaviest, featuring a significant number of two-qubit gates, slices, and the highest gate parallelism.

The execution of the benchmarks was simulated on two distinct topologies: a 1D linear array and a ring structure (Figure <ref>). The selection of a linear topology was based on previous works <cit.>; that is, to be able to accurately compare the STA approach with prior initial qubit placement strategies. The ring structure was chosen as a natural extension of the linear topology, offering enhanced trap connectivity. Moreover, it actually represents a configuration utilized in the Quantinuum H2 QCCD system <cit.>, currently the most promising ion-trap technology.




§ EVALUATION AND SCALABILITY ANALYSIS

The Spatio-Temporal Aware Qubit Allocation Algorithm (STA) will be first evaluated against the Greedy policy used in <cit.>  as well as a Random approach, wherein qubits are randomly assigned to the traps, in a linear architecture. Moreover, comparisons will be conducted between the 1D-linear array and a ring structure using the STA algorithm. In addition, an architectural scalability analysis will be conducted, in which we will also explore different excess capacities. For all experiments, the QCCD simulator <cit.> will be utilized, alongside the qubit routing approach proposed in <cit.>. The principal used metric is the execution time, which includes the duration required to execute all essential operations such as gates, shuttling, merge, and split. Additionally, specific experiments provide the number of inserted SWAPs and performed shuttles. For further details on how these metrics are computed, refer to <cit.>.



 §.§ Qubit allocation strategies: Random vs Greedy vs STA 





The STA algorithm is compared with the Greedy policy outlined in a prior work <cit.>, as well as a random approach in which qubits are randomly distributed among various traps. It should be noted that this comparison does not include the qubit allocation method proposed by Upadhyay et al.  <cit.> as its performance is similar to the Greedy approach. 

Building upon prior works <cit.>, an equivalent configuration of qubits and QCCD topology has been selected for comparison. Specifically, all benchmarks will be executed utilizing 64 logical qubits (program qubits) within a 1D-linear array comprising 6 traps, each containing 17 ions. Furthermore, all traps will be equipped with an initial excess capacity of two (i.e. at the beginning of the execution, 15 qubits will be located per trap).

Table <ref> shows the outcomes of such comparisons in terms of the number of shuttles required, SWAP operations necessary for ion relocation within the trap, and the total execution time of the circuit derived from the duration of all operations, including shuttling, gates, split and merge. Note that the execution time reveals how fidelity might vary across different approaches, with longer execution times potentially resulting in lower circuit fidelity due to increased operational durations. Furthermore, it offers insights into the potential heat accumulation within the device, as prolonged execution times correlate with heightened heat generation. 

STA improves the overall execution time for all considered circuits compared to the random (Δ R) and the greedy (Δ G) strategies. A notable enhancement is observed compared to random qubit initial placement, reaching up to 76.22% for the Quantum Fourier Transform (QFT) algorithm. When compared to the Greedy policy, STA exhibits execution time improvements of up to 50%, as can be seen in the Cuccaro Adder (CA). However, these are less pronounced for the random algorithms.

An important observation is that qubit allocation plays a critical role in light algorithms (few number of two-qubit gates), as the positioning of qubits significantly influences potential movements within the device. Conversely, in heavy algorithms (high number of two-qubit gates), the impact of qubit allocation diminishes. This trend is reflected in our results. For instance, CA exhibits the most significant enhancement as it is the lightest algorithm. Contrarily, heavier algorithms, such as Quantum Volume (QV), possess a notably higher number of operations and a higher average of two-qubit gates per slice, thus diminishing the influence of initial qubit placement.
This logic suggests that reducing the number of qubits or gates of the algorithm could further amplify the benefits of the STA approach. Conversely, as the number of qubits and gates increases, the impact of such improvements becomes less pronounced, a phenomenon applicable to any qubit allocation method. For subsequent experiments, only the STA approach will be employed, given its demonstrated effectiveness.




 §.§ Topological Performance Analysis: Linear vs Ring


In this section,  a comparison between two different topologies has been performed: a 1D-linear array and a ring. As in the previous experiment, both configurations consist of 6 traps and 17 ions per trap, with an initial excess capacity of two for all traps within the devices.






Table <ref> presents the metrics obtained for the ring topology. In addition, Figure <ref> shows the overall execution time for both the linear and ring devices. It can be observed that for most benchmarks, except those with a random component, the linear device generally outperforms the ring counterpart. This observation can be attributed to the fact that the routing scheme proposed in <cit.> has been tailored for linear devices, potentially restricting possible optimizations for ring topologies. This explanation is further supported by subsequent experiments when conducting scalability analyses, where ring devices typically outperform linear ones.




 §.§ Scalability analysis




A scalability analysis was conducted for both topologies, linear and ring, employing two distinct methodologies: strong and weak scaling. This approach enabled the examination of how the various topologies behave as both the number of qubits per trap and the number of traps vary.

Due to the similarities between QAOA and QFT, only results for QFT will be presented. Additionally, given that QV and random circuits incorporate random components, solely the results for QV will be shown for simplicity. For all experiments, an initial excess capacity of 2 was established, allowing for the allocation of additional ions per trap in the subsequent mapping process. The metric utilized across all experiments is the total execution time, which considers all operations such as gates, splits, merges, and shuttles.

The initial experiment conducted is a strong scaling, in which the number of physical qubits is increased by gradually adding more traps, each of them having a fixed number of physical qubits. Each trap contains 17 ions, and the number of traps is raised from 2 to 14, reaching a total of 238 physical qubits within the device. However, considering the excess capacity set at the beginning of the execution, benchmarks are chosen to use 210 ions (logical qubits). For QV, being the heaviest quantum algorithm, strong scaling was performed until it reached 140 physical qubits due to the very long simulation time.


Figure <ref> (top row) shows the results of the strong scaling experiment. Across all benchmarks, a consistent trend is observed: the Cuccaro Adder (CA) exhibits linear behavior, while the remaining benchmarks display exponential growth. As the number of qubits and traps increases, the execution time correspondingly rises, as anticipated, due to the increased number of operations required to execute the algorithms. Additionally, both topologies demonstrate similar trends, with the ring topology slightly outperforming the linear topology for DA, whereas the linear topology exhibits marginally better performance for CA and QV.

The subsequent experiment is a weak scaling, in which the total number of physical qubits in the device is kept constant at 180 ions; meanwhile, the number of traps increases, redistributing the physical qubits among the traps accordingly. The benchmarks executed utilize only 128 qubits due to the initial excess capacity allocated for each trap -2 ions. The experiment begins with two traps of 90 ions each. Subsequently, the number of traps is incremented by one each time, while the total number of ions in the device remains unchanged (i.e. the number of traps is increased while decreasing the number of ions per trap). This results in the distribution of ions across all traps until reaching 26 traps, with 6 ions per trap, always keeping 2 ions for the excess capacity, except for the QV, where the experiment was performed until it reached 17 traps due to the long simulation time.

The results are depicted in Figure <ref> (bottom row). It can be seen that, in half of the cases, the ring topology outperforms the linear one (DA and QFT). For CA, the execution times are similar. This can be attributed most probably to the easy execution of the algorithm as it does not require a lot of complexity in the mapping procedure, thereby resulting in negligible differences between the two topologies.

In the DA and QFT cases, the execution time initially decreases before subsequently increasing again. This phenomenon arises from increased parallelism; specifically, a higher number of traps allows for more concurrent executions. It is important to recall that operations within a single trap must be executed sequentially, thus a higher number of traps leads to enhanced parallelism. However, with more traps, the number of qubits per trap decreases, resulting in a higher frequency of shuttle and SWAP operations, consequently increasing the total execution time. In these cases, the optimal QCCD architecture comprises approximately 14 to 16 traps and between 11 to 13 ions per trap, considering an initial excess capacity of two.

 In the QV results, the execution time decreases when the number of traps increases; this behavior can be attributed to its high level of gate parallelization, as shown in Table <ref>. With more traps, more operations can be concurrently executed, leading to a notable reduction in the overall execution time, as occurs in DA and QFT cases. 

It is crucial to note that despite the variations in results, a consistent trend is observed for all benchmarks. This consistency suggests that the device design can be refined to achieve an overall optimization across a wide range of quantum algorithms. Additionally, it should be considered that these results may be influenced by the chosen initial excess capacity and mapping approaches, and there is potential to optimize the outcomes accordingly.



 §.§ Evaluating the impact of the excess capacity


The excess capacity was analyzed to determine the optimal number of free spaces for each trap at the start of executions for both the 1D-linear array and ring topologies. Similar to previous experiments, the Cuccaro Adder (CA), Draper Adder (DA), Quantum Volume (QV), and Quantum Fourier Transform (QFT) were utilized. In all cases, benchmarks were executed using 64 logical qubits, with the total execution time serving as a metric.

Two distinct approaches were implemented for this experiment. In the first one, the excess capacity was augmented by adding additional ions, from 1 to 10, to a fixed number of traps. There are 5 traps, initially each of them containing 14 ions. The experiment concluded with 24 ions per trap, with 10 ions allocated as excess capacity at the start of the execution. In the second approach, the total number of qubits per trap decreased as the excess capacity increased. Specifically, with a total of 14 ions per trap, the excess capacity was gradually increased to 10, resulting in 4 qubits per trap. Consequently, this approach utilized more traps compared to the first approach, totaling 16 traps. However, at the beginning, only 5 traps were utilized.






The results for both experiments are shown in Figure <ref>. In the case of CA, for both topologies, when the number of ions is fixed in a trap and the excess capacity varies, a trend is observed where the results improve when an even number of excess capacities is allocated. This is caused by the STA algorithm and the CA structure: a qubit from the ratio list is placed when one spot is left in a trap, and the excess capacity is even; in contrast, a pair from the time interaction list is placed when a trap is becoming full, and the excess capacity is odd, leading to the observed behavior. As the excess capacity increases alongside the ions per trap, a linear trend is observed. In this case, the number of shuttles and SWAPs is equal to zero, as qubits do not need to move traps, as shown in Figure <ref> (only the results for shuttles are shown due to similar trends). However, the time required to execute these operations increases due to the higher number of ions per trap. Specifically, a greater execution time is needed to perform gates as the number of qubits per trap increases.

For DA, the ring topology with a fixed number of ions per trap emerges as the most optimal approach, whereas the linear topology with a fixed number of ions per trap ranks as the least favorable. In general, a linear topology with excess capacity equal 1 and a ring topology with an excess capacity of 2 and 14 qubits per trap appears to be best performing configuration. Conversely, although the number of SWAPs and shuttles decreases as the number of ions increases with respect to the excess capacity, similar to the CA scenario, the execution time increases due to the greater number of ions within a trap, rendering it a less optimal approach. An intriguing observation is that a total fixed number of ions per trap for both topologies yields opposite results. This can be attributed to the superior connectivity of the ring topology compared to the linear one, resulting in fewer shuttles and SWAPs movements without increasing the execution time.

A similar trend to the results obtained for DA is observed in QFT. The ring topology with a fixed number of ions per trap emerges as the most efficient approach, while the linear topology with an increasing number of ions per trap proves to be less optimal. However, a notable difference between QFT and DA is noted in the behavior regarding excess capacity in the ring topology. Initially, increasing the excess capacity leads to improved efficiency until reaching a maximum at an excess capacity of 7, after which it diminishes. This phenomenon can be attributed to QFT's higher degree of gate parallelization compared to DA, as shown in Table <ref>. As excess capacity increases, qubits are more evenly distributed across traps, enabling more parallel executions. However, beyond a certain point, execution time starts to increase. This is due either to the high number of ions in a single trap (limited parallelism and increase in gate execution time) or to too few qubits placed in each trap due to the high excess capacity (increasing the number of SWAPs and shuttles). It is important to highlight that the routing process does not prioritize parallelization when relocating ions. Consequently, distributing ions evenly across traps naturally enhances parallelization, a factor not explicitly considered in the routing algorithm.
Lastly, in the QV benchmark, a different trend is observed compared to all other benchmarks: the most optimal approach is to increase the excess capacity with the number of ions per trap. Through the routing process, ions are stored within the fewest number of traps possible, thereby reducing the total number of shuttling and SWAP operations. While this approach does not yield favorable results for other benchmarks, it proves effective for QV by significantly reducing the overall number of operations. Note that in this case is more efficient to keep all qubits in fewer traps as QV shows a high degree of qubit interactions. Through experimentation with excess capacity, it becomes evident that the optimal initial free spaces depend on the quantum algorithm to be executed. Additionally, it is observed that, overall, the ring topology tends to show better results compared to the linear one. However, careful consideration must be given when adding extra ions per trap, as this can significantly diminish the quality of executions, even if those ions are not utilized during the execution of the quantum algorithm on the device.


§ CONCLUSION AND FUTURE WORK

This paper introduced a new qubit allocation strategy for traped-ion QCCD architectures, the "Spatio-Temporal Aware Qubit Allocation" (STA) algorithm, which demonstrates an enhancement of up to 50% in execution time compared to previous methods. Additionally, we conducted a scalability analysis (weak and strong scaling) for two distinct topologies: a 1D-linear array and a ring topology. Results indicate that as the number of qubits and traps increases, so does the execution time. However, ion distribution across an optimal number of traps enhanced parallelism, thereby reducing execution time. Our findings suggest that current ion trap qubit routing techniques could be further optimized for ring topologies. Furthermore, the consistent trends observed across all algorithms in the scalability analyses indicate that architectural designs and mapping procedures can be optimized to enhance the overall performance across a diverse range of quantum algorithms. Lastly, the initial excess capacity was explored, showing a very different behavior for each quantum algorithm.

Further research should be done to improve the qubit routing procedure in modular QCCD devices for both topologies, linear and ring, with a particular focus on optimizing the excess capacity.  In addition, there is potential to refine mapping techniques specifically tailored for ring topologies, capitalizing on their superior connectivity relative to 1D linear arrays.



§ ACKNOWLEDGMENTS

The authors acknowledge financial support from the European Union’s Horizon Europe research and innovation program through the project Quantum Internet Alliance under grant agreement No. 101102140. CGA also acknowledges support from the Spanish Ministry of Science, Innovation and Universities through the Beatriz Galindo program 2020 (BG20-00023) and the European ERDF under grant PID2021-123627OB-C51.



















IEEEtran














 Paper Title: Redefining Lexicographical Ordering: Optimizing Pauli String Decompositions for Quantum Compiling.




Rotational influence on fermions within negative curvature wormholes
     Abdullah Guvendi^ 1[ E-mail: abdullah.guvendi@erzurum.edu.tr  ] ,  Semra Gurtas Dogan^ 2[ E-mail: semragurtasdogan@hakkari.edu.tr (Corr. Auth.) ] ,  R. L. L. Vitória^ 3[ E-mail: ricardo-luis91@hotmail.com ]   

	^ 1 Department of Basic Sciences, Erzurum Technical University, 25050, Erzurum, Türkiye

	^ 2 Department of Medical Imaging Techniques, Hakkari University, 30000, Hakkari, Türkiye

	^ 3 Faculdade de Física, Universidade Federal do Pará, Av. Augusto Corrêa, Guamá, Belém, PA 66075-110, Brazil
    
============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================


















§ ABSTRACT 
In quantum computing, the efficient optimization of Pauli string decompositions is a crucial aspect for the compilation of quantum circuits for many applications, such as chemistry simulations and quantum machine learning. In this paper, we propose a novel algorithm for the synthesis of trotterized time-evolution operators that results in circuits with significantly fewer gates than previous solutions. Our synthesis procedure takes the qubit connectivity of a target quantum computer into account. As a result, the generated quantum circuit does not require routing, and no additional CNOT gates are needed to run the resulting circuit on a target device. We compare our algorithm against Paulihedral and TKET, and show a significant improvement for randomized circuits and different molecular ansatzes. We also investigate the Trotter error introduced by our ordering of the terms in the Hamiltonian versus default ordering and the ordering from the baseline methods and conclude that our method on average does not increase the Trotter error.



Circuit synthesis, Pauli string, Lexicographical ordering, Phase gadgets, Trotter decomposition, compilation, optimization





§ INTRODUCTION

With the increasing size and quality of current quantum computers comes a need for efficient compilation algorithms that can process more complicated circuits in polynomial time without generating more operations than necessary. In this paper, we focus on the synthesis of Hamiltonian time evolution operators to quantum circuits constrained under the qubit connectivity of a target quantum computer. These operators can be found in the ubiquitous Ising model as well as algorithms such as the unitary-coupled cluster ansatz (UCC) <cit.> or the Harrow–Hassidim–Lloyd (HHL) algorithm <cit.>, among others. Similarly, sequences of multiple controlled rotations can also be represented as a sequence of Pauli string operations which are common in e.g. Amplitude Encoding <cit.>, and Linear Combinations of Unitaries <cit.>, among others.
Moreover, they form a universal representation, so any quantum algorithm can be rewritten into a product formula that contains only time evolution operators. 
While time evolution operators in algorithms can be easily determined for classical simulation, transpiling the same operations to gates on a quantum device is not obvious. 

In this context, our proposed method synthesizes a quantum circuit from the first-order Suzuki-Trotter decomposition of the target Hamiltonian's time evolution operator (i.e. a product of exponentiated Pauli strings). Additionally, the algorithm takes qubit connectivity into account during synthesis. As a result, the synthesized circuit is already routed and no additional SWAP gates are required to run the circuit on the target device.  
Along the way, we extend a previous method of ours <cit.> to more effectively synthesize Clifford tableaus by allowing a final permutation either implemented on the quantum circuit or by a permutation of swaps.

Our experiments show that the introduced algorithm significantly improves the number of CNOTs, the circuit depth, and the CNOT depth with respect to Paulihedral <cit.> and TKET <cit.>, both in the constrained and unconstrained case. In the case of targeting a restricted architecture, our proposed method improves these metrics up to an order of magnitude for UCCSD ansatzes.



§ PRELIMINARIES AND RELATED WORK



 §.§ Pauli Gadgets and Pauli Polynomials


In this work, we consider Trotterized time evolution operators written in the form 
    U = e^-i ∑_n (θ_n/2⊗_m P_m )≈(∏_n e^-i θ_n/2r⊗_m P_m)^r,

where P_m is a single-qubit Pauli operator, and r the number of repetitions of the circuit.  The tensor of Pauli gates can also be written as a sequence of the letters representing each Pauli gate i.e. representing X ⊗ I ⊗ X ⊗ Y ⊗ Z as XIXYZ. This sequence is also called a Pauli string because it is a word s of arbitrary length over the alphabet {I, X, Y, Z}, or the four Pauli letters. 

Following the conventions in <cit.> and <cit.>, we define a Pauli gadget ϕ with a Pauli string s and angle θ:

    ϕ(θ, s) = exp(-iθ/2 s )

Additionally, we also utilize their convenient graphical representation of the Pauli gadget in the ZX-calculus:

    exp(-iθ/2IXYZ) = figures/PauliExpDef


 In this representation, each non-identity Pauli letter in s maps to a leg on the corresponding qubit or wire, each attached to a central "spider" containing the angle information, as shown in <ref>.
 Then, given a Hamiltonian described by a sum of weighted Pauli strings, we can approximate the corresponding time evolution operator in terms of Pauli gadgets. We refer to such a representation as a Pauli polynomial. For example, given one such arbitrary Hamiltonian H = α IXYX + β IZZI + α IXYY + β IYYI, where α, β∈ℝ, we could approximate the time evolution operator over a time step t as follows:

    figures/PauliExpTrotter

This notation is equivalent to that used in <cit.> with the difference being that our notation can be considered a dialect of the ZX-calculus <cit.> and, therefore, all equalities presented in this work can be considered mathematical equations. 
Additionally, this representation includes well-defined rules for propagating Clifford gates through Pauli gadgets <cit.> that are key for the proposed methods.
The graphical representations of the Clifford operators relevant to this work are then as follows:

    V = figures/VDef  
    V^†   = figures/VDDef  
    H = figures/HDef  
    CNOT = figures/CNOTDef
    
    S    = figures/SDef   S^† = figures/SDagDef ,

where V := √(X).
Then, propagating a CNOT through gadgets with ZZ-legs or ZY-legs is represented as follows:

    figures/PauliExpCXZZra = figures/PauliExpCXZZrb


This equality is used to remove legs from Pauli gadgets with a set formulation in the proposed method. The remaining propagation rules used in this work to manipulate legs are:


    figures/PauliPropva = figures/PauliPropvb


    figures/PauliPropha = figures/PauliProphb


    figures/PauliPropsa = figures/PauliPropsb








    


 §.§ Steiner-Gray Synthesis

Does this belong in related work?

In this work, we extend a specific compilation algorithm introduced by <cit.>,  which we will refer to as steiner-gray synthesis. The algorithm synthesized Z polynomials. The method relies on the fact that Z-phase gadgets commute, allowing for an arbitrary ordering of the gadgets without loss of generality. Following <cit.>, we can find a process to make this algorithm architecture-aware, grouping it into two main processes: The base recursion and the one recursion. With this, the overall goal is to place as few CNOTs as possible in an architecture-aware fashion.


  §.§.§ Base Recursion

During the base recursion, the overall goal is to select a row q_0 of the Z Polynomial to maximize the number of gadgets, with a leg present or non-present. The key concept hereby is dividing the Z Polynomial into as many regions as possible. 


  §.§.§ One Recursion

The goal of the one recursion step is to remove as many legs from row q_0 as possible. Note, therefore <ref>; we observe that if our Pauli-gadget contains only Z-legs, we will either introduce or remove one leg among propagation of a CNOT. 
At last, the particular case remains, where we encounter no legs at all in the selected row q_1; in this case, we can swap the two rows by applying a combination of CNOTs with control and target (q_0, q_1) and (q_1, q_0).



 §.§ A general compilation scheme for the unitary coupled cluster ansatz


<cit.> provide a general compilation scheme for synthesizing the unitary coupled cluster ansatz, which describes a totterized time evolution operator. The core assumption in this algorithm is that any gadget can be swapped if the angles are small. This allows one to neglect the commutation rules required for swapping the gadgets, consequently allowing a regrouping of commuting regions of Pauli Polynomials. Such regions can then be diagonalized in a second step, meaning that it is allowed to rewrite them as a series of Cliffords, combined with a pure Z Polynomial followed by the conjugated series of Clifford gates.
The Z polynomial can then be synthesized using the gray synth algorithm <cit.>. Clifford regions require no further synthesis step.
In combination, the steps of the algorithm can be summarized as follows:

    
  * Regrouping of the Pauli polynomial to commuting regions
    
  * Diagonalization of the commuting regions
    
  * Synthesis of the Z polynomial regions










 §.§ Trotter error

Once a Pauli polynomial has been specified, an ordering of the Pauli gadgets needs to be determined. 
Normally, Pauli gadgets can only be reordered in a circuit if they commute, but since the gadgets emerge from a sum in the exponent (see <ref>), we can reorder the Pauli strings in the sum and, likewise, the corresponding Pauli gadgets, arbitrarily.

However, the transformation in <ref> is approximate and results in a small Trotter error.
Although arbitrarily reordering the individual Pauli gadgets does not worsen the order of the approximation error in trotterized decompositions, investigations by <cit.> indicate that the variance in Trotter error when reordering non-commuting terms is not trivial if interaction terms are large.

Investigations by <cit.>, <cit.>, <cit.>, and <cit.> also indicate that the ordering of terms also impacts the number of entangling gates generated during synthesis, each proposing ordering strategies that reduce both synthesized gate count and theoretical upper bounds for the resultant Trotter error. Additionally, lexicographical ordering, in which the Pauli strings of each component are treated as words and sorted using a strict letter ordering, is shown to be a good heuristic for obtaining gate cancellations between synthesized Pauli gadgets <cit.>, <cit.>.


In this work, similar to the assumptions made in <cit.> in their Paulihedral implementation, we ignore the Trotter error for arbitrarily ordering Pauli gadgets, focusing mainly on reducing entangling gate count. This choice is grounded in the assumption that entangling gates introduce greater error than that introduced by poor ordering. Furthermore, the effects of poor ordering can be mitigated by limiting time step size and utilizing higher order decomposition methods <cit.>. 


Nevertheless, the proposed algorithm can also be used to synthesize the Pauli gadgets in groups. Although, we do not use this feature in our experiments, nor do we provide a grouping heuristic.





 §.§ Related algorithms

Once a grouping of Pauli gadgets is decided, various algorithms exist to synthesize these sets of gadgets. We use two of these as a baseline to the proposed algorithm.

First, we have <cit.>, which assumes a user-specified grouping and uses the coupling map to group the Pauli gadgets based on cumulative qubit distance and attempts to synthesize each layer such that the gates between layers are maximally canceled.

Secondly, we compare against <cit.> where Pauli gadgets are optionally re-grouped using graph coloring. Afterwards, each group is synthesized by diagonalizing the Pauli gadgets and then synthesizing them using GraySynth <cit.>. The idea of <cit.> is to synthesize the gadgets in order of the Gray code <cit.>, such that each subsequent gadget only requires a single CNOT to be synthesized. 


In this paper, we propose an alternative method for ordering and synthesizing a sequence of Pauli gadgets that is inspired by GraySynth <cit.> and its architecture-aware version <cit.>. 







 §.§ Clifford Tableaus

A final component required by our proposed algorithm is Clifford tableau synthesis. The Clifford tableau is a compact representation introduced by <cit.> that specifies a Clifford circuit or operator. More specifically, an n qubit Clifford circuit can be designated by a 2n × 2n matrix and a vector of size 2n, all in GF(2) and the tableau of an empty circuit is an identity matrix.
Appending and prepending Clifford gates to a Clifford operator corresponds to row and column operations on its Clifford tableau. The prepending operations are visualized in <ref>, and the appending operations act transposed with respect to prepending: they act on the respective columns, rather than the rows. 

Given a Clifford tableau, one can synthesize the equivalent quantum circuit as follows, starting with an empty circuit:

    
  * Generate a Clifford gate and its adjoint.
    
  * Append the gate to the circuit and prepend it to the tableau.
    
  * Continue until the Clifford tableau is an identity matrix.

In this paper, we adapted one of our previous algorithms <cit.>, which uses the structure of the RowCol algorithm <cit.> for CNOT synthesis and applies it to the Clifford tableau synthesis from <cit.>, <cit.>.



    






§ METHODS


The proposed algorithm combines the ideas from the related algorithms into a single architecture-aware strategy for synthesizing these trotterized Hamiltonians. 
The algorithm expects a set of Pauli gadgets that can be decomposed in arbitrary order, and returns a quantum circuit implementing those Pauli gadgets.
The structure of this algorithm is an adaptation of GraySynth <cit.> and its architecture-aware variant <cit.>. The outline of the algorithm is as follows, with the functions  and  defined in <ref>:

    
  *  Choose a qubit q_pivot to remove from the problem, according to heuristic h_pivot.
    
  *  Synthesize any Pauli gadgets that are connected to only a single qubit:  them and place the R_Z(α) gate. Then, remove these gadgets from the problem.
    
  *  Split the Pauli gadgets into groups based on the letters on the legs on q_pivot.
    
  *  Take the Pauli gadgets that are not connected to q_pivot, i.e. with an I on the leg, and go to step <ref> with q_pivot removed from the problem.
    
  *  Choose another qubit q_neighbor that is highly entangled with the chosen qubit according to the remaining Pauli gadgets, according to heuristic h_n.
    
  *  If there are Pauli gadgets not connected to q_neighbor,  as many Pauli gadgets as possible with respect to the first qubit, and place two CNOT gates between the two qubits to  the first qubit.
    
  *  Otherwise,  as many remaining Pauli gadgets as possible by placing at most two single qubit Clifford gates to  some Pauli gadgets (i.e. convert legs to {Y, Z}, see <ref> for specifics) and a CNOT gate between the two chosen qubits to disconnect the gadgets (see <cit.> for an overview).
    
  *  Repeat step <ref>: split the remaining Pauli gadgets.
    
  *  Repeat step <ref>: go to step <ref> with disconnected Pauli gadgets and q_pivot removed from the problem.
    
  *  Go to step <ref> with the remaining remaining Pauli gadgets.

The adjoint of all synthesized Clifford gates are aggregated into a Clifford tableau and later synthesized (e.g. using <cit.>) and all unfinished Pauli gadgets are updated according to the Clifford gate commutation rules <cit.> whenever a Clifford gate is synthesized.
Additionally, the algorithm can be adjusted to synthesize the Pauli gadgets in the given order by iteratively using this procedure on sets of mutually commuting Pauli gadgets. In which case, the Clifford tableau collects the Clifford gates from all iterations and the Clifford gates are propagated through all subsequent Pauli gadgets, including those in future iterations.
Moreover, this algorithm can be trivially made architecture-aware by choosing a q_pivot in step <ref> that does not disconnect the connectivity graph when removed, and restricting  q_neighbor  in step <ref> to neighbors of the chosen qubit. 

Intuitively, this algorithm tries to greedily find an optimal lexicographical order for the letters, i.e. colors, on each Pauli gadget, per qubit. Because of the greedy nature, we assume that I always comes first in the alphabet, such that we never unnecessarily introduce entanglement with a new qubit. The exception is when this is needed for routing purposes in the architecture-aware case. Moreover, as explained in the remainder of this section, when placing the Clifford gates, we essentially partially decompose the Pauli gadgets and assume that the gates on the other side of the Pauli gadget will cancel out. This is done by propagating the adjoint of the added Clifford gates through all remaining Pauli gadgets and gathering them in a Clifford tableau to be synthesized as a final optimization step. Thus, all Clifford gates are generated as late as possible, on an as-needed basis. As a result, the algorithm does not need to guess a good gadget ordering a priori. Instead, the ordering emerges from the structure of the synthesis procedure, while grouping gadgets with the same letters on the same qubits together. A simple example that steps through the synthesis process is shown in <ref>.















 §.§ Gate placement


To be precise, we  gadgets with respect to a single qubit as follows: if the gadget has no leg or a green leg on the qubit (i.e. I, or Z), we are done, otherwise, we place a single qubit Clifford gate and its adjoint according to the following transformations:[Here, we use the dashed lines to separate the diagram into different regions: the left-most region contains the gates added to the circuit, the middle region is the updated Pauli gadgets, and the right-most region contains gates that are stored in a Clifford tableau for re-synthesis.]

    Xfigures/DiagonalizeXa = figures/DiagonalizeXb = figures/DiagonalizeXc



    Yfigures/DiagonalizeYa = figures/DiagonalizeYb = figures/DiagonalizeYc


The introduced gate is propagated through all following Pauli gadgets, and gathered in the Clifford tableau, following the rules in <ref>.
Note that we choose the rule that s as many Pauli gadgets as possible. Typically, not all affected legs are mapped to a green leg and their updated type must be considered.

After diagonalizing, we  as many Pauli gadgets as possible by placing up to a single qubit Clifford gate and two CNOTs. By construction, the first qubit will always be diagonalized.
The simplest case is when the second qubit is already disconnected. Then, we can swap the two qubits. However, because we know the type of the Pauli gadget, we can remove one of the CNOTs since it commutes with the Pauli gadget and cancels out. Thus, we have:^<ref>

    Ifigures/Swapa = figures/Swapb


If the second qubit is connected, we can match the legs of the Pauli gadgets under consideration as one of three cases, each corresponding to a different Clifford operation to be applied:^<ref>

    Z or Yfigures/DisentangleYZa = figures/DisentangleYZb



    Z or Xfigures/DisentangleXZa = figures/DisentangleXZb



    X or Yfigures/DisentangleXYa = figures/DisentangleXYb


Each introduced rule only removes entanglement for two types of legs in the second wire. However, the second wire can contain X, Y or Z legs. Thus, we choose the option where the most Pauli gadgets are matched and place the corresponding Clifford gates. Then, we propagate these gates through all remaining Pauli gadgets and prepend them to the Clifford tableau.
[figure]font=small, skip=0pt



    







    figures/DisentangleYZa = figures/DisentangleYZb


We can convert any two Pauli rotations to Y and Z rotations before remove operations from q_pivot by propagating a single Clifford gate as follows:


    figures/DisentangleXZa = figures/DisentangleXZb



    figures/DisentangleXYa = figures/DisentangleXYb


Finally, if P_0 = P_1 = I,  moves the legs on q_pivot to q_neighbor using four CNOTs.


    figures/Swapa = figures/Swapb


Up to this point, the operations are similar to synthesis steps in prior Pauli polynomial synthesis algorithms. However, unlike in previous work, where the introduced gates sandwich the remainder of the gadget, the proposed algorithm propagates G through the remainder of the unsynthesized Pauli polynomial and prepends it to a Clifford tableau. Thus, each operation returns a series of G^† operations appended to the synthesized circuit and a series of G operations that are collected in a Clifford tableau, which is synthesized only as the final step.[For brevity, only G^† is returned in the psuedocode as G is easily recovered as the adjoint.]




 §.§ Heuristics

The proposed algorithm relies on choosing a good qubit to perform the splitting heuristic on. For our experiments, we have used the following heuristic and cost function:

    c_q(s_q, g)    = k · P_I(s_q[g]) 
       = +  P_max(s_q[g])  -  P_min(s_q[g]), 
    
        h(g, o)    = arg max_q ∈o(c_q(s_q, g)),  
    
        h_pivot(g, G)    = h(g, {q ∈(G)  |  (q,G)}),  
    
        h_n(q_p, g, G)    = h(g, {q ∈(G)  |   (q_p, q)∈(G)}),

where q_p is the chosen pivot qubit, s_q is the string of Pauli letters of all legs on qubit q, g the set of Pauli gadgets examined, and k describes a weighting factor which we parametrized with k=10. P_I(s), P_max(s) and P_min(s) are all functions acting on the Pauli string and returning a number in 𝐍:
P_I(s) counts the amount of  Pauli I's in s, P_max(s) counts the size of the largest non-I region in s, P_min(s) determines the size of the smallest non-I region in s.

It is possible to tweak the cost function in our heuristic to be more specific to the target architecture. For example, the cost can be weighted based on the fidelity of the qubits or the gates, or adjusted to prefer gate parallelisation. In principle, any cost function would result in a correct algorithm, but with a different performance in terms of resulting gate count. We expect that the design of an optimal cost function is architecture dependent and leave analysis to future work.






















 §.§ Clifford tableau synthesis up-to-permutation

The last step of our algorithm is the synthesis of the Clifford tableau with the propagated Clifford gates. For this, we use an adaptation of the method proposed by <cit.>. We extend this algorithm to include the trick from <cit.> that allows us to synthesize the tableau up-to-permutation; emulating the movement of qubits on the quantum computers. Thus, the resulting circuit might have qubits that are moved to a different register than they were originally placed in. This is a normal phenomenon in quantum circuits that have been routed using SWAP gates and it allows us to reduce CNOT count.

Similar to the proposed algorithm, the Clifford tableau synthesis algorithm <cit.> uses a heuristic for picking which qubit to synthesize and remove from the problem. While the final Clifford tableau is typically an identity matrix, we can alternatively end the synthesis procedure when the Clifford tableau is a permutation matrix. The permutation then corresponds to a relocation of the qubits on the quantum computer. We do this by introducing a secondary heuristic for choosing which register (i.e. column in the tableau) the chosen qubit goes to, following the strategy from <cit.>. Lastly, we need to take care that the signs of the qubits are corrected on the new qubit register and not the old one.

When selecting the logical qubit (row), we choose the following heuristic:

    h_row(r)     = ∑_i [𝕀(T_r,i≠ 0 ∨ T_r,i+q≠ 0) .
       .    + 𝕀(T_r+q,i≠ 0 ∨ T_r+q,i+q≠ 0)],

where T is a Clifford tableau, and q is the number of qubits. This heuristic favors the row with the fewest interactions. When choosing a physical qubit (column), we look for the shortest CNOT distance, which is approximated by summing up the distances of rows along the connectivity graph:

    h_col(c)     = ∑_i [𝕀(T_c,i≠ 0 ∨ T_c,i+q≠ 0) .· d_c, i
       .    + 𝕀(T_c+q,i≠ 0 ∨ T_c+q,i+q≠ 0) · d_c, i]

Here d_i,j, describes the shortest path between i and j, something which can be efficiently precomputed using the Floyd-Warshall algorithm <cit.>.




 §.§ Perm Row Col Clifford

In the final step of our algorithm, we use an adaptation of the algorithm proposed by
<cit.> that synthesizes a stabilizer circuit from a Clifford tableau in an architecture-aware fashion. This algorithm, inspired by <cit.>, iteratively selects a non-cutting vertex in the connectivity graph of the underlying architecture and removes its interaction with the rest of the circuit, utilizing Steiner trees to find minimal placements of architecture-conforming CNOTS[We refer the reader to <cit.> for a more in-depth explanation of the algorithm]. However, this scheme requires an initial mapping of logical (rows in the tableau) to physical qubits (columns in the tableau). 

In this work, we utilize an extended variant of the algorithm to synthesize the final Clifford tableau. The extension introduces the PermRowCol scheme from <cit.>, which identifies an improved mapping during the synthesis process. Thus, instead of a single cost function, two are used: one to identify a logical qubit with a small amount of interaction with the rest of the circuit and a second to identify a non-cutting vertex in the connectivity graph to be removed.
When selecting the logical qubit (row), we choose the following heuristic:

    h_row(r)     = ∑_i [𝕀(T_r,i≠ 0 ∨ T_r,i+q≠ 0) .
       .    + 𝕀(T_r+q,i≠ 0 ∨ T_r+q,i+q≠ 0)],

where T is a Clifford tableau, and q is the number of qubits. This heuristic favors the row with the fewest interactions. When choosing a physical qubit (column), we look for the shortest traversal along the resultant Steiner tree, which is approximated by summing up the distances of rows along the connectivity graph:

    h_col(c)     = ∑_i [𝕀(T_c,i≠ 0 ∨ T_c,i+q≠ 0) .· d_c, i
       .    + 𝕀(T_c+q,i≠ 0 ∨ T_c+q,i+q≠ 0) · d_c, i]

Here d_i,j, describes the shortest path between i and j, something which can be efficiently precomputed using the Floyd-Warshall algorithm <cit.>.

The final permutation on the stabilizers can be extracted as follows:

    perm_i = arg max_j T_i, j + 2 T_i+q, j,

which implements the logical to physical qubit mapping.




    
















 §.§ Runtime Complexity Analysis

The following recurrence relation describes the worst-case runtime of our algorithm:

    T(n) = 
    		1     n = 0 
    
    		4T(n-1) + nq    otherwise

where n is the number of gadgets in the Pauli polynomial, and q is the number of qubits. One can see that this is true since our method requires per step at most 𝒪(nq) many operations to compute the heuristics, reorder the gadgets, and propagate Clifford gates through the Pauli polynomial. Secondly, the worst case in splitting the Pauli polynomial is to select one gadget in one of the recursion steps and partition the rest to the others [Note that this would be equivalent to decomposing one gadget and propagating the gates throughout the rest of the Pauli polynomial.]. Furthermore, we can see that <ref> is bound by 𝒪(n^2q + q^4), which yields the closed-form upper bound in terms of runtime. The term 𝒪(q^4) arises from synthesizing the Clifford tableau <cit.>.



 §.§ CNOT Complexity Analysis


Given that per n of the gadgets, at most q many CNOTs are required to synthesize it entirely, and given that there are at most n gadgets, a bound can be found from the fact that there are at most 𝒪(nq) CNOTs required to synthesize the whole Pauli polynomial. Since the remaining gadgets are collected in a Clifford tableau, which requires at most 𝒪(q^2) many CNOTs among synthesis, we arrive at an overall overhead of 𝒪(nq + q^2) CNOTs. We want to emphasize that this bound is quite naive since duplicates are canceled when propagating the CNOTs through the rest of the Pauli Polynomial. So the average bound might be much tighter as indicated by our evaluations.



§ EXPERIMENTS

We investigated the performance of the proposed methods on both architectures with all-to-all connectivity (refereed to as complete architectures) and restricted architectures, namely those available in the IBM Software stack[Our implementation can be found on Github: <https://github.com/daehiff/pauliopt/tree/new_methods>. 
]. In our experiments, we targeted the following IBM backends: quito (5 qubits),  nairobi (7 qubits), guadalupe (16 qubits), mumbai (27 qubits), and ithaca (65 qubits). 

All experiments were run using a process pool where each combination of circuit and synthesis strategy was distributed as tasks in a multi-threaded manner. Each synthesis strategy ran completely serially with no other innate parallelism.

For the randomized experiments, we synthesized 20 random Pauli polynomials with increasing gadget count for each architecture. The leg count was uniformly sampled from {1..q}, where q is the size of the given architecture. The leg positions were randomly selected and its Pauli letter was uniformly sampled from {X,Y,Z}. The gadget angle was uniformly sampled from {π, π/2, π/4, π/8, π/16}. 

Additionally, we synthesized the molecules present in pytket's UCCSD test set. These correspond to Hamiltonians used to estimate the ground state energy of small molecules and were used as benchmarks in <cit.>.

We compare the proposed methods against Paulihedral <cit.>  and the UCCSD pair-wise and set-based synthesis methods in <cit.>. As a baseline, we included a naive Steiner tree-based decomposition which did not use any reordering. An additional routing step utilizing pytket <cit.> is used for methods that are not architecture-aware. To compare apples with apples, all circuits are synthesized to the gateset {CNOT, U3} and evaluated on CNOT count, circuit depth, and CNOT depth. 

Since the gadget ordering found by our algorithm affects the Trotter error, we also calculated the Trotter error of all methods for random circuits and two molecules: H2 (in 631g with parity encoding) and H4 (in sto3g with parity encoding).



 §.§ Random Pauli polynomials

We present the average CNOT count, circuit depth, and two-qubit gate depth of the synthesized circuit in <ref> and <ref>, for restricted and complete architectures respectively.
For both cases, the proposed algorithm improves across the board for deep circuits (>50 gadgets). We hypothesize that this is a consequence of collecting operations in a Clifford tableau, which avoids the construction of the full CNOT ladder (i.e. sequence of CNOTs) in gadget synthesis as we propagate the second half of the ladder through the remainder of the polynomial. Thus, subsequent gadgets can now produce transverse two-qubit gates. 
Interestingly, we see that for shallow circuits (<50 gadgets), the proposed method performs worse on CNOT count, we attribute this to the greedy choosing of rows with large numbers of I legs, effectively routing rotations through long chains of I. Additionally, the proposed algorithm uses a naive initial placement of the qubits while both Paulihedral and TKET are able to find a better qubit mapping as a starting point for synthesis.



 §.§ UCCSD molecular ansatzes

Comparison of the three metrics for the various examined algorithms can be found in <ref>, <ref>, and <ref>. In this experiment, we synthesized each molecule for the smallest IBM device that would fit the circuit. We applied a time-out of 60 minutes but we found that in some cases the threads had been suspended resulting in incorrect runtimes. For a fair comparison, we decided to only present the molecules for which all methods finished and to omit runtime results.

From the tables, we can see that the proposed algorithm shows a clear improvement over the other examined methods; the only exception is the three smallest molecules. We hypothesize that this is, once again, due to the lack of qubit mapping and the greedy heuristic .




 §.§ Evaluation of the Trotter error

As the chosen ordering of Pauli gadgets is determined heuristically, the new order may increase the resulting Trotter error. Hence, we investigated the unitary overlap of the exact time evolution operator U_exact = exp( -i θ/2 H)  and the synthesized circuit U_circuit, as in  <ref>:

    overlap = ⟨0| U_exact U_circuit^†|0⟩

We used 20 randomly generated Pauli polynomials with six qubits and 160 gadgets. Each gadget had uniformly distributed legs across the six qubits with a type in [X, Y, Z]. The rotation angle was uniformly sampled from: [ π/32, π/64, π/128]. Additionally, we performed this investigation on molecules  and  from pytket's UCCSD test set <cit.>. For those molecules, we again picked the angles of the individual gadgets at random from the set [ π/32, π/64, π/128] to facilitate a fair comparison among the experiments.

We examined the overlap for all cases over multiple timesteps in [0:2π], and the results are plotted in <ref>. We omit TKET UCCSD (pair) from the figure as it does not change the ordering of the Pauli gadgets and the Trotter error would be equal to that of naive synthesis.

Across the various cases, we see that the proposed method is comparable to default Trotterization. Still, the best possible performance of each algorithm depends on the molecule's structure.
myTableStyletabular=|l|l|c|c|r|r|r|r|r|,
        table head=Molecule.     Backend.     # Qubits.     Naive     Paulihedral     TKET (set)     TKET (pair)     Proposed 
,
        late after line=
,
        head to column names,
        respect underscore=true
        














§ CONCLUSION

In this work, we introduce a novel method for architecture-aware Pauli polynomial synthesis, which propagates synthesized gates through the remainder of the polynomial. As a consequence, we introduce an alternative lexicographical grouping methodology that emerges from the structure of the proposed algorithm. We additionally describe an extension to the architecture-aware synthesis introduced in <cit.> that allows Clifford tableau synthesis up-to-permutation. We observe that the introduced method outperforms other examined methods on both restricted and complete architectures and, for the small set of test cases observed, does not seem to introduce significant Trotter error when compared with similar algorithms. 

Although investigation of the Trotter error associated with the proposed method is preliminary, we note that the error introduced by the noisy gates in the resulting circuits of alternative methods vastly outweighs the potential difference in Trotter error in cases where the proposed method worsens Trotter error. Considering the scale of improvement in gate counts and depth as observed with our method, reducing Trotter error will become a factor in fault tolerant devices.

Nevertheless, there are several areas for future improvement. One key weakness of our introduced algorithm is the cost function for pivot selection; for shallow regions, it removes interactions inefficiently. This means that the final recursions in the algorithm likely introduce unnecessary two-qubit operations. This can be addressed by improving or tuning the cost function or by introducing methods such as reverse traversal <cit.> to improve the initial qubit mapping. The current algorithm also does not find a good qubit placement during synthesis, incorporating a scheme similar to PermRowCol may be further improve its overall performance.

We additionally note that the time to solution for the proposed method is not ideal, taking up to 30 minutes on a local workstation for the largest, deepest circuits in our experiments. Although this can be partially attributed to unoptimized python code, algorithmic improvements may be necessary for future larger devices. Methodologies such as `parallel-in-time' may be interesting to investigate if the cost of propagated Clifford tableaus through synthesized regions is low and does not impact performance significantly. The current algorithm can also beextended to support the synthesis of higher-order symmetric product formulas mentioned in <cit.> or other higher-order decompositions 
<cit.>. Additionally, the algorithm can be represented as row additions in a 2q× g matrix, where q is the number of qubits and g the number of Pauli gadgets, which suggests that an efficient GPU implementation is possible.

In summary, the architecture-aware synthesis of Pauli gadgets offers a powerful alternative to existing methods to synthesize Pauli-based product formulas on restricted architectures; architecture-conforming algorithms reduce routing overhead tremendously, a key step to enable effective quantum hardware usage in the near term.







IEEEtran

  





 Paper Title: Engineering photonic band gaps with a waveguide-QED structure containing an atom-polymer array.



APS/123-QED 



School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China

wenzjia@swjtu.edu.cn
School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China



             

§ ABSTRACT 
We investigate the generation and engineering of photonic band gaps in waveguide quantum 
electrodynamics systems containing periodically arranged atom-polymers. We first consider the configuration 
of a dimer array coupled to a waveguide. The results 
show that if the intra- and inter-cell phase delays are properly designed, the center and the 
width of the band gaps, as well as the dispersion relation of the passbands can be modified by adjusting the 
intra-cell coupling strength. These manipulations provide  ways to control the propagating modes in  the waveguide, leading to some interesting effects such as slowing or even stopping a single-photon pulse. Finally, we take the case of the tetramer chain as an example to show 
that, in the case of a larger number of atoms in each unit cell, tunable multi-gap structures and more 
sophisticated band-gap engineering can be realized. Our proposal provides efficient ways for photonic 
band-gap engineering in micro- and nano-quantum systems, which may facilitate the manipulation of photon transport in future quantum networks.


Engineering photonic band gaps with a waveguide-QED structure containing an atom-polymer array
    W. Z. Jia
    August 5, 2024
==============================================================================================







§ INTRODUCTION


Waveguide quantum electrodynamics (wQED) systems are realized by strongly coupling quantum 
emitters (e.g., atoms, artificial atoms, cavities, and so on) to a one-dimensional (1D) waveguide and 
are ideal platform for the manipulation of strong emitter-photon interactions <cit.>.
The high coupling efficiency between emitters and photonic modes makes wQED setups optimal for single-photon level quantum devices <cit.>.
In a 1D wQED setup containing multiple emitters, the coherent and dissipative interactions between 
them can be mediated by their common electromagnetic environment, giving rise to a large number of 
intriguing phenomena, including superradiant and subradiant phenomena <cit.>, 
long-range quantum entanglement between atoms <cit.>,
cavity QED with atomic mirrors <cit.>, 
decoherence-free interactions in giant artificial atoms <cit.>, 
topologically protected scattering spectra <cit.>, 
multiple Fano interferences <cit.>, 
control-field-free atomic coherence effects <cit.>,
and so on.

In recent years, considerable attention has been devoted to photonic band-gap materials in micro- and 
nano-quantum structures. One potential avenue for achieving this objective is through the use of a 
conventional photonic crystal waveguide (PCW) <cit.>,
in which the periodicity of the dielectric structure is employed to generate a set of Bloch bands. 
Moreover, in the microwave domain, PCW has a series of circuit versions, including, for example, 
coplanar waveguide with periodically modulated impedance <cit.>, resonator array waveguide <cit.>,
and circuit-crystal metamaterials with Josephson junction arrays <cit.>. 
Another method to generate photonic band gaps is based on wQED structures with periodically 
arranged emitters (either an atom array or a cavity array) side coupled to a 1D linear waveguide <cit.>. 
Achieving highly tunable band-gap modulation in periodic structures that have already been prepared  
is of significant importance. This type of band-gap engineering can facilitate the manipulation of 
photon transport and the control of light-matter interactions in a variety of ways. In the case of band-gap structures produced by wQED containing multiple atoms, existing schemes for band-gap modulation are mainly based on the adjustment of the atomic frequency <cit.>
and the application of an external control field <cit.>.

In micro- and nano-quantum systems, such as superconducting quantum circuits, the tunability of the 
coupling strength between artificial atoms is a prominent feature <cit.>. 
It seems reasonable to posit that the tunable coupling between atoms can be harnessed to engineer 
the band-gap structure formed by wQED systems containing an atomic array. In 
this paper we present a proposal to achieve this goal, as shown schematically in Fig. <ref>. In 
our proposal, a periodic atomic array is coupled to a 1D waveguide, with each unit cell 
containing at least two atoms with tunable couplings between the nearest neighboring atoms, forming 
a so called polymer. The transfer matrix of the system is derived using the real-space formalism, and 
then the corresponding scattering coefficients are expressed analytically in terms of Chebyshev 
polynomials. Using these results, we further analyze the band-gap structures in the scattering spectra and demonstrate the potential for band-gap engineering through the use of interatomic couplings.
Specifically, for the case where each cell is an atom-dimer (i.e., a cell with two coupled atoms), band-gap structures with highly tunable  center frequencies and widths can be realized by properly adjusting the intra-cell coupling strength and designing the  interatomic phase delays. Moreover, these kinds of modulations can also be used to engineer the propagating modes in the passband. In particular, if a band gap is closed 
at a specific frequency, the Bloch modes in the vicinity of this 
frequency will become slow modes with a linear dispersion relation. 
Furthermore, if the width of a passband is dynamically compressed by modulating the coupling strength, the formation of a flat band can enable the setup to slow down or even stop a single-photon pulse at will. 
Finally, the results for a dimer array can be generalized to the case of a polymer chain with a greater 
number of atoms in each unit cell. This enables the realization of a tunable multi-band-gap structure 
and the implementation of more sophisticated band-gap engineering.

The remainder of this paper is organized as follows. In Sec. <ref>, we present a theoretical 
model that includes the system Hamiltonian and corresponding single-photon transport equations. 
Furthermore, using the transfer matrix approach, we derive the expressions of the single-photon scattering coefficients. In Sec. <ref>, we present some 
general discussions on the band-gap structure produced by a wQED system 
containing a polymerized atomic chain. In Sec. <ref>, we analyze tunable band-gap structures 
and propagation-mode control in a wQED system 
coupled by an atom-dimer chain.
In Sec. <ref>, we further examine band-gap engineering in a wQED system 
containing an atom-tetramer chain.
Finally, further discussions and conclusions are given in Sec. <ref>.




§ THEORETICAL DESCRIPTION OF SINGLE-PHOTON SCATTERING





 §.§ Hamiltonian and single-photon transport equations





In this study, we concentrate on the wQED structures comprising M periodically spaced polymers, each containing N 
two-level atoms with identical transition frequency 
ω_a, coupled to photonic modes in a 1D waveguide with linear dispersion, as schematically shown in Fig. <ref>. 
The atoms are located at position x_m,s (m=1,2,⋯,M labels the cells and s=1,2,⋯,N denotes the atoms
in each cell). The lattice constant of the polymer array is L=x_m,s-x_m-1,s. The distance between neighboring atoms in a cell
is a=x_m,s-x_m,s-1.
The strength between the sth atom
and the (s+1)th atom in a cell is J_s. The coupling strength between each atom and the waveguide is g. Under the rotating-wave approximation, 
the Hamiltonian of the system in the real space can be written as (ħ=1):


    Ĥ=   ∑_ρ∫dxâ_ρ^†(x)(-iτ_ρv_g∂/∂ x)â_ρ( x )
       +∑_m=1^M∑_s=1^Nω_aσ_m,s^+σ_m,s^-+∑_m=1^M∑_s=1^N-1(J_sσ_m,s^+σ_m,s+1^-+H.c.)
    
       +∑_ρ∑_m=1^M∑_s=1^Ng∫dxδ(x-x_m,s)[â_ρ^†(x)σ_m,s^-+H.c.],


where ρ=R, L
 and τ_R,L = ±1. â_R^†(x) [â_R(x)] and
â_L^†(x) [â_L(x)] are the bosonic creation (annihilation) operators of the right- and left-going wave at position x. v_g is the group velocity of photons in the waveguide.
σ_m,s^+ (σ_m,s^-) is the raising (lowering) operator of the sth atom in the mth cell. 

In the single-excitation manifold, the scattering eigenstate can be written as


    |Ψ⟩=∑_ρ∫dxΦ_ρ(x)â_ρ^†(x)|∅⟩
    +∑_m=1^M∑_s=1^Nf_m,sσ_m,s^+|∅⟩,


where |∅⟩ is the vacuum state with no photons propagating in the waveguide 
and the atoms occupying their ground states. Φ_ρ(x) is the single-photon 
wave function in the ρ mode. f_m,s is the excitation amplitude of the sth atom 
in the mth cell. 

We assume that a single photon with energy ω=v_g k is initially incident from the left, where k is the wave vector of the photon. The corresponding right(left)-going wave function Φ_R(x) [Φ_L(x)] takes the following Ansatz:



    Φ_R(x)   =   e^ikx{θ(x_1,1-x)
    
          +∑_m=1^M∑_s=1^Nt_m,s[θ(x-x_m,s)-θ(x-x_m,s+1)]},


    Φ_L(x)=e^-ikx∑_m=1^M∑_s=1^Nr_m,s[θ(x_m,s-x)-θ(x_m,s-1-x)].



Here, we define x_1,0=-∞ and x_M,N+1=+∞. t_m,s (r_m,s) is the transmission (reflection) amplitude of the sth atom in the mth cell. And the transmission (reflection) amplitude of the last (first) atom t_M,N≡ t (r_1,1≡ r) is defined as the transmission (reflection) amplitude of the atom array. θ(x) denotes the Heaviside step function. 

From the eigenequation Ĥ|Ψ⟩=ω|Ψ⟩, we can obtain the following set of equations:



    t_m,s-t_m,s-1+i√(Γ/2)f̃_m,se^-iϕ_m,s=0,


    r_m,s+1-r_m,s-i√(Γ/2)f̃_m,se^iϕ_m,s=0,


    J_s-1f̃_m,s-1-Δf̃_m,s+J_sf̃_m,s+1
          +1/2√(Γ/2)[(t_m,s+t_m,s-1)e^iϕ_m,s
    +(r_m,s+1+r_m,s)e^-iϕ_m,s]=0.



Note that in the above equations the inter-cell connection relations t_m,0=t_m-1,N, r_m,N+1=r_m+1,1 and the boundary
conditions t_1,0=1, r_1,1=r, t_M,N=t, r_M,N+1= 0 are satisfied. f̃_m,s=f_m,s/√(v_g) is the atomic 
excitation amplitude scaled by √(v_g). Γ=2g^2/v_g is the decay rate of the atoms into the guided modes.  Δ=ω-ω_a is the detuning between the photon and the atom. The phase factor corresponding to position x_m,s is 
defined as ϕ_m,s=kx_m,s.




 §.§ Transfer matrix and scattering coefficients



From Eqs. (<ref>) and (<ref>), after iteration, we can obtain



    t_m,s=t_m-1,N-i√(Γ/2)∑_j=1^se^-iϕ_m,jf̃_m,j,
	

    r_m,s=r_m+1,1-i√(Γ/2)∑_j=s^Ne^iϕ_m,jf̃_m,j.



From now on, we rewrite the phase factor of the sth atom in the mth cell as ϕ_m,s=β_m+ϕ_s. β_m=(m-1)β 
is the reference phase of the mth cell (without loss of generality, we have chosen β_1=0), where 
β=ϕ_m,s-ϕ_m-1,s=kL is the phase delay between neighboring cells. ϕ_s=[s-(N+1)/2]α is the relative phase 
of the sth atom in each cell with respect to the reference phase, where 
α=ϕ_m,s-ϕ_m,s-1=ka is the phase delay between neighboring atoms in a unit cell. 
Note that the phase delays 


    α=ka=(1+Δ/ω_a)α_0,  β=kL=(1+Δ/ω_a)β_0


is detuning dependent, where α_0=ω_aa/v_g and β_0=ω_aL/v_g
are detuning-independent phase delays defined in terms of the frequency of the resonant photons.

Using the above definitions and substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>), we have


    (Δ𝐈-𝐇_𝐜)𝐟̃_m=e^iβ_m𝐕t_m-1,N+e^-iβ_m𝐕^*r_m+1,1,


where  𝐈 is the N-dimensional identity matrix. 𝐇_𝐜 is the effective non-Hermitian Hamiltonian of each cell, with elements


    (ℋ_c)_ss'=J_sδ_s,s'-1+J_s'δ_s-1,s'-iΓ/2e^i|ϕ_s-ϕ_s'|.


𝐕 and 𝐟̃_m take the form 



    𝐕=√(Γ/2)(e^iϕ_1,e^iϕ_2,⋯,e^iϕ_N)^⊤,


    𝐟̃_m=[ f̃_m,1, f̃_m,2,      ⋯,  f̃_m,N ]^⊤.




According to Eqs (<ref>) and (<ref>), we can read the relations satisfied by the transmission (reflection) amplitudes of neighboring cells: 



    t_m,N=t_m-1,N-ie^-iβ_m𝐕^†𝐟̃_m,


    r_m,1=r_m+1,1-ie^iβ_m𝐕^⊤𝐟̃_m.



On the other hand, Eq. (<ref>) allows us to express the vector 𝐟̃_m, which is composed of the 
excitation amplitudes of the atoms in the mth cell, in terms of the corresponding input amplitudes 
t_m-1,N, r_m+1,1. Substituting this result into Eqs. (<ref>) and (<ref>), one can 
obtain the connection relation between the 
input and the output amplitudes of the mth unit cell


    t_m,Nr_m,1=ℬ^-1_m𝒮ℬ_mt_m-1,Nr_m+1,1,


where ℬ_m=diag(e^iβ_m,e^-iβ_m) and the scattering matrix 𝒮 takes the form


    𝒮=
    [   1-i𝐕^†(Δ𝐈-𝐇_𝐜)^-1𝐕  -i𝐕^†(Δ𝐈-𝐇_𝐜)^-1𝐕^*;    -i𝐕^⊤(Δ𝐈-𝐇_𝐜)^-1𝐕 1-i𝐕^⊤(Δ𝐈-𝐇_𝐜)^-1𝐕^* ].


Alternatively, the connection relation can be expressed as 


    t_m-1,Nr_m,1=ℬ^-1_m𝒯ℬ_mt_m,Nr_m+1,1.


Here 


    𝒯=
    [                  1/1-i𝐕^†(Δ𝐈-𝐇_c)^-1𝐕 i𝐕^†(Δ𝐈-𝐇_c)^-1𝐕^*/1-i𝐕^†(Δ𝐈-𝐇_c)^-1𝐕;  -i𝐕^⊤(Δ𝐈-𝐇_c)^-1𝐕/1-i𝐕^†(Δ𝐈-𝐇_c)^-1𝐕             Det[𝒮]/1-i𝐕^†(Δ𝐈-𝐇_c)^-1𝐕 ]


represents the transfer matrix, which establishes a relationship between the light field on either side of the mth unit cell. As indicated by Eq. (<ref>), the connection 
relation for a single-polymer setup (i.e., the case of M=1) with the reference phase set to zero acquires the following form 


    ([  1; r̃ ])
    =𝒯([ t̃;  0 ]),


where t̃ and r̃ are the corresponding transmission and reflection amplitudes. 
Furthermore, by inserting Eq. (<ref>) into Eq. (<ref>), it can be observed that



    t̃=1-i𝐕^†(Δ𝐈-𝐇_c)^-1𝐕,


    r̃=-i𝐕^⊤(Δ𝐈-𝐇_c)^-1𝐕.



In addition, the atomic chain under consideration satisfies time reversal symmetry. Consequently, 
the transfer matrix can also be expressed in terms of t̃ and r̃ as follows <cit.> 


    𝒯=
    [      1/t̃ r̃^*/t̃^*;     r̃/t̃    1/t̃^* ].



Using Eq. (<ref>) iteratively M times in succession and noticing the conditions t_0,N=1, r_1,1=r, t_M,N=t, r_M+1,1=0, and β_1=0, we finally obtain the following connection relation between the reflection and transmission amplitudes for the entire atomic chain:


    ([ 1; r ])=𝒯̃^M([ te^iMβ;      0 ]),


with


    𝒯̃=𝒯𝒯_β,  𝒯_β=[ e^-iβ     0;     0  e^iβ ].


Thus the transmission and reflection amplitudes for the polymer chain read


    t=e^-iMβ/[𝒯̃^M]_11,  r=[𝒯̃^M]_21/[𝒯̃^M]_11,
	


Notice that the determinant of the matrix 𝒯̃ is 1. Based on Abeles's theorem <cit.>, 
we can write the matrix 𝒯̃^M in terms of Chebyshev polynomials of the second kind:


    𝒯̃^M=U_M-1(y)𝒯̃-U_M-2(y)ℐ,


with


    y=1/2Tr(𝒯̃).


ℐ is the two-dimensional identity matrix. According to Eqs. (<ref>) and (<ref>), one can 
obtain the transmittance and the reflectance of the polymer chain (see Appendix <ref> for details)



    T=|t|^2=1/1+ζ U_M-1^2(y),
	

    R=|r|^2=ζ U_M-1^2(y)/1+ζ U_M-1^2(y),



where ζ≡|𝒯̃_12|^2=|𝒯̃_21|^2.  It should be pointed out that the relation T+R=1 is satisfied as a consequence of the 
conservation of photon number. Therefore, in what follows, we will focus on the reflectance R alone when analyzing the band-gap structures.

Moreover, we will focus on the regime where the phase-accumulated effects for detuned photons can be ignored, i.e., 
one can approximate the detuning-dependent phase delays defined in Eq. (<ref>) as α≃α_0 and β≃β_0. Or 
equivalently, we can say that the Markovian approximation can be performed, i.e., the time delays between the coupling points can be neglected. 
In Appendix <ref>, we present the Markovian conditions for our system, which are easily satisfied under typical parameters 
of wQED experiments <cit.>.
In what follows, we will use this approximation in our analysis to keep the physics transparent, while in the full numerical calculations we still make the 
phase delay α and β depend on the detuning in order to obtain accurate results.









§ FORMATION OF BAND-GAP STRUCTURE



In this section, we present some general discussions on the band-gap structure produced by a polymerized atomic chain.  
The existence of band gaps is a common feature of periodic structures. For our
system, it is not surprising that band-gap structures can form as the number of cells M increases.
To illustrate this point, we present in Figs. <ref>(a)-<ref>(c) the reflection spectra for an atom-dimer array for varying 
values of M. As M increases, the spectrum with two total reflection points for small M evolves 
into a band-gap structure for relatively large M. Moreover, it can be observed from
Eq. (<ref>) that the solutions to the equation for Δ


    y(Δ)=cos(lπ/M)≡ y_l


are the zero points of the function R(Δ), since y_l (l=1,2,⋯,M-1) is a root of the the Chebyshev polynomial of the second kind
U_M-1(y) <cit.>. Importantly, these zero reflection points 
outside the band gap indicate the frequencies of the propagating modes that are permitted  
by the passbands.

In order to gain insight into the formation of the band gap, we consider a periodic chain comprising an 
adequate number of cells and present the dispersion relation based on Bloch's theorem (see 
Appendix <ref> for details):


    cos(qL)=y(Δ).


Here q represents the Bloch wave number, which is used to label the propagating 
modes that are permitted by the wQED system with atomic chain. Clearly, Eq. (<ref>) tells us that for 
frequencies satisfying the inequality 


    |y(Δ)|>1,


no propagating mode exists, resulting in band-gap structures with R=1 in the 
reflection spectrum, as shown by the shaded region in Fig. <ref>(a). The corresponding dispersion relation, which can only
be defined in the pass bands, is shown in Fig. <ref>(b). 
 
On the other hand, the formation of a spectrum with band-gap structures can be understood in terms of interference effects between 
different scattering channels corresponding to the excitations of the collective modes. To this end, we decompose the reflection spectrum into the 
superpositions of several Lorentzian-type amplitudes contributed by the collective excitations (see Appendix <ref> for details):


    r=∑_j=1^N'ℒ_j,  ℒ_j=d_j/Δ-Δ̃_j+iΓ̃_j/2.
	

Here N'=MN is the total number of atoms, Δ̃_j represents the detuning between the jth collective mode and the atoms,
Γ̃_j denotes the effective decay of the jth collective mode, and d_j determines the weight of each Lorentzian 
component.  
In Figs. <ref>(c)-<ref>(e), we illustrate this decomposition by taking the case of a dimer array
(with α_0=π/2, β_0=π) as an example.
It can can be seen that the Lorentzian-type amplitudes ℒ_j are symmetrically distributed around Δ=0. The band-gap region with near total reflection is mainly attributed to the excitation of the most superradiant states [see Fig. <ref>(c)]. While the two narrowest excitation amplitudes 
corresponding to the most subradiant states are situated at the innermost [see Figs. <ref>(d) and <ref>(e)], and the corresponding width, 
denoted as Γ̃_ms, decreases sharply with increasing number of cells M [see Fig. <ref>(f)]. 
The Fano-type destructive interferences between these two most subradiant excitations and the superradiant ones result in the formation of 
a pair of innermost reflection minima and the corresponding steep band-gap walls [see Figs. <ref>(c)-<ref>(e)]. This phenomenon becomes more remarkable as the parameter M increases. Consequently, as the value of M increases to approximately
M>5, the distance w between the centers of the two innermost amplitudes approaches the width of the band gap W [which is 
obtained from the condition (<ref>)], as shown in Fig. <ref>(g). This result indicates that the condition (<ref>) can even be used to calculate the gap width of a chain containing a relatively small number of cells.





§ TUNABLE PHOTONIC BAND GAP WITH A DIMER CHAIN


Let us first consider the simplest case of N=2, namely a dimer array coupled to a waveguide, 
as shown in Fig. <ref>(a). 
As demonstrated in the preceding sections, once the expressions for the quantities ζ and y 
[from which we can determine the scattering coefficients (<ref>) and (<ref>), 
as well as the dispersion relation (<ref>)]
have been derived, a more comprehensive examination of the scattering spectrum and the band-gap structure is possible. Specifically, in the Markovian regime with α≃α_0 and β≃β_0, the quantities ζ and y for a dimer array take the form 



    ζ=Γ^2(2Δcosα_0+2J+Γsinα_0)^2/4[Δ^2-J(J+Γsinα_0)]^2,
	

    y   =   1/2Γ^2sin(β_0-α_0)sinα_0+Γ(Jcosα_0+Δ)sinβ_0/Δ^2-J(J+Γsinα_0)
          +cosβ_0,




which are strongly influenced by the intra-cell phase difference α_0 and the
inter-cell phase differences β_0.
Here we focus on the cases that the intra- and inter-cell phase delays satisfy either the Bragg or the anti-Bragg condition. The following analysis will demonstrate that, with the exception of the case where both the intra- and inter-cell phase delays satisfy the Bragg condition (see Sec. <ref>), spectra with highly tunable band-gap structures can be generated in the remaining cases by varying the intra-cell coupling strength J (see Secs. <ref>-<ref>).







 §.§ α_0=nπ and β_0=mπ: spectrum with Lorentzian line shape
 

We begin with the assumption that both the intra- and inter-cell phase differences satisfy the Bragg condition, with α_0=nπ (n∈ℕ^+) and β_0=mπ (m∈ℕ^+, m>n). Accordingly, Eqs. (<ref>) and (<ref>)
can be simplified as



    ζ=Γ^2/[Δ-(-1)^nJ]^2,
	

    y=(-1)^m.



Plugging these expressions into Eq. (<ref>) and using the identity U^2_M-1[(-1)^m]=M^2,  the reflectance can be further written as


    R=M^2Γ^2/[Δ-(-1)^nJ]^2+M^2Γ^2.


Obviously, similar to an uncoupled atom array (with J=0) under Bragg condition, the reflection spectrum shows the well-known phenomenon of superradiance, with a line width 2MΓ (scaled as the number of atoms 2M). However, the center frequency is shifted to Δ=(-1)^nJ
instead of at Δ=0. 
These properties are shown in Figs. <ref>(a) and <ref>(b), where we take the case of α_0=π (n=1) and α_0=2π (n=2) as examples. 








 §.§ α_0=(n-1/2)π and β_0=mπ: band gap with tunable width


We proceed to assume that α_0=(n-1/2)π (n∈ℕ^+) and β_0=mπ (m∈ℕ^+, m≥ n), i.e., 
the intra(inter)-cell phase difference satisfies the anti-Bragg (Bragg) condition.
Under these conditions, Eqs. (<ref>) 
and (<ref>) can be simplified as



    ζ=Γ^2[2J-(-1)^nΓ]^2/4{Δ^2-J[J-(-1)^nΓ]}^2,
	

    y=(-1)^m(1-1/2Γ^2/Δ^2-J[J-(-1)^nΓ]).



After substituting Eq. (<ref>) into the band-gap condition (<ref>), we find that except for the case of n∈𝔼^+,J=Γ/2 (where the gap is closed), the region of the band gap is


    |Δ|<|J±Γ/2|.


Namely, we can obtain a single band gap centered about Δ=0, with width


    W=|2J±Γ|.


Here “+” (“-”) corresponds to n∈𝕆^+ (n∈𝔼^+). It should be emphasized that the value of m do not influence the solution set of the inequality (<ref>), thus the range and width of the band gap are only dependent on the parity of n. A similar conclusion can be drawn with regard to the situations discussed in the following subsections (Sec. <ref> and <ref>). Importantly, the results obtained in Eqs. (<ref>) and (<ref>) indicate that the width of the band gap can be manipulated by changing the coupling strength J between the atoms 
within a cell, which is a significant advantage over a chain of non-interacting atoms.

Without loss of generality, we will consider the cases of α_0=π/2 (n=1) and α_0=3π/2 (n=2) 
as illustrative examples to demonstrate the modulation effect of J on the width of the band gap.
To this end, we plot the reflection spectra and corresponding gap width under 
these phase delays for varying J in Fig. <ref>.  Specifically, when α_0=π/2 and β_0=3π, the bandwidth increases 
linearly with J. Particularly, if J≫Γ, we can obtain a wide band gap with width W≃2J 
[see Figs. <ref>(a) and <ref>(b)]. On the other hand, when α_0=3π/2 
and β_0=3π, the bandwidth W first decreases linearly, reaching a minimum value of zero 
at J=Γ/2, and then increases linearly with J to a wide band gap [see Figs. <ref>(c) and <ref>(d)].
Additionally, the dispersion relations for this case around J=Γ/2 are plotted in Figs. <ref>(a)-<ref>(c).
It is shown that at J=Γ/2 the gap is closed, so the atom-coupled waveguide, like a bare waveguide, 
allows photons of all frequencies [see Fig. <ref>(b)].
But its dispersion relation has been greatly modified compared to a bare waveguide. 
In particular, around the resonance point Δ=0, one can 
obtain an effective linear dispersion |Δ|≃ṽ_g|q|, with ṽ_g=Γ L/4 [see the 
dashed lines in Fig. <ref>(b)]. 
Moreover, the ratio between the group velocities of the perturbed and the bare waveguides can be estimated as 
ṽ_g/v_g∼Γ/ω_a≪1, which means that under these parameters, the atom-coupled 
waveguide can be considered as a linear slow-light waveguide.      



 §.§ α_0=nπ and β_0=(m-1/2)π: band gap with tunable center frequency





We subsequently discuss the case of α_0=nπ (n∈ℕ^+) and β_0=(m-1/2)π (m∈ℕ^+, m>n), i.e., 
the intra(inter)-cell phase difference satisfies the Bragg (anti-Bragg) condition.
Under these conditions, Eqs. (<ref>) and (<ref>) can be simplified as



    ζ=Γ^2/[Δ-(-1)^n J]^2,
	

    y=(-1)^m+1Γ/Δ-(-1)^n J.



Similar to Sec. <ref>, with the help of the condition (<ref>), we can find that the region of the band gap is


    ∓ J-Γ<Δ<∓ J+Γ,


indicating that the center of the band gap is located at Δ=∓ J (where “-” corresponds to n∈𝕆^+ and “+” corresponds to n∈𝔼^+), 
and the corresponding width is


    W=2Γ.


This result shows that the center of the band gap can be tuned by changing the intra-cell coupling strength J. 
As shown in Figs. <ref>(a) (with α_0=π and β_0=5π/2) and <ref>(b) (with α_0=2π and β_0=5π/2), due to the existence of the intra-cell coupling, the center of the band gap is red-shifted or blue-shifted by J in comparison with a non-interacting atomic chain [see the red dashed lines in Figs. <ref>(a) and <ref>(b)], while the width of the band gap remains unchanged.






 §.§ α_0=(n-1/2)π and β_0=(m-1/2)π: tunable double band gaps and center passband





  §.§.§ Tunable band-gap structures


Finally, we discuss the case of α_0=(n-1/2)π (n∈ℕ^+) and β_0=(m-1/2)π (m∈ℕ^+, m>n), 
i.e., both the intra- and inter-cell phase differences satisfy the anti-Bragg condition.
For this case, Eqs. (<ref>) and (<ref>) can be simplified as



    ζ=Γ^2[2J-(-1)^nΓ]^2/4{Δ^2-J[J-(-1)^nΓ]}^2,
	

    y=(-1)^m+1ΓΔ/Δ^2-J[J-(-1)^nΓ].




For n∈𝕆^+, different from the previous cases, there are always two regions satisfying the condition (<ref>), with 


    J<|Δ|<J+Γ.


These regions with no propagating modes will form two band gaps centered at Δ=±(J+Γ/2) with equal width 


    W=Γ.


We refer to them as the right and left band gaps, respectively [see Fig. <ref>(a)]. And the region -J<Δ<J between the two gaps forms a
passband whose width is proportional to J 


    W'=2J.


In a word, we can obtain a symmetrical spectral structure with a passband sandwiched between two band gaps of equal width. In addition, 
the centers of the gaps, and thus the width of the passband, show a linear relationship with J, as shown in Fig. <ref>(a). 
To show these more clearly, the widths W and W' as functions of J are given in Fig. <ref>(b).

If n∈𝔼^+, as in the case of n∈𝕆^+, two band-gap regions symmetric about Δ=0 
appear in the reflection spectra for different J, except for the cases of  
J=Γ/2 (where the gaps are closed) and J=Γ
(where the two gaps merge into one). But a richer modulation of the band-gap structure can be obtained by varying the 
intra-cell coupling strength J [see Fig. <ref>(c)]. According to the condition (<ref>), we 
can obtain the regions of the gaps, the width W of each gap, and the width W' of the passband between the gaps, as 
summarized in Table <ref>. Correspondingly, the widths W and W' as functions
of J are plotted in Fig. <ref>(d). Specifically, in the region 0<J<Γ, the two band gaps are centered
at Δ=±Γ/2 and the width of each gap first decreases linearly (with W=Γ-2J) to zero at 
J=Γ/2 [where the gaps vanish, see the dashed lines in Figs. <ref>(c) and <ref>(d)] and then 
increases linearly (with W=2J-Γ). Finally the two gaps merge into a single band of width 2Γ 
at Δ=Γ [see the dash-dotted lines in Figs. <ref>(c) and <ref>(d)]. 
Accordingly, the width of the center passband first increases linearly (with W'=2J) to a local maximum 
when J→Γ/2 (note that at J=Γ/2, the center passband cannot be defined because 
the gaps are closed) and then decreases linearly [with W'=2(Γ-J)] to zero at 
Δ=Γ. In the region J>Γ, the gap splits again into two, 
centered at Δ=±(J-Γ/2) and of equal width J. And the width of the corresponding center passband increases 
linearly [with W'=2(J-Γ)].    





  §.§.§ Center passband engineering






Compared to the previous subsections, the most significant feature of the situation now discussed is the 
possibility of realizing a highly engineered center passband. As we have discussed earlier, its width is easily tunable 
by the intra-cell coupling strength J. Here we will further discuss its other properties.    
As discussed in Sec. <ref>, the solutions to Eq. (<ref>) can give the 
zero reflection points in the reflection spectrum. For present case, as shown in Fig. <ref>, 
the center branch of y(Δ) can be used to fix the M-1 zero reflection points within the passband, 
which correspond to the propagating modes permitted by this band. According to Eqs. (<ref>) 
and (<ref>), one can find that the wave numbers of these propagating modes are q=± lπ/(ML) (l=1,2,⋯,M-1).

If the frequencies of photons are concentrated within the center passband, the atom-perturbed system can be regarded
as a nonlinear waveguide with finite bandwidth, with the group velocity being considerably less than that of the 
propagating photons in a bare waveguide. By changing J, the band width and thus the dispersion relation of the center 
passband can also be easily engineered. To illustrate, we consider the case of n∈𝔼^+ and start with a 
coupling strength that falls within the range of Γ/2 < J < Γ [see Fig. <ref>(a)]. 
Subsequently, the value of J is increased until it reaches J≃Γ, during which the bandwidth of the 
center passband is compressed and ultimately reduces to zero. Consequently, the center passband becomes a flat 
band, as illustrated in Fig. <ref>(b). It is evident that, in this situation, the density of the traveling modes 
becomes exceedingly high, while the group velocity for photons in this band is close to zero. As a result, it is
possible to slow or even stop a single-photon pulse at will in this setup by dynamically controlling J. 
If the coupling strength J is precisely equal to Γ, the center passband will disappear, 
as the two gaps will merge into a single gap of width 2Γ. As the value of J is increased to satisfy 
J>Γ, the group velocity of the center passband undergoes a change in sign, as illustrated in 
Fig. <ref>(c). This indicates that the sign of the group velocity (i.e., the direction of wave-packet 
propagation) for the passband can also be controlled by selecting an appropriate J. We note that similar 
modulations can also be achieved in a wQED structure containing a resonator array <cit.> or a  chain of superconducting qubits <cit.> through the appropriate adjustment of the detuning between the emitters in a cell.

Similar to the second case discussed in Sec. <ref>, 
when n∈𝔼^+ and J=Γ/2 [where the gaps are closed, indicated by the dashed lines in 
Figs. <ref>(c) and <ref>(d)], the atom-coupled waveguide can be utilized as 
a linear waveguide when the modes in the vicinity of Δ=±Γ/2 are of interest. When
α_0=3π/2, β_0=5π/2 [the parameters used in Fig. <ref>(c)], 
the corresponding dispersion relations in the linear regions can be approximated as 
|Δ-Γ/2|=ṽ_g|q| and |Δ+Γ/2|=ṽ_g|q±π/L|, respectively. 
Here ṽ_g=Γ L/2 is the group velocity at Δ=±Γ/2 of the perturbed waveguide. 
We also have ṽ_g/v_g∼Γ/ω_a≪1, which indicates the 
slow-light effect. 






§ TUNABLE PHOTONIC BAND GAPS WITH A POLYMER CHAIN



The discussions in last section can be generalized to the case of a polymer chain, wherein each unit cell contains a greater number of atoms than two. Naturally, as the number of atoms in a cell increases, multi-band-gap structure
can be realized. Furthermore, the design of intra-cell interactions enables the implementation of more sophisticated 
band-gap engineering. In this section, we utilize the case of a tetramer (N=4) array coupled to a waveguide, as shown schematically in Fig. <ref>(b), as an example
to discuss these properties. The coupling strengths between neighboring atoms in a cell
are set as  J_1=J_3=J and J_2=η J, where the dimensionless parameter η is 
introduced to enhance the tunability of the system. 
The phase factors are set as α_0=π/2 and β_0=2π. 
Correspondingly, the function y(Δ) under the Markovian approximation takes the form 


    y(Δ)=Δ^4-𝒜Δ^2+ℬ/Δ^4-𝒞Δ^2+𝒟,


where


    𝒜=2Γ^2+(η+2)Γ J+(η^2+2)J^2,
    
       ℬ=(Γ^2+J^2)[(η+2)Γ J+J^2]
    +1/2Γ^2[Γ^2+(η+2)^2J^2], 
    
       𝒞=𝒜-2Γ^2,  𝒟=[(Γ+J)^2+ηΓ J]J^2.


 
Substituting Eq. (<ref>) into the band-gap condition (<ref>), 
we can obtain the band-gap regions for different values of J and η. To
classify the solutions to the inequality (<ref>), 
one can define the following discriminant function 


    f(J̃,η)=3J̃^2η^2+2(4J̃^2+J̃)η-1,


with J̃=J/Γ. For 
f(J̃,η)≠ 0, the reflection spectra indicate that a triple-band-gap structure, 
comprising a center gap and two side gaps of equal width, is distributed symmetrically about Δ=0, 
as shown in Figs. <ref>(a) (for fixed η=5 and varying J) 
and <ref>(d) (for fixed J=Γ/2 and varying η). 
The details of these spectra around the regions satisfying f(J̃,η)= 0 are provided
in Figs. <ref>(b) and <ref>(e).
Furthermore, both the centers and the widths of the two side gaps and the two passbands 
between the gaps are highly tunable by varying the values of J and η.  
The regions of the gaps, the width W_c of the center gap, the width W_s of the two side gaps, 
and the width W_p of the two passbands between the gaps are 
summarized in Table <ref>. The quantities δ and δ̃_1,2 in the table take the form

 

    δ=1/2[Γ+(η+2)J],


    δ̃_1,2=1/2[±η J+√((Γ+2J)^2+(Γ+η J)^2)].



It can be demonstrated that the relations 0<δ̃_2<δ̃_1, 0<δ<δ̃_1, and
 

    .δ-δ̃_2{[       >0, f(J̃,η)>0;       =0, f(J̃,η)=0;       <0, f(J̃,η)<0 ]..


are satisfied. In accordance with the reflection spectra illustrated in Figs. <ref>(a) and <ref>(d), the widths W_c, W_s, and W_p are plotted as functions of J (for fixed η) and of η (for fixed J)  in Figs. <ref>(c) and <ref>(f), respectively.
These results show that a tetramer chain can provide greater flexibility in band-gap engineering.




In the case of f(J̃,η)=0, the three band gaps merge into a single one centered about Δ=0, with width 2δ̃_1, as 
illustrated by the dashed lines in Figs. <ref>(a)-<ref>(f). Importantly, if
we select an appropriate path in the parameter space spanned by (η,J) [i.e., from the area satisfying |f(J̃,η)|>0
to a point on the curve f(J̃,η)=0], the two passbands
between the gaps can be easily compressed and ultimately becomes a flat band when f(J̃,η)→ 0. 
This property is illustrated in Figs. <ref>(a)-<ref>(c), in which η is held constant 
while J is allowed to vary.  Furthermore, the values of η and J employed in these figures have been selected to 
satisfy the conditions f(J̃,η)<0, f(J̃,η)≃ 0, and  f(J̃,η)> 0, respectively.
Thus, like the case discussed in Sec. <ref>, it is possible to slow or even stop a single-photon pulse 
by modulating the relevant parameters. However, this time we are able to control the propagation of photon pulses in two 
distinct passbands simultaneously. Finally, it can be seen from Figs. <ref>(a) and <ref>(c) 
that the group velocities of the two passbands between the gaps can also be designed
by selecting suitable values of the parameters η and J. 





§ CONCLUSIONS



In this paper, we have analyzed the generation and engineering of photonic band gaps in wQED systems containing multiple atom-polymers.
We first consider the configuration of a dimer array coupled to a waveguide with the intra- and inter-cell phase 
delays satisfying either the Bragg or anti-Bragg condition. It was found that, with the exception of both the 
intra- and inter-cell phase delays satisfying the Bragg condition, the photonic band-gap structures can be 
observed in the spectra for the remaining three cases. More importantly, the band-gap structures in these 
cases can be manipulated in a variety of ways (e.g., the center and  the width of the gaps, as well as the 
dispersion relation of the passbands) by adjusting the intra-cell coupling strength. 
Furthermore, modulation of the dispersion relation and bandwidth provides an effective way to manipulate the 
propagating modes in the waveguide, leading to some interesting phenomena such as slowing or even 
stopping a single-photon pulse. Finally, the results for a dimer array can be generalized to the case of a 
greater number of atoms in each unit cell. This enables the realization of a tunable multi-gap structure and the 
implementation of more sophisticated band-gap engineering. 

In our proposal, the key point is the use of tunable couplings between atoms to realize band-gap engineering. 
This can be easily achieved in micro- and nano-quantum systems such as superconducting quantum circuits 
<cit.>, where superconducting qubits or superconducting quantum interference devices (SQUIDs) are usually used as tunable coupling elements 
<cit.>. Thus, in summary, we 
present an experimentally feasible proposal to produce tunable band-gap structures in 1D wQED setups, 
which may provide novel ways to control single-photon propagation. Our proposal may have potential 
applications in long-distance quantum networking and quantum communication.



This work was supported by the National Natural Science Foundation of China (NSFC) under Grants No. 61871333.





§  DERIVATION OF EQS. (<REF>) AND (<REF>)


According to Eq. (<ref>), one can obtain 



    [𝒯̃^M]_11=𝒯̃_11U_M-1(y)-U_M-2(y),
	

    [𝒯̃^M]_21=𝒯̃_21U_M-1(y).



Substituting these results into Eq. (<ref>) and taking the square of the modulus yields the transmission and reflection 
coefficients 



    T=|t|^2=1/|𝒯̃_11U_M-1(y)-U_M-2(y)|^2,
	

    R=|r|^2=|𝒯̃_21|^2U_M-1^2(y)/|𝒯̃_11U_M-1(y)-U_M-2(y)|^2.



Note that the elements of the transfer matrix 𝒯̃ satisfy the relation 
Det(𝒯̃)=1, 𝒯̃_11^*=𝒯̃_22 and 
𝒯̃_12^*=𝒯̃_21 [see Eq. (<ref>)]. Thus we have 
|𝒯̃_11|^2-|𝒯̃_21|^2=1 and 
𝒯̃_11+𝒯̃_11^*=𝒯̃_11+𝒯̃_22=Tr(𝒯̃)=2y. 
Using these results and the identity


    U^2_M-1(y)+U^2_M-2(y)-2yU_M-1(y)U_M-2(y)=1,


the dominator of T and R can be further simplified as


    |𝒯̃_11U_M-1(y)-U_M-2(y)|^2
    
       =   |𝒯̃_11|^2U^2_M-1(y)+U^2_M-2(y)-(𝒯̃_11+𝒯̃_11^*)U_M-1(y)U_M-2(y)
    
       =   (1+|𝒯̃_21|^2)U^2_M-1(y)+U^2_M-2(y)-2yU_M-1(y)U_M-2(y)
    
       =   1+|𝒯̃_21|^2U^2_M-1(y).


Finally, by defining the quantity ζ=|𝒯̃_21|^2=|𝒯̃_12|^2, one can obtain the transmittance and reflectance 
described by Eqs. (<ref>) and (<ref>). 




§ THE MARKOVIAN CONDITIONS


In this paper, we focus on the regime in which the detuning-dependent phase delays defined in Eq. (<ref>)
can be approximated as α≃α_0 and β≃β_0. 
Here we will present the condition that allows us to make this approximation. 
The accumulated phase factor between two ends of the atomic chain 
under consideration in this paper is 
(M-1)β=(1+Δ/ω_a)(M-1)β_0. Obviously, for a not so small M, if the 
term caused by the detuning effects (M-1)β_0Δ/ω_a≃ Mβ_0Δ/ω_a≪1, it can be safely neglected. Note that for the band-gap cases (the cases in Secs. <ref>-<ref> and <ref>), the detuning range we are interested in is approximately Δ∼Γ, 
and the order of magnitude of β_0 we care about is β_0∼ 1, so we end up with 


    M≪ω_a/Γ,


which is easily satisfied under typical parameters for wQED systems <cit.>. And for the case discussed in Sec. <ref>), we have Δ∼ MΓ, the Markovian condition becomes


    M≪√(ω_a/Γ).


Furthermore, under these conditions the detuning-dependent term corresponding to any other pair of coupling points can also be ignored,
since the distance between them is smaller than the length of the atomic chain.
In a word, once the condition (<ref>) (valid for the cases discussed in 
Secs. <ref>-<ref> and <ref>) or (<ref>)
(valid for the case discussed in Sec. <ref>) is satisfied, we can safely let α≃α_0 and β≃β_0.




§ DERIVATION OF DISPERSION RELATION


Consider an infinite periotic polymer chain, we write the photonic wave functions to the immediate left and right of the mth unit cell as



    u^(k)_m(x)=t_m-1,Ne^ikx+r_m,1e^-ikx,


    u^(k)_m+1(x)=t_m,Ne^ikx+r_m+1,1e^-ikx.



According to Bloch's theorem, we have u_m+1,k(x+L)=u_m,k(x)e^iqL, where q is the Bloch wave number, which
leads to the following relations


    t_m,N=t_m-1,Ne^i(qL-β),   r_m+1,1=r_m,1e^i(qL+β).


Substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>) yields the following linear equations for  
t_m-1,N and r_m,1:


    (ℬ^-1_m𝒯ℬ_m-1-e^-iqLℐ)
    [ t_m-1,N;   r_m,1 ]
    =0.


The non-zero solutions for t_m-1,N and r_m,1 requires the determinant of the coefficient matrix to be zero, resulting in 
the following dispersion relation


    cos(qL)=Re(1/t̃e^-iβ)
    =1/2Tr(𝒯̃)=y(Δ).





§ EXPRESSIONS OF SCATTERING AMPLITUDES IN TERMS OF COLLECTIVE MODES


By starting from the transport equations (<ref>)-(<ref>) and performing some algebra,
we can obtain the scattering amplitudes of the atomic chain <cit.>



    t=1-i𝐖^†(Δ𝕀-𝐇)^-1𝐖,


    r=-i𝐖^⊤(Δ𝕀-𝐇)^-1𝐖.



Here 𝕀 is the N'-dimensional identity matrix. N'=MN is the total number of atoms. 𝐖 takes the form 


    𝐖=√(Γ/2)(e^iϕ_1,e^iϕ_2,⋯,e^iϕ_N')^⊤.


𝐇 is the effective non-Hermitian Hamilton matrix of the atomic array, with elements 


    ℋ_nn'=𝒥_nδ_n,n'-1+𝒥_n'δ_n-1,n'-iΓ/2e^i|ϕ_n-ϕ_n'|,


with n,n'=1,2,⋯,N'. Here the coupling strength 𝒥_n between neighboring atoms 
is defined as 𝒥_mN+s=J_s and 𝒥_mN=0 (m=0,1,2,⋯,M-1 and s=1,2,⋯,N-1).
The first two terms of Eq. (<ref>) describe the direct atom-atom coupling, and the last term
summarizes the atom-waveguide decay (for n=n') as well as the coherent and dissipative atom-atom interactions mediated by the 
waveguide modes  (for n≠ n'). 

To better understand the physical properties of the scattering process, we rewrite the scattering amplitudes in terms of collective modes of the 
atomic chain:



    t=1-i∑_j=1^N'(𝐖^†𝐔_j^ℛ)
    (𝐔_j^ℒ^†𝐖)/Δ-λ_j,


    r =-i∑_j=1^N'(𝐖^⊤𝐔^ℛ_j)
    (𝐔^ℒ_j^†𝐖)/Δ-λ_j,
	


where 𝐔^ℛ_j and 𝐔^ℒ_j are the right and left eigenvectors of the non-Hermitian Hamilton matrix 
𝐇, and λ_j and λ^*_j are the corresponding complex eigenvalues, satisfying 
𝐇𝐔^ℛ_j=λ_j𝐔^ℛ_j, 
𝐇^†𝐔^ℒ_j=λ^*_j𝐔^ℒ_j,
and 𝐔^ℒ_j^†𝐔^ℛ_j'=𝐔^ℛ_j^†𝐔^ℒ_j'=δ_jj' <cit.>. 

In the Markovian regime we are interested in, the phase accumulation effect caused by the detuned photons can be ignored, 
thus the quantities 𝐖, the non-Hermitian Hamiltonian 𝐇, and the corresponding eigenvalue λ_j 
and eigenvector 𝐔_j^ℒ, ℛ are independent of the detuning Δ of photons. 
The transmission and reflection amplitudes [Eqs. (<ref>) and (<ref>)] can be further written as



    t=1+∑_j=1^N'c_j/Δ-Δ̃_j+iΓ̃_j/2,


    r=∑_j=1^N'd_j/Δ-Δ̃_j+iΓ̃_j/2.
	


These results show superpositions of several Lorentzian-type amplitudes contributed by the collective excitations. 
Δ̃_j=Re(λ_j)|_Δ=0 is the detunning between the jth collective mode and the atoms, and 
Γ̃_j=-2Im(λ_j)|_Δ=0 is the effective decay of the jth collective mode. 
c_j=-i[(𝐖^†𝐔_j^ℛ)(𝐔_j^ℒ^†𝐖)]|_Δ=0 
and d_j=-i[(𝐖^⊤𝐔_j^ℛ)(𝐔_j^ℒ†𝐖)]|_Δ=0
determine the weight of each Lorentzian component.  









 Paper Title: The fermionic linear optical extent is multiplicative for 4 qubit parity eigenstates.





Center for Theoretical Physics, Polish Academy of Sciences, Warsaw

§ ABSTRACT 
The Fermionic linear optical (FLO) extent is a quantity that serves two roles, firstly it serves as a measure of the “quantumness” (or non-classicality) of quantum circuits. Secondly it controls the runtime of a class of classical simulation algorithms, which are state-of-the-art for simulating quantum circuits formed mostly of FLO unitaries and promoted to universality by the addition of “magic states”. It is therefore interesting to understand the scaling behaviour of the extent as magic states are added to a circuit. In this work we solve this problem for the case of 4-qubit parity eigenstates. We show that the FLO extent of a tensor product of any pure state and a 4 qubit parity eigenstate is the product of the extents of the two tensor factors. Applying this result recursively one proves a conjecture that the extent is multiplicative for arbitrary tensor products of 4 qubit magic states.



The fermionic linear optical extent is multiplicative for 4 qubit parity eigenstates
    Oliver Reardon-Smith
    
====================================================================================






  
Introduction.

In the last few years work on classical simulation algorithms for quantum circuits has brought attention to a quantity known as the stabilizer extent <cit.>. While this quantity serves as a measure of the non-classicality[More specifically non-stabilizerness.] of a quantum computation, of more practical interest is its role as a bound on the exponential component of the runtime of practical classical simulation algorithms <cit.>. These simulation algorithms are state-of-the-art for quantum circuits consisting primarily of Clifford gates, promoted to university by some non-Clifford gates or non-stabilizer input “magic states”. The exponential behaviour of the runtime of these simulation algorithms follows directly from the fact that the stabilizer extent is multiplicative for tensor products of magic states <cit.>. This link may be summarised in the slogan - the more quantum the computation is, the slower a classical simulation of it will be.

Stabilizer quantum mechanics is not the only efficiently classically simulable subtheory of quantum mechanics <cit.>, another well-known example is fermionic linear optics (or FLO, for brevity). FLO circuits are equivalent, under the Jordan-Wigner transformation, to those formed of nearest-neighbour matchgates <cit.>[In this work we will discuss our results in terms of matchgates for notational clarity however all of our results are valid for any qubit encoding of FLO and are not limited to the Jordan-Wigner transformation.]. In recent years attention has been brought to applications fermionic linear optics including in the context of classical shadows <cit.>, estimation of fermionic observables <cit.>, and Heisenberg limited learning of fermionic observables <cit.>. They have also been a fruitful source of complexity theoretic results including the quantum advantage scheme of Ref. <cit.>, and the unexpected result that, in stark contrast to stabilizer states, PAC learning fermionic Gaussian states is NP-hard <cit.>.

In addition to being an central model of quantum computation, fermionic linear optics naturally represents the physics of a system of spinless fermions under a quadratic Hamiltonian. The circuits our results apply to, consisting of mostly FLO gates, with the addition of non-FLO magic naturally represent the evolution of such fermions with limited interaction terms perturbing the free Hamiltionians <cit.>. Our results therefore apply to a class of operations which have already been applied in the context of quantum simulations of chemistry and many-body systems <cit.>.

These applications may involve the application of FLO or matchgate operations input states which are not themselves fermionic Gaussian states, thus escaping from the efficient simulability results of <cit.>. This motivates the generalization of the simulation algorithms of the first paragraph from stabilizer quantum mechanics to matchgate quantum mechanics. Recently this goal has been achieved for the subtheory known as fermionic linear optics (FLO) <cit.>. These works directly generalize the results known from the stabilizer case, obtaining classical simulation algorithms for circuits consisting of FLO unitaries promoted to universality by the addition of fixed parity magic states (or, equivalently, controlled rotation gates with arbitrary phases)<cit.>. The non-polynomial component of the runtimes of these algorithms is comprised of a linear dependence on the fermionic linear optical extent. Under the natural conjecture that the extent is multiplicative for magic states, just as the stabilizer extent is, this non-polynomial component is exactly the exponential scaling we expect to obtain when simulating quantum computations with a classical computer.

However, while Refs. <cit.> obtain the FLO extent for a single magic state, and show that the extent of a product of two magic states is the product of the individual extents, the behaviour of the extent for products of more magic states is unknown. In this work we remedy this situation and show that the extent of an arbitrary tensor product of n FLO magic states is simply the product of the individual extents of the tensor factors. A proof of this result was announced in Ref. <cit.> however it appears to contain an error which renders it incomplete. See Supplemental Material for details.

In fact our main result, theorem <ref> is slightly stronger than this. We show that the FLO-extent of a tensor product of an arbitrary n qubit pure state and any fixed-parity 4 qubit state is simply the product of the extents of the tensor factors. While the first side of this, the arbitrary quantum state, is a considerable generalization; the second side, the fixed parity 4 qubit state is less of a generalization than it would appear since all fixed-parity 4 qubit states are FLO magic states. It has been noted in <cit.> that multiplicativity results for the FLO fidelity imply those for the FLO-extent. Interestingly, however, while our proof of theorem <ref> does involve a multiplicativity result for the fidelity we do not need anything close to as general as theorem <ref> for the FLO fidelity. Indeed we are only able to show the fidelity is multiplicative for a specific class of maximally magic 4 qubit states. 



  
Setting. We first briefly introduce Fermionic linear optics from the point of view of the Jordan-Wigner transformation, since this embeds the state-space into that of an n-qubit system, and is hence easily understood by quantum information theorists. Label the n qubits with integers from 0 to n-1 and define the Majorana fermion operators

    c_2j   = (∏_k=0^j-1 Z_k) X_j
    
      c_2j+1   = (∏_k=0^j-1 Z_k )Y_j,

where X_i, Y_i and Z_i are the Pauli operators on the i^th qubit. The Majorana fermion operators obey the standard anticommutation relation {c_i, c_j} = 2δ_ij I and hence define fermionic creation and annihilation operators by the relations

    c_2j   = (a_j + a_j^†)
    
      c_2j+1   = -i(a_j - a_j^†).

One identifies the all-zero computational basis state with the fermionic vacuum state, and a computational basis state containing a |1⟩ on any of the qubits with a fermionic state with those modes occupied. A unitary U is called fermionic linear optical (FLO) if it obeys the relation

    U c_i U^† = ∑_j R_ij c_j,

for some real matrix R, a consequence of (<ref>) is that R is orthogonal. Fermionic Gaussian states are just those which are obtained from a computational basis state by applying a FLO unitary, we will also call these FLO states for conciseness, and call the class of n qubit FLO states FLO(n). Fermionic linear optics is a restricted model of quantum computation, but it may be promoted to universality by the addition of non-FLO magic-states <cit.>. Since the term “magic” is primarily used in Clifford/stabilizer quantum mechanics we will call the magic states we are interested in FLO magic states. From Ref. <cit.> we know that FLO magic-states must be fermionic or eigenstates of the parity operator. Since all fixed-parity states of one, two and three qubits are FLO states the first interesting case is that of four qubits. We will label the 4 qubit magic states of Ref. <cit.> as

    |M_ϕ⟩   = 1/2(|0000⟩+|0011⟩+|1100⟩ +e^iϕ|1111⟩),

in fact any fixed parity 4 qubit pure state is proportional to U|M_ϕ⟩ for a FLO unitary U and some ϕ∈ℝ, while |M_ϕ⟩ with different ϕ (mod 2π) are inequivalent by FLO unitaries. Finally we define the two resource theoretic quantities relevant in this work, are the FLO fidelity and FLO extent, definitions <ref> <ref>, respectively.
[FLO fidelity]
  For an n qubit quantum state |ψ⟩, the FLO fidelity is
  
    F_FLO(|ψ⟩)    = max_|ω⟩∈FLO(n)|⟨ω|ψ⟩|^2.


[FLO extent]
  For an n qubit quantum state |ψ⟩, the FLO extent is
  
    ξ_FLO(|ψ⟩)    = inf{a^2_1| ∑_ja_j|ω_j⟩ = |ψ⟩},

  where the states |ω_j⟩ are FLO states. Note that we do not put a restriction on the dimension of the complex vector a, other than that it is finite.

These two definitions are linked by a very useful characterisation of the extent for general resource theories in terms of the convex dual problem,

    ξ_FLO(|A⟩)    = max_|ω⟩|⟨ω|A⟩|^2/F_FLO(|ω⟩),

where the max is over all pure states |ω⟩ of appropriate dimension. See Ref. <cit.> or, for an application to the stabilizer extent, Ref. <cit.>.




  
Results. We begin with what is arguably the main technical result of our work, lemma <ref>. We leave proofs to the Supplemental Material but give a summary of the idea here. We employ the characterization of entanglement of FLO states from Ref. <cit.> to reduce the problem to one of 4 qubit systems, then apply a decomposition of each space of fixed-parity 4-qubit states into orbits under FLO unitaries due to Ref. <cit.>.

  Let
  
    |a_8⟩   = 1/√(2)(|0000⟩ + |1111⟩),

  and let |ϕ⟩ be an n qubit state of fixed parity, then
  
    F_FLO(|ϕ⟩|a_8⟩)    = F_FLO(|ϕ⟩)F_FLO(|a_8⟩)
       = F_FLO(|ϕ⟩)1/2










Here we note a simplification of the convex dual problem for the extent (<ref>) in the case of fermionic linear optics. 

  Let ℋ_± be the subspaces of the Hilbert space corresponding to positive and negative eigenvalues of the parity operator. Let x∈{+,-} and |ψ⟩∈ℋ_x then 
  
    ξ_FLO(|ψ⟩)    = max_|ω⟩∈ℋ_x|⟨ω|ψ⟩|^2/F_FLO(|ω⟩).


Combining Lemmas <ref> and <ref> with the parametrization of FLO magic states due to <cit.> one obtains lemma <ref>.

  Let |ψ⟩ be an n qubit state of fixed parity, and let |M⟩ be an even 4 qubit state. Then
  
    ξ_FLO(|ψ⟩|M⟩)    = ξ_FLO(|ψ⟩)ξ_FLO(|M⟩).



  Let |ψ⟩ be an n qubit state, and let |M⟩ be an even 4 qubit state. Then
  
    ξ_FLO(|ψ⟩|M⟩)    = ξ_FLO(|ψ⟩)ξ_FLO(|M⟩).





  
Outlook An interesting feature of our proof is that while we rely on a multiplicativity result for the fidelity to prove one for the extent, but the result for the extent we obtain is substantially more general than the one for the fidelity. We are only able to show multiplicativity for the fidelity for a relatively narrow class of states, those equivalent to |a_8⟩ under FLO unitaries. It so happens that this class contains the witness states required for the extent (the optimizing |ω⟩ in equation (<ref>)). It remains to be seen whether the FLO fidelity is multiplicative for 4 qubit fermionic states.
While the extent and fidelity are both interesting and practically relevant quantities in their own right, the ultimate theoretical runtime scaling of classical simulation algorithms such as <cit.> is given by the (approximate) rank, for which the extent provides an upper bound. Little is known about the behaviour of the FLO rank for products of FLO magic states, with the investigations in Ref. <cit.> being the state-of-the-art. In this work we have employed the results of Ref. <cit.>, which completely characterize the orbits of 4 qubit fixed parity states under FLO unitaries. It remains to be seen whether the techniques we have employed here imply relevant constraints on the FLO-rank or approximate rank.



  
Acknowledgements O. R.-S. acknowledges funding from National Science Centre, Poland under the grant OPUS: UMO2020/37/B/ST2/02478.












	Supplemental Material





§ THE “MAGIC BASIS”


In this section we will summarize the results of Refs. <cit.> which are relevant for this work. There is a “magic basis” {|η_i⟩}_i=1^8

    |η_1⟩   = 1/√(2)(|0000⟩ + |1111⟩),    |η_2⟩   = i/√(2)(|0000⟩ - |1111⟩)
    |η_3⟩   = 1/√(2)(|0011⟩ - |1100⟩),   |η_4⟩   = i/√(2)(|0011⟩ + |1100⟩)
    |η_5⟩   = 1/√(2)(|0101⟩ + |1010⟩),    |η_6⟩   = i/√(2)(|0101⟩ - |1010⟩)
    |η_7⟩   = 1/√(2)(|1001⟩ - |0110⟩),   |η_8⟩   = i/√(2)(|1001⟩ + |0110⟩),

with the property that for any even, 4-qubit FLO state |M⟩ there exists real, 8 dimensional vectors r and s such that r_2 = s_2 = 1, r· s  = 0 and

    |M⟩   ∝∑_j (cos(a)r_j + isin(a) s_j) |η_j⟩,

for some a∈ (-π,π]. We will neglect the ∝ and (without loss of generality) assume the overall phase of |M⟩ is such that (<ref>) is an equality. Note that the action of 4 qubit FLO unitaries is transitive on (r,s) pairs, in particular for any (r,s) there exists a FLO unitary U such that

    U(∑_j (cos(a)r_j + isin(a) s_j) |η_j⟩)    = cos(a)|η_1⟩ + isin(a)|η_2⟩.

The parameter a in equation (<ref>) may therefore be seen as labelling the orbits of 4-qubit FLO unitaries, since we may multiply the whole of (<ref>) by a factor of i to swap the role of the cos and sin and absorb factors of -1 into the vectors r and s, the orbits are uniquely determined if we restrict to a∈[0,π/4]. The decomposition (<ref>) for the magic states |M_ϕ⟩ takes the form

    |M_ϕ⟩   = 1/2(|0000⟩ + |0011⟩ + |1100⟩ + e^iϕ|1111⟩)
    
    e^iπ-ϕ/4|M_ϕ⟩   = cos(ϕ+π/4)( cos(p/2)|η_1⟩ - sin(p/2)|η_2⟩ + |η_4⟩) +
          i sin(ϕ+π/4)( cos(p/2)|η_1⟩ - sin(p/2)|η_2⟩ -|η_4⟩),

i.e.

    [ r_1; r_2; r_4 ]   = 1/√(2)[  cos(ϕ/2); -sin(ϕ/2);         1 ]
    [ s_1; s_2; s_4 ]   = 1/√(2)[  cos(ϕ/2); -sin(ϕ/2);        -1 ],

and the other 5 components of each vector are 0.

It is easily verified that the FLO-fidelity of a single 4 qubit even state is given by the following expression

    F_FLO(cos(a)|η_1⟩ + isin(a)|η_2⟩)    = max_r,s1/√(2)∑_j (r_j -i s_j)⟨η_j|(cos(a)|η_1⟩ + isin(a)|η_2⟩)^2
       =  1/2max_r,s∈ℝ^8 (r_1cos(a) + s_2sin(a))^2 + (r_2sin(a) - s_1cos(a))^2
       =1/2(cos(a) + sin(a))^2
       =1/2(1 + sin(2a))




§ PROOF OF LEMMA <REF>



  We will prove the result assuming the fixed parity state |A⟩ is of even parity. The proof is almost identical for |A⟩ of odd parity so we will omit it. In addition to the results of Ref. <cit.>, summarized in section <ref> we will need the main result of Ref. <cit.>. For any even-parity FLO state |ψ⟩ in a Hilbert space partitioned into subsystems A⊗ B with n = (A) ≤(B) = m , there exist FLO unitary operators U_A and U_B acting on the respective subsystems and angles {θ_j}_j=1^n such that

    |ψ⟩ = U_A⊗ U_B ∏_j=1^n (cos(θ_j)|0⟩_A_j|0⟩_B_j  + sin(θ_j)|1⟩_A_j|1⟩_B_j) |0⟩^⊗(m-n)_B,

We may assume that in addition to being FLO unitaries, the operators U_A and U_B are both even operators. A FLO unitary must be either even or odd (but not a mixture of both). First note that if one is odd, then both must be so that the left and right hand sides of (<ref>) have the same parity, then observe that if both are odd we may factor a single Majorana fermion operator (e.g. c_1 = a_1 + a_1^†) out of each to obtain two even operators. Applying these Majorana fermion operators to the states in parentheses in (<ref>) swaps the role of |0⟩_A_1 with |1⟩_A_1 and |0⟩_B_1 with |1⟩_B_1. We then bring the expression back to one of the same form as (<ref>) by changing θ_1 to swap the cos with the sin.

It will be convenient for us to define the notation

    t_θ(y)    = ∏_j=1^ncos(θ_i)^1-y_isin(θ_i)^y_i,

for a binary vector y. Multiplying out the brackets in equation (<ref>) we obtain

    |ψ⟩ = U_A⊗ U_B ∑_y∈{0,1}^n t_θ(y)|y⟩_A|y⟩_B|0⟩^⊗(m-n),

which we may substitute directly into the expression for the fidelity

    F_FLO(|a_8⟩|ϕ⟩)      = max_U_A, U_B, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩⟨ϕ|U_B |y⟩|0⟩^⊗(m-4)^2.

At this point it is convenient to consider instead √(F_FLO(|a_8⟩|ψ⟩)), we are going to apply the triangle inequality and then Hölder's inequality, with p=1, q=∞

    √(F_FLO(|M⟩|ψ⟩))   = max_U_A, U_B, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩⟨ψ|U_B |y⟩|0⟩^⊗(m-4)
       ≤max_U_A, U_B, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩⟨ψ|U_B |y⟩|0⟩^⊗(m-4)
       ≤max_U_A, U_B, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩max_y ⟨ψ|U_B |y⟩|0⟩^⊗(m-4)
       = max_U_A, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩max_U_B⟨ψ|U_B |0⟩^⊗ m
       = max_U_A, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩√(F_FLO(|ψ⟩)).

Having recognised the second term as √(F_FLO(|ψ⟩)) we seek to bound the first term. We recall that the FLO-fidelity of a_8 is 1/2 so we are attempting to show the first term is less than 1/√(2). We introduce the real, 8 dimensional, 2-norm normalized vector r such that and

    U_A|M⟩ = ∑_j r_j|η_j⟩.


    max_U_A, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩ = max_r, θ   ∑_y∈{0,1}^4t_θ(y) ∑_j r_j⟨η_j|y⟩
    
                                                                                 = max_r, θ   (t_θ(0000)+t_θ(1111))1/√(2)r_1 + ir_2 +
       (t_θ(1100)+t_θ(0011))1/√(2)r_3 + ir_4 +
       (t_θ(1010)+t_θ(0101))1/√(2)r_5 + ir_6 +
       (t_θ(1001)+t_θ(0110))1/√(2)r_7 + ir_8.

Defining two 4 dimensional vectors

    τ(θ)    = [ t_θ(0000)+t_θ(1111); t_θ(1100)+t_θ(0011); t_θ(1010)+t_θ(0101); t_θ(1001)+t_θ(0110) ],
    ρ   = 1/√(2)[  r_1 + ir_2;  r_3 + ir_4;  r_5 + ir_6; r_7 + i r_8 ],

we observe that

    ρ_2^2 = 1/2∑_j r_j^2 = 1/2r_2^2 = 1/2,

and

    τ(θ)_2^2    = 1/2(1 + sin(2θ_1)sin(2θ_2)sin(2θ_3)sin(2θ_4) + cos(2θ_1)cos(2θ_2)cos(2θ_3)cos(2θ_4) )
       ≤max_θ1/2(1 + sin(2θ_1)sin(2θ_2)sin(2θ_3)sin(2θ_4) + cos(2θ_1)cos(2θ_2)cos(2θ_3)cos(2θ_4) )
       ≤max_θ1/2(1 + sin(2θ_1)sin(2θ_2)+ cos(2θ_1)cos(2θ_2))
       ≤max_θ1/2(1 + cos(2θ_1 - 2θ_2))
       ≤ 1.

The expression (<ref>) for τ(θ)_2^2 may be demonstrated by a tedious calculation, while that for ρ_2^2 is a consequence of the fact that r is a real vector so

    r_2j-1 + i r_2j^2 = r_2j-1^2 + r_2j^2.

Applying Hölder's inequality to equation (<ref>), this time with p=q=2 we obtain

    max_U_A, θ∑_y∈{0,1}^4t_θ(y) ⟨a_8|U_A|y⟩≤1/√(2),

this value is achieved by the choice of parameters θ_1 = θ_2 = θ_3 = θ_4 = 0, U_A = I. We also note that if |ω⟩∈FLO is the witness state achieving

    ⟨ω|ψ⟩^2 = F_FLO(|ψ⟩),

then we have the obvious witness state |0⟩|ω⟩ achieving

    ⟨0|⟨ω||a_8⟩|ψ⟩^2 = 1/2F_FLO(|ψ⟩),
  





§ PROOF OF THEOREM <REF>























We first prove lemma <ref>.

  As mentioned in the main text the following characterisation of the extent is well known
    
    ξ_FLO(|ψ⟩)    = max_|ω⟩|⟨ω|ψ⟩|^2/F_FLO(|ω⟩),

    where the max is over all pure states |ω⟩ of appropriate dimension. Take a witness state |ω⟩ and split it into even and odd parts
    
    |ω⟩   = a_+|ω_+⟩ + a_-|ω_-⟩.

    Now because |ψ⟩∈ℋ_x and FLO states have fixed parity we have
    
    ξ_FLO(|ψ⟩)    = max_|ω⟩a_x^2|⟨ω_x|ψ⟩|^2/max_y∈{+1,-1}a_y^2 F_FLO(|ω_y⟩).

    If we have
    
    a_x^2 F_FLO(|ω_x⟩) ≤a_x̅^2 F_FLO(|ω_x̅⟩),

    where x̅ = -x, then reducing a_x̅^2 and increasing a_x^2 to maintain normalisation will only increase the overall max the above equation. We may therefore assume that the max in the denominator is given by a_x^2 F_FLO(|ω_x⟩) so the overall expression is
    
    ξ_FLO(|ψ⟩)    = max_|ω⟩a_x^2|⟨ω_x|ψ⟩|^2/a_x^2 F_FLO(|ω_x⟩)
       =max_|ω⟩∈ℋ_x|⟨ω|ψ⟩|^2/F_FLO(|ω⟩).



The proof of lemma <ref> follows easily from the characterization given in lemma <ref>, combined with the decomposition (<ref>).

  First note that the extent is trivially submultiplicative so
  
    ξ_FLO(|ψ⟩|M⟩)    ≤ξ_FLO(|ψ⟩)ξ_FLO(|M⟩).
 
  We use the characterization of the extent in terms of the fidelity
  
    ξ_FLO(|ψ⟩|M⟩)    = max_|ω⟩|⟨ω|ψ⊗ M⟩|^2/F_FLO(|ω⟩).

  Clearly restricting the max to be over a smaller set of ω can not increase the value so we obtain the inequality
  
    ξ_FLO(|ψ⟩|M⟩)    ≥max_|ω_1⟩, |ω_2⟩|⟨ω_1|ψ⟩|^2|⟨ω_2|M⟩|^2/F_FLO(|ω_1⟩|ω_2⟩)
       =max_|ω_1⟩|⟨ω_1|ψ⟩|^2/F_FLO(|ω_1⟩)max_|ω_2⟩|⟨ω_2|M⟩|^2/F_FLO(|ω_2⟩)

  where |ω_1⟩ has fixed parity and |ω_2⟩ is a 4 qubit even state of minimal FLO fidelity, i.e. |ω_2⟩ =1/√(2) U(|0000⟩ + |1111⟩) for some FLO unitary U, and the equality is an application of lemma <ref>. Applying lemma <ref> shows that the first term is just the FLO-extent of |ψ⟩. Recalling the parametrization from equation <ref>, it remains only to show
  
    ξ_FLO(|M⟩)    = max_r|∑_jr_j⟨η_j|M⟩|^2/F_FLO(∑_jr_k|η_j⟩).

  Without loss of generality assume
  
    |M⟩   = cos(a)|η_1⟩ + isin(a)|η_2⟩,

  then
  
    max_|ω_2⟩|⟨ω_2|M⟩|^2/F_FLO(|ω_2⟩)   = 2 max_r| r_1cos(a) + ir_2sin(a) |^2
       = 2 max{cos^2(a), sin^2(a) }
       =1+cos(2a),

  recalling the decomposition (<ref>) we have that 
  
    max_|ω_2⟩|⟨ω_2|M⟩|^2/F_FLO(|ω_2⟩)   = 1+sin(ϕ/2)
       =ξ_FLO(|M_ϕ⟩).



We now have all the ingredients required to prove theorem <ref>

  If |ψ⟩ is a state of mixed parity it has a unique decomposition into orthogonal even and odd parts (this follows, for example, from the eigen-decomposition of the parity operator). Since all FLO states have fixed parity the only possible decompositions of |ψ⟩ in terms of FLO states are formed by independently decomposing the even and odd parts. Let |ψ⟩ = a|ψ_e⟩ + b|ψ_o⟩ be the orthogal decomposition into even and odd parts respectively and let

    |ψ_e⟩   = ∑_j a_j|x_j⟩
    |ψ_o⟩   = ∑_j b_j|y_j⟩,

be the extent optimal decompositions of the two parts. Then

    ξ(|ψ⟩)    = (a∑_ja_j + b∑_jb_j)^2
       = (a∑_ja_j+ b∑_jb_j)^2
       = (a√(ξ(|ψ_e⟩)) + b√(ξ(|ψ_o⟩)))^2.

Since |M⟩ has fixed parity the parity-eigenspace decomposition of |ψ⟩⊗|M⟩ is given by that of |ψ⟩

    |ψ⟩|M⟩ = a|ψ_e⟩|M⟩ + b|ψ_o⟩|M⟩,

therefore

    ξ(|ψ⟩|M⟩)    = ξ(a|ψ_e⟩|M⟩ + b|ψ_o⟩|M⟩)
       = (a√(ξ(|ψ_e⟩|M⟩)) + b√(ξ(|ψ_o⟩|M⟩)))^2
       = (a√(ξ(|ψ_e⟩)ξ(|M⟩)) + b√(ξ(|ψ_o⟩)ξ(|M⟩)))^2
       = (a√(ξ(|ψ_e⟩)) + b√(ξ(|ψ_o⟩)))^2ξ(|M⟩)
       = ξ(|ψ⟩)ξ(|M⟩).







  If |ψ⟩ is a state of mixed parity it has a unique decomposition into orthogonal even and odd parts (this follows, for example, from the eigen-decomposition of the parity operator). Since all FLO states have fixed parity the only possible decompositions of |ψ⟩ in terms of FLO states are formed by independently decomposing the even and odd parts. Let |ψ⟩ = a|ψ_e⟩ + b|ψ_o⟩ be the orthogonal decomposition into even and odd parts respectively and let

    |ψ_e⟩   = ∑_j a_j|x_j⟩
    |ψ_o⟩   = ∑_j b_j|y_j⟩,

be the extent optimal decompositions of the two parts. Then

    ξ(|ψ⟩)    = (a∑_ja_j + b∑_jb_j)^2
       = (a∑_ja_j+ b∑_jb_j)^2
       = (a√(ξ(|ψ_e⟩)) + b√(ξ(|ψ_o⟩)))^2.

Since |M⟩ has positive parity the parity-eigenspace decomposition of |ψ⟩⊗|M⟩ is given by that of |ψ⟩

    |ψ⟩|M⟩ = a|ψ_e⟩|M⟩ + b|ψ_o⟩|M⟩,

therefore

    ξ(|ψ⟩|M⟩)    = ξ(a|ψ_e⟩|M⟩ + b|ψ_o⟩|M⟩)
       = (a√(ξ(|ψ_e⟩|M⟩)) + b√(ξ(|ψ_o⟩|M⟩)))^2
       = (a√(ξ(|ψ_e⟩)ξ(|M⟩)) + b√(ξ(|ψ_o⟩)ξ(|M⟩)))^2
       = (a√(ξ(|ψ_e⟩)) + b√(ξ(|ψ_o⟩)))^2ξ(|M⟩)
       = ξ(|ψ⟩)ξ(|M⟩).







§ THE APPARENT ISSUE IN THE PROOF OF CUDBY AND STRELCHUK 2023


The approach of Ref. <cit.> is to attempt to cast the problem of computing the FLO fidelity of a product of un-normalized magic states

    |M_4⟩^⊗ k   = (|0000⟩ + |1111⟩)^⊗ k,

as a semidefinite program, and use the powerful duality theory of such programs. Unfortunately this is complicated for the following reason. The semidefinite program must include two sets of constraints, the state being optimized over must first be a valid density operator (i.e. both positive-semidefinite and normalized) and secondly must be a fermionic Gaussian state. The first set of constraints is easily expressed as a semidefinite program in the density operator

    ρ = |ψ⟩⟨ψ|,

while the second set of constraints is instead expressed as a homogeneous quadratic in the computational-basis coefficients of |ψ⟩, i.e. linear constraints in the operator

    Z_ij = ⟨i|ψ⟩⟨j|ψ⟩.

Since

    ρ_ij = ⟨i|ψ⟩⟨ψ|j⟩≠ Z_ij,

there is not an obvious way to express the two sets of constraints simultaneously in a semidefinite program. In Ref. <cit.> appendix C. There appears to be an error where the complex conjugate which is the difference between Z_ij and ρ_ij is lost, specifically in the currently version of the manuscript (v3) on arXiv, the semidefinite program given in equation (51) is incorrect.



 Paper Title: Silicon Double-Disk Optomechanical Resonators from Wafer-Scale Double-Layered Silicon-on-Insulator.



SOISilicon-on-Insulator

WGMWhispering Gallery Mode
EBPGElectron-Beam Pattern Generator
EBLElectron-Beam Lithography
RIEReactive-Ion Etching
IPAisopropyl alcohol
HFhydroflouric acid
VPEVapor Phase Etching
FSRFree Spectral Range
SNRSignal-to-Noise Ratio
PECProximity Effect Correction
SEMScanning Electron Microscope
FEMFinite Element Method
Sisilicon
SiO_2silica
PDphotodector
PSDPower Spectrum Density
FWHMFull Width at Half Maximum
DRIEDeep Reactive Ion Etching
BOXburied oxide
PCPolarization Controller
HWHMHalf Width at Half Maximum





1School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia.

2ARC Centre of Excellence for Engineered Quantum Systems, St Lucia, Queensland 4072, Australia.

3Centre for Quantum Dynamics, Griffith University, Nathan, Queensland 4222, Australia.

4ARC Centre of Excellence in Quantum Biotechnology, St. Lucia, Queensland 4072, Australia

*w.bowen@uq.edu.au 



 
WGM optomechanical resonators are a promising technology for the simultaneous control and measurement of optical and mechanical degrees of freedom at the nanoscale. They offer potential for use across a wide range of applications such as sensors and quantum transducers. Double-disk WGM resonators, which host strongly interacting mechanical and optical modes co-localized around their circumference, are particularly attractive due to their high optomechanical coupling. Large-scale integrated fabrication of silicon double-disk WGM resonators has not previously been demonstrated. In this work we present a process for the fabrication of double-layer silicon-on-insulator wafers, which we then use to fabricate functional optomechanical double silicon disk resonators with on-chip optical coupling. The integrated devices present an experimentally observed optical quality factors of the order of 10^5 and a single-photon optomechanical coupling of approximately 15 kHz.





§ INTRODUCTION

	
Optomechanical WGM resonators have applications in many fields including precision sensors <cit.>, biosensors <cit.>, memory <cit.>, communications <cit.> and for quantum information devices <cit.>. Such resonators support optical modes confined to their periphery by the refractive index contrast at their outer boundary. They also support mechanical degrees of freedom that strongly overlap spatially with the confined optical field <cit.>, leading to significant optomechanical coupling. WGM optomechanical resonators have been experimentally demonstrated in structures such as microspheres <cit.>, microtoroids <cit.>, (single) microdisks <cit.>, and double-disks <cit.>.

A double-disk WGM resonator is made by fabricating two coaxial disks separated by a narrow air gap. Evanescent coupling between the two disks results in hybridization of the WGMs, forming modes that have a high field intensity in the gap<cit.>. This is illustrated in Fig. <ref>(a). Modulation of the air gap by out-of-plane mechanical motion results in large optomechanical coupling; particularly for the symmetrical out-of-plane oscillation (as shown in Fig. <ref>(b)). This is compounded by the tendency for out-of-plane deformations to be much larger than in-plane deformations (of equal energy), due to bending typically requiring less energy than stretching or compressing a material <cit.>. As such, double-disk systems achieve some of the largest optomechanical coupling rates in WGM optomechanical devices demonstrated to date <cit.>.

The optomechanical coupling can also be increased by improving the confinement of the optical mode and ensuring strong co-localization of the optical and mechanical modes<cit.>.
Both objectives can be achieved by moving to smaller resonators fabricated from materials with higher refractive indices. Double-disk optomechanical resonators have previously been fabricated from materials such as silica (SiO_2, n≈ 1.44) <cit.>,  silicon nitride (Si_3N_4, n≈ 1.98) <cit.>, and lithium niobate (LiNO_3, n≈ 2.21)    <cit.>. There is much interest in developing Si optomechanical devices for many applications <cit.>. Silicon's larger refractive index (n=3.48) permits tighter confinement of optical modes and hence higher coupling rates and further miniaturization of devices. This potential is further bolstered by silicon's compatibility with electronics fabrication processes and operation with light in the telecommunications C-band, leading to opportunities for integration with a range of technologies and systems. In particular, Si optomechanical resonators present ready compatibility with silicon photonic, semiconducting, and superconducting circuits. Further, recent studies have demonstrated control of defect-induced single photon emissions in silicon substrates <cit.> offering the potential for integration with quantum information systems utilizing such photon sources. 

Previous reports have demonstrated chip-scale bonding to produce high-purity Si double-layers with the necessary oxide spacing layer (∼ 100 nm) for production of free-standing Si double-disks <cit.> or oxide-clad ring resonators <cit.>. However, due to the complicated nature of their production, demonstration of large-scale Si double-disk WGM resonator fabrication and experimental investigations of their optomechanical performance remained elusive.

In this work we present a truly wafer-scale approach for double layer SOI substrate fabrication, with subsequent chip-scale fabrication and characterization of free-standing double-silicon-disk WGM optomechanical resonators. We achieve this by bonding two entire 4-inch SOI wafers prior to device fabrication. This allows for both chip- and wafer-scale double-layer SOI photonics, which are compatible with the aforementioned technologies. Further, we present FEM modeling and comparison to experimental results of the fabricated devices. We observe optical and mechanical resonances of the disks under vacuum, and characterize the single-photon optomechanical coupling rate (g_0) through measurements of the optomechanical spring effect.




§ DEVICE DESIGN AND FABRICATION


As described above, double-disk resonators consist of two coaxial Si microdisks separated vertically by a SiO_2 sandwich layer of thickness z_g, acting as a spacer. This SiO_2 layer is selectively etched around the periphery, resulting in an air gap between the disks. When the gap is sufficiently small, evanescent coupling results in the emergence of a hybridized optical mode that exists in both disks and the gap simultaneously; this is depicted in Fig. <ref>(a). As expected, our optical FEM simulation (COMSOL Multiphysics) shows that the fundamental (transverse) WGMs are confined to the outer ∼ 2 m of the disks, which is commensurate with the region of greatest deformation caused by mechanical modes. The fundamental symmetric flapping mode is shown in Fig. <ref>(b).

To characterize the impact of the motion on the optical WGM, we first consider the effective refractive index (n_eff), which can be determined from the weighted integral method<cit.>:

    n_eff = (∫_V E(r⃗) n^2(r⃗)  dV/∫_V E(r⃗)  dV)^1/2.

Here, E and n are the electric field amplitude and material refractive index, respectively, at position r⃗ within the simulation volume V. The effective index of the WGMs with varying disk separation is presented in Fig. <ref>(c). When there is no air-gap (i.e. z_g = 0) the WGM behaves as if contained within a single 400-nm disk, with the field well-confined inside the Si; thus, n_eff approaches n_Si. Conversely, when the gap is very large (e.g. z_g ≥ 300 nm) the mode resembles two independent WGMs with negligible shared field (see Fig. <ref>(c) insets). The effective refractive index is lower than that of the 0-nm gap because a larger fraction of the optical field extends evanescently beyond the Si.

The angular resonance frequencies of an optical WGM (ω_m) with a radius of r are approximately given by ω_m = mc/r n_eff, where m is the mode number associated with the number of wavelengths within the cavity and c is the speed of light in vacuum. The dependence of ω_m on n_eff, which is modulated by mechanical motion, ensures that changes in the disk separation will result in a shift in optical resonance frequency. For small deformations, the dependence can be summarised by the optomechanical coupling  (we suppress the subscript for the mth mode),

    G_OM = δω_m/δ z_g.

A high G_OM value is advantageous for applications such as sensing and transduction <cit.>. Our FEM model is presented in Fig. <ref>(d). Here, it can be seen that the optomechanical coupling diminishes with increasing separation. The maximum coupling rate occurs as z_g approaches zero; however, this regime is experimentally impracticable due to large van der Waal's and Casimir forces that can promote collapse of the disks, effectively destroying the mechanical modes of interest. For these reasons, and to ensure fabrication consistency, a separation of 60 nm was chosen. With a disk separation of 60 nm we predict an optomechanical coupling of G_OM/2π≈150 GHz/nm, which is among the highest reported in optomechanical cavities <cit.>.



A second key figure of merit is the single-photon optomechanical coupling rate, which 
is given by:


    g_0 = x_zpfG_OM,


where x_zpf = √(ħ/2m_effΩ) is the zero point motion of the oscillation with resonance frequency of Ω/2π, ħ is the reduced Planck's constant, and

    m_eff =∫_Vρ(r⃗) |X⃗(r⃗)|^2/| X⃗_max|^2  dV
 
is the effective mass of the mechanical oscillation. Here ρ is the material's density, X⃗ is the displacement, and X⃗_max is the maximum displacement amplitude of the excitation. The single-photon coupling rate is equal to the frequency shift per zero-point displacement of the oscillator, and also determines the optical force per photon <cit.>.

Using the data extracted from both optical and mechanical FEM simulations (see S.1 for the mechanical FEM), a 120 m radius Si double–disk resonator with an air gap of 60 nm and undercut of 3.35 m yields a single photon coupling rate of g_0/2π = 116 kHz. 



 §.§ Further design considerations


The design of our double-disk devices broadly follows the concepts of those presented in <cit.>. However, unlike previous reports—which demonstrated only chip-scale double-layer SOI—we present a wafer-scale fabrication process utilizing two whole wafers of commercially available SOI (see Sec. <ref> & S.2.1). Furthermore, unlike many previous works that have coupled light directly from tapered optical fibers into the WGM cavity <cit.>, our work utilizes coupling via integrated on-chip waveguides, enabling superior flexibility and scalability.

Our wafer-scale double-layer SOI approach allows for rapid production of many devices, allowing for a range of devices of varying radii and waveguide-to-disk coupling distance to be fabricated and explored on the same chip. The radii of our devices range from 15 m to 120 m. A minimum radius of 15 m is used to ensure radiative optical losses in the devices are negligible <cit.>. This range of radii presents free spectral ranges (FSRs) in the range of 1–7 nm, allowing for expedition of experimental probing.

The coupling distance ranges from 100–400 nm. The on-chip waveguides are 500 nm wide, which ensures single mode operation around 1550 nm wavelength. The waveguides' two right-angled bends form an S-shaped configuration, minimizing the potential for stray light (e.g., light reflected off the substrate) to couple into the output. The waveguides are tapered down to 100 nm over the course of 50 m, allowing for mode-matched coupling to and from tapered optical fiber tips in a similar manner to that presented in <cit.>. Thin tethers along the length of the waveguides minimize buckling due to stress in the Si films. These tethers are narrower than the waveguide to suppress their scattering cross-section.





§ FABRICATION







 §.§ Double-layer silicon-on-insulator wafer fabrication


Wafer-scale fabrication of double-disk devices requires double-layer SOI wafers. However, such wafers are not commercially available and so were produced in-house in a similar—though scaled-up—manner to that which is presented in <cit.>. Here SOI wafers (4 inch) consisting of a 215-nm upper layer of crystalline Si, a 3-m buried oxide (SiO_2) layer, and a 750-m Si carrier layer were purchased from WaferPro. The wafers were thermally oxidized in a dry oxidation furnace (HiTech benchtop oxidation furnace) at 900^∘C for two hours, forming an SiO_2 layer at the surface of the wafers. This oxide layer forms one half of the spacer layer in the final double-disk devices, as depicted in Fig. <ref>(b & c).  Thin-film reflectometry thickness analysis (SCI FilmTek 2000m) revealed an oxide thickness of 30 nm and underlying silicon layer thickness reduced to 200 nm. This increase in overall thickness is consistent with prior work <cit.>, and occurs because of silica's lower density than Si. The oxidized SOI wafers were then inspected for cleanliness with darkfield microscopy; pristine surfaces are a requirement for successful wafer bonding. After determining the surfaces were free of contaminants, the wafers were surface activated by exposure to oxygen plasma (200 sccm, 50 W, Oxford Instruments PlasmaPro80). The bonding process was then performed using an EVG520 IS Hot Embosser. The process consists of two pistons being pushed together. Mounted onto the top piston is a piece of quartz and the bottom piston is flat steel. When the process starts the temperature is slowly ramped up to 600^∘C and a force of 3.1 kN is applied. The high temperature reduces the buildup of stress between the wafers, and increases the bonding strength <cit.>. Bond strength was then assessed by a razor blade test, as described by <cit.>.

Following the bonding, one side of the stack is exposed to HF VPE in order to remove any SiO_2 grown on the back of the wafer during the thermal oxidation process (as the underside of the wafers were also exposed in the oxidation furnace).
Afterwards, DRIE (Plasma-Therm Versaline DRIE) is used to remove the 750-m silicon handle layer on one side of the bonded stack. Here a two-step Bosch process is used, consisting of C_4F_8 gas, which forms a passivating layer, followed by an isotropic Si etch with SF_6. This process was repeated and the progress of the etch was monitored periodically by removing the wafer stack and visually inspecting. Photographs of the etch progression throughout the DRIE process can be found in the Supplementary Material (Sec. S.2, Fig. S3), showing non-homogeneous removal of material. As a result, a final liquid HF etch, used to remove the 3-m BOX layer that is now partially exposed, leaves approximately 60% of the wafer area usable for fabrication of devices. Finally, the double-layer SOI substrate was diced into chips of 20 mm× 20 mm for the subsequent fabrication of optomechanical silicon double-disks.



 §.§ Device fabrication


The double-disks are fabricated using standard EBL and RIE. A schematic of the fabrication process is depicted in Fig. S4. 


Electron-resist AR-P 6200.13 (CSAR, AllResist) was spincoated  onto the double-layer SOI chips (1500 RPM), followed by 180^∘C bake producing a ∼580-nm coating (confirmed with thin-film reflectometry). As the total thickness of material that needs to be etched is relatively thick (2×200 nm Si and 60 nm SiO_2), and the selectivity of the RIE is quite low (approaching 1:1) a thicker resist than typical SOI fabrication was found to be desirable to protect the top device layer. To compensate for the thicker resist and heterogeneous stack, electron scattering simulations and resultant PEC were performed with GenISys Tracer and GenISys Beamer software respectively. The chips were then patterned with a Raith EBPG-5150 electron beam pattern generator (100 kV accelerating voltage, beam current 1–100 nA) for a total dosage of approximately 300 C/cm^2, followed by developing  with AR600-546 developer, O-xylene and IPA for 90, 5, and 15 seconds respectively.

After patterning, the chips were then etched with RIE in a three-stage process to remove the Si, SiO_2 and Si device layers sequentially. Here, the Si etches consist of SF_6/CHF_3 20/35 sccm (82 W RF power) for 450 s, and SiO_2 etch of CHF_3/Ar 25/45 sccm (200 W) for 120 s.
Afterwards, any residual resist was removed by soaking the chips in Remover-PG at 60^∘C until the resist was visibly removed upon inspection with an optical microscope. The final step in the fabrication process is release of the design with HF VPE (at 35^∘C).



The devices were inspected with optical microscopy, due to concerns surrounding degradation of their optical Q-factors when subjected to scanning electron microscopy (SEM). Any devices imaged with SEM were not used for subsequent measurements. SEM pictures of a fabricated double-disk device and waveguide are included in Fig. <ref>(b–d). The pictures are taken at an oblique angle of 45^∘. The high contrast between the edges of the waveguides and disks indicate the existence of an air gap.




§ DEVICE CHARACTERIZATION






 §.§ Optical properties of Si double-disk resonators


Optical transmission spectra were utilized to confirm the functionality of the waveguides and access the optical interaction with the double-disk resonators. For this, a fiber-coupled laser (EXFO T100S-HP model CU) was connected to a single sided taper, which was aligned with the waveguide using a 6-axis positioning stage. Another single sided taper was similarly aligned on the other end of the S-bend waveguide and connects to a PD (New focus 1811-FC-AC, Newport Inc.). The PD is connected to an oscilloscope (Tektronix DPO 3034 Digital Phosphor Oscilloscope), a computer (through a National Instruments Data Acquisition (DAQ) board) and a spectrum analyser (Liquid instruments, Moku:Pro), as depicted in Supplementary Fig. S6. The emission wavelength of the laser was swept whilst monitoring the transmitted intensity, as is presented in Fig. <ref>. Here, a high density of sharp dips are observed from the spectrum. These correspond to a FSR of ∼ 2.8 nm.
The FSR  (Δλ) can be approximated as 

    Δλ≈λ^2/2π r n_eff,

which for an r = 45-m,  z_g = 60-nm double-disk  results in an FSR of 2.88 nm, in good agreement with experiment.

The fundamental WGMs of the double-disks are accompanied by higher order resonances, as observed from the additional families of sharp dips with similar FSRs. The broader features of the transmission spectrum are attributed to the polarization-dependent coupling efficiencies between the fiber tapers and on-chip waveguides <cit.>. These can be compensated for by manipulating the PC. However, it is evident from Fig. <ref> (inset) that these polarization effects are negligible over the  line-width of individual resonance dips, so we will neglect them in the following characterization.


 

Characterization of the optomechanical interaction was performed using a slow wavelength sweep with the devices held in vacuum (see section <ref> for details). An exemplar transmission trace is shown in Fig. <ref>. The asymmetrical lineshape is unexpected, as most WGMs exhibit symmetrical Lorentzian profiles <cit.>. This asymmetry is independent of the PC, optical power, and sweep rate, which suggests that it is not caused by thermal broadening <cit.>. Rather, the asymmetry appears to be due to interference between the high-Q optical mode of the resonator and low-Q optical resonances supported by the waveguide, i.e., a Fano resonance. The Fano-type lineshape is given by <cit.>:


    T(ω) = T_0+(T_∞-T_0)(q-ϵ)^2/1+ϵ^2,


where T_0 and T_∞ are the transmissions at resonance and far off-resonance respectively, q is the Fano asymmetry parameter, and ϵ = 2Δ/κ is the detuning Δ = ω_0-ω_L normalised by the cavity (amplitude) loss rate, κ/2. Here, ω_0 is the cavity resonance frequency and ω_L is the laser frequency.
This lineshape was found to be in excellent agreement with the measured data (Fig. <ref>(b)). The extracted energy decay rate of the cavity is κ = 2π× 1.71 GHz, which is in good agreement to the estimate based upon the the FWHM of the transmission dip. This yields an optical Q-factor of (Q = ω_0/κ) ∼ 2×10^5,
which is comparable to reports of other double-disk photonic resonators (which range from 2.5 × 10^4<cit.> to ∼10^6<cit.> for Si and SiN devices respectively). The fitted Fano parameter is q=+0.1744.



 §.§ Optomechanical properties of Si double-disk resonators


Initial investigations of the mechanical resonances of the disks were performed by detuning the laser around the HWHM point (sideband detuning) of one optical mode and monitoring the PSD of the transmitted light with a spectrum analyzer. The mechanical displacement is transduced into intensity fluctuations of the detected signal. The resulting power spectrum shows, as expected, that squeeze-film damping greatly degrades the mechanical Q-factor under ambient pressures, resulting in mechanical Q-factors of less than 10 (see S.3.1, & Fig. S7). This is consistent with prior works <cit.>. Despite this, we also see the existence of a mechanical resonance at 24 MHz, as predicted by our FEM simulation for double-disks with an undercut of ∼3.35m (Fig. S1).

To overcome the effects of squeeze-film damping and study the intrinsic optomechanical properties of the disks, further measurements were performed under vacuum (∼ 10^-6 mbar). The vacuum apparatus is depicted in Fig. <ref>. Here, a free-space laser (New Focus 6300 High-Power Velocity Tunable Diode Laser) is coupled into a single-mode fiber. A 99:1 beamsplitter is used to monitor the input power during the experiment prior to insertion into the chamber, and the output fiber is connected to a  PD (New Focus 1811-FC-AC, Newport Inc.). An oscilloscope (Tektronix, DPO3034) is used for data acquisition.

Single-sided tapered fibers are inserted into the vacuum chamber using feedthroughs. The fiber tips are glued on aluminum supports and mounted on positioning stages (SmarAct Linear Stages). Two windows (top and side view) allow for monitoring of the coupling of the fiber tips to the waveguide. 
Fig. <ref> shows the vacuum chamber experimental set-up with a mounted chip. The chamber is pumped down with a roughing-pump-backed turbomolecular pump.

 

Fig. <ref>(a) depicts the PSD of the transmitted signal when sideband-detuned (with detuning Δ = -0.744 GHz). Here, what was a very broad and nondescript feature in the spectrum at ambient pressure is revealed to be four distinct mechanical modes. Fitting Lorentzian lineshapes to each peak shows that the mechanical Q-factors of the four modes are on the order of ∼10^3, which is comparable to previous reports on silica double-disks in vacuum conditions <cit.>; this is, however, the first measurement of silicon double-disk resonators under vacuum. The complete fit model and parameters are summarised in S.3.2 and Table S.1.



Determining the optomechanical coupling rate is less straightforward, but can be achieved by exploiting the optical spring effect, wherein a detuned laser drive causes a shift of the mechanical resonance frequency <cit.>. The expected shift of angular frequency δΩ_i is given by


    δΩ_i/Ω_i =  [ 1 - C_i(0)/Q_i·2ϵ/(1+ϵ^2)^2]^1/2-1,


where Q_i is the zero-detuning mechanical Q-factor[We use the optomechanically shifted Q-factor in the calculation of C, which introduces an error on the order of 0.5%. This is far less than the uncertainty in the fit, and therefore negligible. See S.3.2 for details.], and κ is the optical linewidth (as discussed in section <ref>). We have introduced the (resonant) optomechanical cooperativity, C_i(0), which is related to the single-photon optomechanical coupling rate by


    g_0 = √(C_i(0) κΓ_i/4n(0)),


where n(Δ) is the number of intracavity photons when the cavity is pumped at detuning Δ. Thus, we can recover g_0 by fitting multiple mechanical spectra recorded at different detunings.

Detuning-dependent power spectra are shown in Fig. <ref>(b). The corresponding optical spring effect fittings (Eq. <ref>) are shown in Fig. <ref>(c), with numerical values quoted in Table <ref>. Two of these modes (1 and 4) achieve cooperativities greater than unity, which is an important threshold for onset of instability in optomechanical amplification <cit.>.

To convert C_i(0) to g_0 we require the resonant intracavity photon number (n(0) ≃ 5 × 10^4), which can be calculated from the approximate waveguide photon number and the coupling strength into the cavity (see S.3.3 & S.3.4 for details). We find that the optomechanical coupling rates lie between g_0/2π = 7.78 kHz and g_0/2π = 15.47 kHz for the modes presented in this study (see Table <ref>). These are consistent with reports for hybridized silica WGM resonators such as those presented in <cit.>. Note that we were unable to observe optical resonances in our low-radius devices (likely due to fabrication imperfections), which were expected to have the largest optomechanical coupling rates. Our devices can therefore be further optimized to improve these g_0 values beyond those obtained in silica. The maximum experimentally determined value of g_0 is ≈ 1/8 of the expected valued based on the FEM (Sec. (<ref>)), which is reasonably congruent. The disparity likely resulting from fabrication inconsistencies or imperfections.
 





§ CONCLUSION

We have demonstrated the first wafer-scale fabrication of double-layer silicon-on-insulator wafers. These custom wafers were then utilized in the fabrication of freestanding Si double-disk WGM optomechanical resonators with associated on-chip silicon waveguides, enabling integration with an enormous variety of existing Si-based technologies. 

Further, we presented the first observation of the optomechanical spring-effect in silicon double-disks. Four mechanical modes were identified that show a strong change in mechanical resonance frequency as a function of detuning. The optomechanical cooperativity for each mode is extracted from fitting to the experimental data. Using this, we determine the single-photon coupling rates in these devices.

Future devices could be improved by increasing the undercut between the disks and reducing the disks' diameters. Increasing the undercut makes the mechanical modes less sensitive to fabrication imperfections, and also tends to improve the mechanical quality factor. Reducing the disk radius is also predicted to lead to higher optomechanical coupling rates.

Further research with these devices could involve the integration of these devices with a wide range of applications including single-photon defect emitters, opto-electronics or microwave resonators which could prove useful for the fields of quantum computing and quantum state transfer. 











Funding
This work was financially supported by the Australian Research Council (ARC) Centres of excellence for Engineered Quantum systems (EQUS, Grant No. CE170100009) and Quantum Biotechnology (QUBIC, Grant No. CE230100021). This project was in part funded under the Next Generation Technologies Fund, which is now administered through the Advanced Strategic Capabilities Accelerator.


Acknowledgments
The authors acknowledge the facilities, and the scientific and technical assistance, of the Australian Microscopy & Microanalysis Research Facility at the Centre for Microscopy and Microanalysis, The University of Queensland. This work was performed in part at the Queensland node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano- and microfabrication facilities for Australia's researchers. This work was financially supported by the Australian Research Council (ARC) Centres of excellence for Engineered Quantum systems (EQUS, Grant No. CE170100009) and Quantum Biotechnology (QUBIC, Grant No. CE230100021). This project was in part funded under the Next Generation Technologies Fund, which is now administered through the Advanced Strategic Capabilities Accelerator.

Disclosures
The authors declare no conflicts of interest.

Supplemental document
See Supplemental Material for supporting content, including further FEM modeling, greater detail about the double-layer SOI wafer production process and the device fabrication, results on optomechanics at ambient conditions, and investigations into the cavity photon number and photon coupling rates. 










