Paper Title: Minimal Quantum Circuits for Simulating Fibonacci Anyons.



















































 

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    |ττ̅⟩1|τ1⟩t|1τ̅⟩|11⟩⟩1/ϕ1/√(ϕ)Department of Physics, University of Illinois at Urbana-Champaign,
    1110 West Green Street, Urbana, IL 61801, USAInstitute for Quantum Information,
    RWTH Aachen University, D-52056 Aachen, GermanyCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USARudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford, OX1 3PU, UKhormozi@bnl.govComputational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973, USA§ ABSTRACT 
    The Fibonacci topological order is the prime candidate for the realization of universal topological quantum computation. We devise minimal quantum circuits to demonstrate the non-Abelian nature of the doubled Fibonacci topological order, as realized in the Levin-Wen string net model.  Our circuits effectively initialize the ground state, create excitations, twist and braid them, all in the smallest lattices possible. We further design methods to determine the fusion amplitudes and braiding phases of multiple excitations by carrying out a single qubit measurement.  We show that the fusion channels of the doubled Fibonacci model can be detected using only three qubits, twisting phases can be measured using five, and braiding can be demonstrated using nine qubits. These designs provide the simplest possible settings for demonstrating the properties of Fibonacci anyons and can be used as realistic blueprints for implementation on many modern quantum architectures.
    
    Minimal Quantum Circuits for Simulating Fibonacci Anyons
        Layla Hormozi
        August 5, 2024
    ========================================================
    
    
    
    § INTRODUCTION
    
    The emergence of topological order in interacting quantum systems is one of richest phenomena in modern condensed matter physics <cit.>. The possibility of creating long-range entangled states <cit.> and quasi-particle excitations with non-Abelian statistics has opened the door to new vistas in fault-tolerant quantum information processing, both in the form of (passively) error-tolerant topological quantum processors, and (active) topological quantum error correcting codes (QECCs) <cit.>. For almost three decades, this two-way relationship has been a wellspring of creative work, from quantum algorithms with super-polynomial speedups <cit.> to topological codes such as the surface code, which are among the most promising QECCs <cit.>. 
    
    In recent years, rapid progress in simulating topological phases has been achieved, and observation of Abelian statistics has been reported in a variety of platforms. This includes minimal implementations using optical qubits <cit.> to intermediate-scale experiments in superconducting qubits <cit.>. More recently, progress has been reported on quantum simulation of certain non-Abelian phases, including dislocation-induced non-Abelian excitations in Abelian states <cit.>, Majorana states <cit.>, as well as non-Abelian states, such as those with D_4 and S_3 topological order <cit.>.
    
    Despite this progress, some of the most interesting topological phases, particularly those which can be used for universal topological quantum computation and quantum error correction, have remained out of reach. Fortunately, rapid advances in the science and engineering of controllable arrays of synthetic qubits, in various platforms, are increasingly enabling more advanced digital quantum simulation. Given efficient circuits which prepare and manipulate the ground states and quasiparticle excitations of topological phases of matter, it is now possible to directly probe their fascinating properties on a variety of nascent quantum hardwares.
    
    
    
    
    
    In this article, we describe minimal quantum circuits which demonstrate the non-Abelian properties of Doubled Fibonacci (DFib) anyons, arising as excitations of Levin and Wen's string-net model <cit.>. Fibonacci topological order <cit.>, is particularly interesting, since it is the simplest candidate for universal topological quantum computing, having braiding operations which are universal and can be implemented fault tolerantly <cit.>. As suggested above, our approach is based on digital quantum simulation: applying a series of quantum gates which effectively project into the desired eigenstate of the string-net Hamiltonian. <cit.>. 
    
    In constructing these circuits, we identify the minimal settings in which the intrinsically “topological” properties  of Fibonacci anyons (i.e., the building blocks for topological quantum computation) can be verified. In particular, each property is realized and measured using the fewest number of qubits possible, and with circuits that are sufficiently shallow to be accessible on virtually any modern quantum hardware. Though our constructions generalize to larger system sizes, asymptotic scaling is not the focus of this paper. Rather, we aim to build a specific set of primitive subroutines to enable further progress in the experimental understanding of topological phases of matter on programmable quantum devices.
    
    
    This paper is organized as follows. In Sec. <ref>, we review the essential features of both the chiral Fibonacci and the achiral DFib models and establish the notation. In Sec. <ref>, we describe our approach to initialization and measurement of DFib anyons. In Sec. <ref>, we discuss the details of constructing minimal circuits for demonstrating fusion, twisting and braiding of DFib anyons in different settings. Our circuits use, three, five and nine qubits, respectively — the smallest number possible in each case. Section <ref> concludes and provides an outlook.
    
    
    
    
    
    
    § THE FIBONACCI TOPOLOGICAL ORDER
    
    
    In this section, we first briefly review the (chiral) Fibonacci anyon model (Fib), then describe the basics of the Levin-Wen string-net model <cit.> that realizes the (achiral) DFib topological order.  
    
    
    
     §.§ The Chiral Fibonacci Model
    
    The Fibonacci anyon model <cit.> can be described by a set of anyon types, here represented by 1 and τ, where the former represents the trivial (vacuum) state and the latter the only non-trivial anyon type, carrying a quantum number known as topological charge. Similar to other quantum numbers (e.g., spin or electric charge), there are rules for combining topological charge. In the case of Fibonacci anyons, these rules are: 
    
    τ× 1 = τ,
    
    1 × 1 = 1,
    τ×τ = 1 + τ.
    
    
    The first two equations mean that combining (fusing) any object of charge 1 or τ with the vacuum state 1 will not affect the charge of that object. The third equation means that combining  the topological charges of two objects with charge τ will result in either an object with trivial charge 1 or an object with charge τ.
    
    An important consequence of the Fibonacci fusion rule is that the Hilbert space of n Fibonacci anyons, each of charge τ, has a dimension which grows according to the Fibonacci sequence (hence the name). In the asymptotic limit, the dimensionality of the Hilbert space grows as ϕ^n, where
    
    
        ϕ = √(5) + 1/2
     
    
    is the golden ratio. Since ϕ sets the growth rate of Hilbert space in this model, it is referred to as the quantum dimension of Fibonacci anyons, d_τ = ϕ. The quantum dimension of the vacuum particle is unity d_1 = 1.
    
    The order of anyon fusion defines a choice of basis for the Hilbert space. For example, given three anyons in a row, we can choose to fuse the first two, then add the third one. Equivalently, we can choose to first fuse the last two, then add the first one. These two choices define two different bases for the three-dimensional Hilbert space of three anyons. These two bases can be mapped to each other by a unitary operation known as F operator, or F-move <cit.>, defined pictorially in Fig. <ref>(a). In its simplest non-trivial form, the F  operator has the following matrix representation:
    
    
    F ≡ F^ττ m_ττ m' = [    1/ϕ 1/√(ϕ); 1/√(ϕ)   -1/ϕ;        ],
    
    
    where m,m' ∈{1, τ}.
    
    
    In the quantum computing community, Fibonacci anyons are best known for the universality of their braiding: their braid operators are generators of SU(2), and hence can be used to approximate all single-qubit gates with arbitrary accuracy <cit.>. In general, the braid operator R^ij_m describes the phases resulting from a counterclockwise exchange of two anyons with topological charges i and j, such that the total charge of the system is m, as depicted in Fig. <ref>(b). The matrix representation of the non-trivial  braiding of two Fibonacci anyons with total charge m is as follows:
    
    
    R 
    ≡ R^ττ_m 
    =  [ e^-4π i/5         0;         0 e^+3π i/5;           ].
    
    The diagonal nature of this matrix indicates that braiding two anyons does not change their total charge. The first and second diagonal elements correspond to the phase acquired by braiding two Fibonacci anyons with total charges 1 and τ, respectively. These are the braiding phases for so-called “right-handed” Fibonacci anyons; we can define a separate left-handed Fibonacci model whose braiding phases are the complex conjugate of those above. 
    
    
    
    
    
    Closely related to braiding is the concept of twisting, or a counterclockwise rotation of  an anyon around itself by 2π, as shown in  Fig. <ref>(c). The action of twisting is encoded in the  topological twist operator, or topological spin, θ_i, which is related to the braid operator as follows, 
    
    θ_i = (R^i i_1)^-1.
    
    
    It follows that braiding and twisting are related to each other as,
    
    
    (R_m^ij)^2 = θ_mθ_i^-1θ_j^-1,
    
    
    provided that 
    i and j   fuse to m.
    Here  i,j,m∈{1, τ}, θ_1 = 1 and θ_τ = e^+4π i/5. A visual depiction of this relation is shown in Fig. <ref>(d).
    
    
    Another element of the Fibonacci anyon model that will be relevant to us is the modular S-matrix, which has the form,
    
    
    S 
    = 1/D[  1  ϕ;  ϕ -1;    ].
    
    
    Here D = √(1 + ϕ^2) is the so-called total quantum dimension of the Fibonacci model. This operator describes the process in which two pairs of τ particles are created out of vacuum, then two anyons from different pairs exchange twice (one making a full wrap around the other), and finally each re-annihilates with its respective partner, forming the so-called “Hopf link.’’ A visual description of this operator is also given in Fig. <ref>(e).
    
    Finally, braiding and fusion rules can be used to derive relations to resolve string crossings. A version of this relation, which will be used extensively in the next section, is given in Fig. <ref>(f).
    
    
    
    
    
    
    There is no known exactly solvable model for the realization of the chiral Fibonacci model. However, Levin and Wen's string-net construction provides a framework for the doubled (achiral) realization of all types of topological order, including DFib. This model can be thought of as two copies of the Fibonacci model with opposite chiralities, where fusion and 
    braiding rules apply independently to the right-handed τ and the left-handed τ̅ excitations. The resulting DFib topological order, Fib⊗Fib,  has four distinct anyon types: {1,τ}⊗{1, τ̅} = { 11, τ1, 1τ̅, ττ̅}. Our realization of the DFib topological order is based on Levin and Wen's original <cit.> and extended string-net models <cit.>. In what follows we will briefly describe the properties of this model. 
    
    
    
     §.§ String-net Realization of the DFib Model
    
    
    
    
    The Levin-Wen (LW) string-net model <cit.> is defined by a set of string types on the edges of a 2D trivalent lattice and a set of self-consistent rules for combining (fusing) different string types. These fusion rules  define the string-net Hilbert space. A family of exactly solvable Hamiltonians can be introduced in this Hilbert space, which realize all achiral topological phases, and can be thought of as a generalization of Kitaev's toric code [Technically,  a string net model can be defined for any “unitary fusion category”C, resulting in an  anyon theory which is the so-called Drinfeld center Z( C).]. These Hamiltonians have the following form, 
    
    
    H = -∑_v Q_v -∑_p B_p,
    
    
    where Q_v and B_p are mutually commuting vertex and plaquette projection operators. The former acts on the three edges surrounding the vertex v, while the latter acts on the edges surrounding the plaquette p. Fig. <ref>(a) shows these operators for a hexagonal lattice. The system is in its ground state when both the vertex and plaquette operators aresatisfied with eigenvalue +1.   We say an operator is  violated if it has eigenvalue 0 rather than +1, and this corresponds to an excited state of the system.
    
    
    The operators Q_v and B_p act on the space of “string types,” which can be encoded as multi-level spins, or qudits, in general. In this picture, the vertex operator enforces certain branching rules, consistent with the fusion rules of the strings. In particular, the branching rules decree that Q_v is always violated if a single non-trivial string ends at a vertex (an open string) [This means a single anyon cannot be created out of vacuum.]. Thus, non-trivial string types that satisfy Q_v always form closed configurations (loops or nets). We will refer to the space of states for which all Q_v's  are satisfied as the “string-net space.”
    
    
     
    The plaquette operator B_p effectively measures the topological charge of plaquette p and is satisfied when the plaquette contains trivial topological charge. This operator, which also controls the dynamics of string-nets, imposes further restrictions on the string-net space. Together with Q_v these operators favor a certain superposition of loops and nets, which defines the ground state of the string-net model. In the original Levin-Wen model, various excitations result from different combinations of Q_v and B_p violations.
    
    
    The DFib topological order <cit.> is a specific instance of the LW string-net model, with two string types, corresponding to {1,τ} in the chiral Fibonacci phase. Here we identify the string types 1 and τ with qubit states |0 and |1, respectively. We will sometimes refer to an edge with string type τ, or equivalently qubit state |1, as an activated edge. The vertex operators, Q_v, essentially enforce branching rules  that are consistent with the fusion rules of the Fibonacci topological order Eq. (<ref>). As a result, Q_v is only violated when a single incoming edge at vertex v is activated. 
    
    
    The plaquette operator in this case can be written as,
    
     
    B_p = 1/D^2(1 + ϕ B_p^τ),
    
    
    where the operator B_p^τ effectively adds a loop of string type τ to the plaquette p. In fact, B_p can be thought as of an operator that adds what is known as a vacuum loop, a weighted sum of vacuum and a τ loop, to the plaquette, which can then be absorbed into the surrounding edges using a series of F-moves [For a detailed description of the process of absorbing strings through lattice edges with a series of F-moves see the original work in <cit.>]. Fig. <ref> depicts  the vacuum loop and the effect of the B_p operator on plaquette p. 
    
    
    As was mentioned above, the DFib model  has four  anyon types, { 11, τ1, 1τ̅, ττ̅}. Here 11 indicates the ground state, corresponding to no topological charge, which results from satisfying all the Q_v and B_p operators. In the original string-net model, the excitations ττ̅, τ1 and 1τ̅ result from combinations of Q_v and B_p violations. 
    
     The ττ̅ excitation consists of four components: ττ̅_11, ττ̅_1τ, ττ̅_τ1 and ττ̅_ττ, all of which have the same topological charge <cit.>. Out of these, the ττ̅_11 component can be realized just by violating B_p, thus it can be realized in the original LW model without leaving the string-net space. This excitation is essentially achiral since the phases resulting from braiding τ  and τ̅ generally cancel each other. However, due to its multi-channel fusion properties, a restricted set of braiding phases can be observed for these anyons. We will discuss this in more detail in Sec. <ref>.
     
    
    
    To demonstrate the full braiding phases of the DFib model, we need to consider the chiral excitations τ 1 or 1 τ̅. These excitations possess braiding statistics that are essentially equivalent to that of chiral Fibonacci anyons (right- and left-handed, respectively) and can therefore realize universal quantum computation <cit.>.
    
    
    To realize these excitations without violating Q_v, we use the extended LW model, which is defined on a tailed lattice with two extra edges added to each plaquette <cit.>, as shown in Fig. <ref>(e-h). The addition of these tails will allow us to correct Q_v violations while preserving the excitations, thus effectively permitting the creation of chiral excitations within the string-net space.
    
    
    
    § INITIALIZATION AND CHARGE MEASUREMENT
    
    
    
    The string-net space is defined by satisfying the vertex operator Q_v on every vertex v.  We can project to this space by measuring this operator at every vertex and then correcting where an error is detected. 
    Relatively simple circuits for measuring the vertex operator and correcting the affected vertices have been introduced <cit.>. Here, we do not explicitly show the vertex measurement/correction circuits, but we assume all Q_v operators are satisfied before applying our circuits. 
    
    
    Projection to the ground state requires further satisfying B_p operators at every plaquette, thus selecting a particular superposition of string-nets, which is a highly entangled state. Since the plaquette operators are projectors, we can ensure they are satisfied by a projective measurement, though the measurement itself is very nontrivial, as we shall now see.
    
    We follow the method introduced in <cit.>, which is inspired by the idea of entanglement renormalization <cit.>. This method is based on the observation that the F-move acting on a given edge e, which effectively redraws the lattice at that edge, and the operator B_p, which effectively measures the  charge of plaquette p,  “commute" with each other in the following sense: 
    
    
    F_e B_p = B_p' F_e.
    
    
    This means that if we measure the charge of an n-sided plaquette p with operator B_p, then apply an F-move to an adjacent edge e to reduce the size of the plaquette, the result is the same as applying the F-move first, followed by B_p' applied to the reduced-size (n-1)-sided plaquette p' (see Fig. <ref>). An important consequence of this observation is that F-moves can be used to locally modify the lattice while preserving the ground state. 
    
    Following <cit.>, we use this fact to both initialize the plaquettes and also to measure their charge. The basic idea is that by repeatedly applying F-moves to the edges of an n-sided plaquette, we can reduce it to a single tadpole: a two-edge graph defined by a a closed edge forming the “head”  and a connecting edge forming the “tail”: 
    
    
        111.
    
    
    We can use the above rule to reduce any lattice to a series of tadpoles, which can be easily initialized to the 11 (B_p = 1) state using single qubit gates , then reverse the F-moves to return to the initial lattice, now in its ground state <cit.>. Similarly, to measure the charge of a plaquette, we can reduce it to a tadpole with the same procedure, then carry out single qubit measurements of the head and tail edges to determine the charge of the original plaquette. In what follows, we discuss how this notion can be  generalized to initialize and measure the charges of other states with anyon types τ1,  1τ̅ and ττ̅. We also generalize this notion to devise a procedure for measuring the total charges of multiple plaquettes. 
    
    
    
     §.§ Excitations
      Not only do F-moves preserve plaquettes satisfying B_p = 1, but the statement that F and B_p operators commute (Eq. (<ref>)) guarantees that this holds for all eigenstates of the LW Hamiltonian. We can use this fact to both create and measure the entire DFib spectrum. 
    
    As noted earlier, in the original string-net model with no tails, the only other anyon type that can be initialized with a tadpole, without leaving the string-net basis, is ττ̅. To realize the chiral excitations τ1 and 1τ̅ without violating the vertex operator, we use the tailed lattice <cit.> in which each plaquette is equipped with two extra edges realizing a tail (see Fig. <ref>).  Similar to the original tail-less plaquette, we can reduce the tailed plaquette using a series of F-moves. The result is a generalized tadpole where an  extra inward tail is added inside the head of the tadpole:
    
    1111,
    
    
    (see also Fig. <ref>(f)). This generalized tadpole is minimally represented by four qubits and  can be used to initialize all four anyon types of the DFib model, including the four components of the ττ̅ anyon. 
    
    
    
    
    Just as the ground state can be created by initializing tadpoles with a vacuum loop, excitations can be created by superimposing τ strings to the vacuum loops <cit.>. This process is shown in Fig. <ref>. If a string passes over a vacuum loop, it results in  a generalized tadpole which generates the τ 1 excitation. The crossings between the τ strings and the vacuum loops can be resolved by using the relation given in Fig. <ref>(f). 
    
    Likewise, if a string passes under a vacuum loop, it leads to a 1τ̅ generating (generalized) tadpole. If two strings pass over and under the vacuum loop, the result is a ττ̅ generating tadpole. Note that the end points of two strings passing over and under the vacuum loop can fuse in four possible ways, resulting in the four components of the ττ̅ anyon. In the tailed lattice, these τ strings must end on the added tails to avoid Q_v violations. These are the same tails that show up in the generalized tadpoles that generate the DFib spectrum, a summary of which is also given below,
    
     =  1/D0000 +  ϕ/D1100
    1 =  1/D0111 +  ξ^2/D1011 +   √(ϕ) ξ/D1111
    t =  1/D0111 +  ξ^-2/D1011 +   √(ϕ) ξ^-1/D1111
    _11 =  ϕ/D0000 -  1/D1100
    _1τ =  1110
    _τ1 =  1101
    _ττ =  √(ϕ)/D0111 +  √(ϕ)/D1011 +   1/D ϕ1111.
    
    
    
    Here the red (shaded) lines indicate  strings of type τ (or qubit state |1), while dotted  lines depict the type 1 strings (qubit state |0), and ξ = e^3π i/5. 
    
    
    
    
    
    
    
    Thus, to create a pair of excitations of any kind, we first reduce the lattice to a series of generalized tadpoles using  F-moves. Then, depending on the desired excitation, we initialize them according the rules given in Eqs. (<ref>)-(<ref>). Finally we return to the original lattice by reversing the F-moves.  The circuits for both initialization and measurement of these generalized tadpoles are shown in Fig. <ref>. The S gate used in the figure is the modular S-matrix (Eq. (<ref>)) while the U gate is defined as,
    
    
    U = [ ξ^-1/√(ϕ)     ξ^2/ϕ;   -ξ^-2/ϕ    ξ/√(ϕ);           ].
    
    
    
    Finally, note that given the spherical boundary conditions, the excitations are always created in pairs — all of the quantum numbers of all the excitations in the entire system must fuse to the identity.   Thus we cannot create a single plaquette excitation from the ground state without also creating its partner.   In Fig. <ref> (and also in Eqs. (<ref>)-(<ref>)) for simplicity, we show only half of the process, where a string is superimposed on a single vacuum loop inside the head of a (generalized) tadpole. In general, we need to form the tadpoles such that plaquettes that share a pair of excitations will be transformed to tadpoles that share tails, which can then be initialized together. We will discuss this in more detail the following section. 
    
    
    
    
    
    
    
     §.§ Multiple Plaquettes
     
    
    In the extended LW model defined on lattices with tails where excitations are created in the absence of Q_v violations, they are essentially created on plaquettes. Thus, to determine the fusion amplitudes resulting from combing several excitations, we need a method to measure the total charge of multiple plaquettes. We discuss this problem from two different perspectives, first by building projective operators that determine the charge of two (or more) plaquettes, then designing quantum circuits which allow us to measure the total charge of multiple plaquettes with a computational basis measurement.
    
    As was noted in Sec. <ref>, the action of the plaquette operator B_p is to determine the charge content of a plaquette (whether it is trivial or not) by inserting a vacuum loop inside the plaquette, and absorbing it through the edges by a sequence of F-moves (see Fig. <ref>). This idea  can be generalized to an operator that determines the charge content of two neighboring plaquettes by inserting  a vacuum loop inside the two plaquettes. However, in this case, there is a choice of whether to insert the loop over or under the dividing edge. It turns out that these two options lead to two different operators, which we call B^o_p_i p_j and B^u_p_i p_j, where the superscripts o and u refer to over and under respectively. A visualization of these operators is depicted in Fig. <ref>. 
    
    
    Satisfying the operator B^o_p_i p_j indicates that the total charge of the right-handed anyons 1 in the two plaquettes p_i and p_j is trivial. Likewise, satisfying  B^u_p_i p_j implies that the total charge of left-handed anyons t on the corresponding plaquettes is trivial. Satisfying both of these the operators B^o_p_i p_j and B^u_p_i p_j  indicates that the total charge content of the two plaquettes is trivial. Note that this can be true even if individual plaquette operators B_p_i and B_p_j are not satisfied. This is analogous to the case where the collective charge of two τ anyons can be trivial. 
    
    
    Similar to single plaquette operators B_p, the double plaquette operators defined above are also projection operators, hence  cannot be implemented directly as a  unitary quantum circuit but instead must be implemented as a (rather complicated) projective measurement. Here we design quantum circuits to directly measure the charge of multiple plaquettes. Our approach is to generalize the single plaquette strategy — initialization and charge measurement by first reducing to a  tadpole — to a charge measurement by reducing multiple plaquettes to a series of concentric circles with each circle tied to the next by a single edge, resembling a series of concentric generalized tadpoles. Examples of this process for small lattices are shown in Fig. <ref>. 
    
    The key idea is that, through a series of F-moves, we reduce one of the plaquettes in the original lattice to the central tadpole in the concentric circles basis. The next-outermost (generalized) tadpole then contains the total charges of the original plaquette and another plaquette, which is reduced to an annulus surrounding the central tadpole. Similarly, with more plaquettes, the outer tadpoles hold the charge of all the original plaquettes which are subsequently enclosed by those tadpoles. Since the system should be thought of as living on the surface of a sphere, the outermost ring can be viewed as a tadpole as much as the innermost ring. 
    
    
    By choosing different series of F-moves, we can select different concentric tadpole bases that correspond to different combinations of plaquettes and use it as a basis for both initialization and measurement of combined charges. As an example, Fig. <ref>(b) and (c) depict two different choices of basis, suitable for measuring the charges of plaquettes p_1 ∪ p_2 or p_3 ∪ p_2. [Note that, in general, different sequences of F-moves can be used to transform a lattice into different concentric tadpoles. The  number of these choices increases in the case of tailed lattices and sometimes there are multiple sequences of F-moves that lead to seemingly similar concentric tadpoles. However, in general different choices can lead to additional twists and sometimes braids. Thus, caution needs to be exercised when selecting the F-move sequence to avoid unwanted phases. ]
    
    The equivalence of this approach to the projective operators above can be understood by recalling the action of the two-plaquette operator B_p_i p_j, which inserts a vacuum loop inside the two plaquettes. By keeping track of the F-moves which modify the lattice into concentric tadpoles, we can see that this vacuum loop effectively covers the area around two concentric tadpoles. 
    
    
    
    
    
    
    
    
    
    
    
    § SIGNATURES OF NON-ABELIAN ANYONS IN MINIMAL LATTICES
    
    
    In this section, we provide an overview of the signatures of the DFib model that we investigate in various settings, postponing the details to the following sub-sections. The basic properties that we are interested in are fusion, braiding and twisting of various excitations within the DFib model using the minimum number of qubits. Fig. <ref> depicts these operations for (chiral) Fibonacci anyons. 
    
    We restrict our study to lattices with spherical boundary conditions. The simplest lattices that can be defined on a sphere are shown in Fig. <ref>. The smallest possible lattice  consists of  two plaquettes each forming a hemisphere with a single common edge forming the meridian (see Fig. <ref>(b)). This two-plaquette lattice can be initialized into ground state or host a pair of ττ̅ = ττ̅_11 excitations, with total charge 11. The former can be realized by by applying an S gate to the single qubit on the edge initialized at |0, while the latter by applying an S gate to the qubit on the edge initialized at |1, as can be seen in Fig. <ref>.
    
    The smallest non-trivial lattice for our purposes consists of three plaquettes, separated by three edges that meet at the two poles of the sphere. A 2D depiction of this lattice, which resembles the Greek letter Θ, is shown in Fig. <ref>(c). We will use this lattice to demonstrate the fusion of two  excitations (Fig. <ref>). 
    
    The next smallest lattice with spherical boundary conditions consists of six edges and four plaquettes and is equivalent to a tetrahedron, as shown in Fig. <ref>(d).  We will use this lattice to realize a different implementation of the  fusion, which reveals additional details about the  structure of fusion channels, and discuss the difference between the two cases. 
    
    To realize the braiding properties of the DFib model (see Fig. <ref>(b)) we need to create the chiral excitations 1 or t without violating the vertex operators Q_v. This can be done using the extended tailed lattice. The simplest tailed  lattices are shown in Fig. <ref>(e-h). These lattices have additional qubits to account for the addition of a tail to each plaquette. The simplest lattice on which the creation and braiding of two pairs of chiral anyons can be demonstrated is the tailed Θ lattice, which consists of nine qubits (see Fig. <ref>(g)).
    
    We will also briefly discuss a simple procedure for revealing the topological twist phase of a single 1 anyon, which is created as part of a pair on the simplest two-plaquette lattice, or a generalized tadpole, consisting of four qubits (Fig. <ref>(c)). A simple generalization of this procedure can then be used to verify the relation between braiding and twisting.
    
    
    A note on notation: For the remainder of the paper, we will use two different notations for our states, depending on the context. The first is the qubit basis, in which we simply refer to the states of the qubits (|0 or |1) that form a particular lattice. The second notation is used when considering the concentric tadpole basis (see Fig. <ref>), where  we use a tensor product of kets that represent the sectors of the DFib model corresponding to each tadpole in this basis. 
    
    For example, in the case of the Θ lattice where the concentric tadpole basis consists of two tadpoles (see Fig. <ref>(a)), we use the notation |inn.|out. to refer to the states of the innermost and outermost tadpoles. Likewise for the tetrahedron lattice, where the concentric tadpole basis consists of three tadpoles (See Fig. <ref>(b) and (c)), we use the notation |inn.|mid.|out., which again refer to specific anyon types of the DFib model represented by each of the innermost, middle and outermost tadpoles in this basis.    
    
    In what follows, we explain the details of how various excitations can be created, braided, twisted and measured in specific lattices, and 
    present the corresponding quantum circuits. The circuit representations of the various F-moves used in this section are given in Fig. <ref>.
    
    
    
    
    
     §.§ Fusion with Three Qubits: The Θ Lattice
    
    
    The simplest setting for detecting the fusion properties of the DFib model is to realize two pairs of  excitations on a minimal lattice consisting of three plaquettes on a sphere, or the Θ lattice. 
    The process of initialization and detection of fusion channels is shown in Fig. <ref>.
    
    
    Initialization:  We start by initializing two pairs of _11 excitations on plaquette pairs p_1, p_3 and p_2, p_3. This can be done by starting from a basis in which each of the plaquettes p_1 and p_2 are reduced to single tadpole, as shown by the trivial state  shown as |ψ_00 in Fig. <ref>. Here  qubits q_1 and q_3 form the heads of the two tadpoles and qubit q_2 serves as the common tail.   We then apply single-qubit gates to the two heads, resulting in the state |ψ_0, which in qubit basis has the form,
    
    
    |ψ_0   =    |q_1q_2q_3
       =   (ϕ|0 - |1/D)|0(ϕ|0 - |1/D).
    
    
    We then apply an F-move to qubit q_2  to restore the original Θ lattice basis, resulting in the state |ψ_1.  At this point, we have created two pairs of  excitations on plaquette pairs p_1, p_3 and p_2, p_3 in the Θ lattice, such that the total charges of plaquettes p_1∪ p_3 and p_2∪ p_3 correspond to the  sector of the DFib model. In other words, we have effectively pulled two  pairs of _11 excitations out of vacuum. 
    
    
    Measurement:  We expect the total charge of plaquettes p_1 ∪ p_2, (or equivalently the charge of plaquette p_3, due to the spherical boundary conditions) to be in a superposition state consistent with the fusion rules of the Fibonacci model. To read this total charge, we need to change the basis into concentric tadpoles (see also Fig. <ref>(a)) such that, either plaquette p_1 or plaquette p_2 will be enclosed by a tadpole, which is nested inside an annulus, containing the charge of the other plaquette. This can be done by applying an F-move, acting on qubit q_3, in |ψ_2 as shown in Fig. <ref> [This could also be done with an F_1 move resulting in a shape where p_2 is enclosed by the central tadpole.]. 
    
    As was discussed in Sec. <ref>, the outer tadpole, which encloses plaquettes p_1 and p_2, can now be measured to ascertain the total charge of p_1 ∪ p_2. Note that in this case this outer boundary is also equivalent to the boundary of the single tadpole that encloses plaquette p_3 (the remainder of the sphere) and contains the same charge as p_1 ∪ p_2. Note also that while we use the term fusion in referring to combining topological charge, we don't actually fuse excitations in plaquettes p_1 and p_2. We merely change the basis such that we can measure the superposition state that contains information on fusion amplitudes, should an actual fusion occur. 
    
    
    
    Using the notation |ψ_2  = |inn.|out., this state can be written as,
    
    
    |ψ_2   =   _11
       +   (_11_11 
    - _1τ_τ1),
    
    
    which represents the anyon types contained by the inner and outer tadpoles, as shown in Fig. <ref>. We see that the inner tadpole, which contains the charge of plaquette p_1, corresponds to a single  anyon in all three terms. This is consistent with our initialization, in which plaquettes p_1 and p_2 each contained a single  excitation. 
    
    Focusing on the outer tadpole, which contains the total charge of plaquettes p_1 ∪ p_2, the first term in |ψ_2 indicates trivial total charge with coefficient 1/ϕ, resulting in the state , while the second and third terms indicate that the total charge corresponds to the charge of a single  anyon with coefficient 1/√(ϕ). Thus, the total charge of of p_1∪ p_2, signifies the following cross fusion relation,
    
    × =  + ,
    
    
    i.e., only two fusion channels appear. We will return to this point at the end of this section.
    
    Note that while we started by initializing the _11 component of the  excitation, cross fusion results in additional components of the  excitation. This can be seen in the third term of |ψ_2 (see Fig. <ref> and Eq. (<ref>)) where the excitation _1τ1 appears in the inner tadpole and _τ1 in the outer one. Since the components of  contain the same charge, this is consistent with Eq. (<ref>), where the second term can contain different combinations of the components of .  These terms can also be obtained by resolving the loops on the right hand side of the equation shown in Fig. <ref>(a) followed by the F-move.
    
    To simplify the measurement of the fusion channels of p_1 ∪ p_2 in a quantum circuit, we apply a series of two-qubit gates to |ψ_2, which effectively maps it to a product state. These two-qubit gates are two controlled-S gates acting on qubits q_2 and q_3 and a controlled-L gate, acting on qubit q_3 where,
    
    
    L = [  1/√(ϕ)    -1/ϕ;    -1/ϕ -1/√(ϕ);         ],
    
    
    resulting in the state
    
    
    |ψ_3   =    |q_1q_2q_3
       =   (|0 + |1)|10.
    
    
    Thus, the information on the fusion channels of  p_1 ∪ p_2 is effectively transferred to qubit q_1 in this state. 
    
    To detect the fusion coefficients, we can perform statistical measurements on q_1 at this stage and observe the corresponding probability distribution. Alternatively, we can apply an F gate (Eq. (<ref>)) to q_1, then use the result as the control for a CNOT gate acting on a target ancilla qubit. This F gate acts as a “filter,” so that if q_1 is in the correct superposition, the F gate maps it to |0, which does not affect the ancilla qubit in the subsequent CNOT. Thus, with this protocol we can verify the coefficient of fusion by carrying out a single measurement of the ancilla qubit. 
    
    Alternatively, we can determine the state of  |ψ_2⟩ by first measuring the state of q_3.  If q_3 is found to be in the |1⟩ state, so should q_1 and q_2 (the last term in the expression for |ψ_2⟩ in the Fig  <ref>.c).  If q_3 is found to be in the |0⟩ state, then q_1 and q_2 can be tomographed, and we should find q_2 to be in the  |ττ⟩_11 state and q_1 to be in the |1 1 ⟩ + |ττ⟩_11 state, which directly reflects the fusion in Eq. <ref>. In cases below we will always favor read-out schemes that minimize the number of measurements, although other read-out schemes are certainly possible.
    
    
    To conclude this section, we return to the observation in  Eq. (<ref>), namely the fact that in the Θ lattice, the result of cross fusing two  excitations is two fusion channels, namely  and . This might look unexpected at first sight, since the two sets of chiral excitations in the DFib model (the right-handed 1 and the left-handed t) must fuse independently from each other, thus in principle we expect the combination of two  excitations to result in four fusion channels: , 1, t and . However, recall that 1 and t are created by passing a τ string over or under the lattice, respectively. On the Θ lattice, two excitations must share the same plaquette and for a single plaquette there is no way to distinguish between “over” and “under.” Thus only the achiral excitations can exist on this plaquette. Consequently,  the combined charge of p_1∪ p_2 lead to only two  fusion  channels:  and .
    
    
    
    
    
    
    
    
    
    
    
    
    
    To explore this idea further, we create two pairs of _11 excitations on the four plaquettes of a  tetrahedral lattice, consisting of six qubits, and measure the cross-fusion channels of opposite plaquettes. We expect to see four fusion channels in this case, including the chiral excitations. We will discuss this in the next section. 
    
    
    
     §.§ Fusion with Six Qubits: Tetrahedron Lattice
    
    
    The process of initialization and measurement of the fusion channels of two pairs of excitations on the four plaquettes of a tetrahedral lattice is shown in Fig. <ref>. 
    
    Initialization:  We start by initializing two pairs of  excitations out of vacuum on pairs of plaquettes p_1, p_2 and p_3, p_4 such that the total charge of each pair corresponds to . This can be done by starting from a basis in which plaquettes p_1 and p_2 form concentric tadpoles, enclosed by a third tadpole containing p_3. In Fig. <ref>, one such starting configuration is depicted as the basis for the state |ψ_00. We then create pairs of _11 excitations on plaquettes p_1, p_2 and p_3, p_4 by initializing the corresponding generalized tadpoles. 
    
    The innermost tadpole, which represents plaquette p_1, consists of qubits q_2 and q_5, forming its head and tail, respectively. This tadpole is initialized as, 
    
    
    |q_2 q_5 = ϕ|0 - |1/D|0.
    
    
    
    The middle (generalized) tadpole, which contains the information about total charges of p_1 ∪ p_2, consists of qubits q_1 and q_6,  together forming its head, and q_5 and q_3, which form its two tails. This tadpole is initialized as,
    
    
    |q_1 q_6 q_5 q_3 = |00 + ϕ |11/D|00
    
    Finally, the outermost tadpole, containing the charge of p_1 ∪ p_2 ∪ p_3, consists of qubit q_4 as its head and qubit q_3 as its inward tail. These qubits are initialized as,
    
    |q_4 q_3 = ϕ|0 - |1/D|0.
    
    
    The resulting state in the qubit basis is the following,
    
    
    |ψ_0   =    |q_2 q_4 |q_1 q_6 |q_5 q_3
       =   (ϕ|0 - |1/D)^⊗2(ϕ|00 +ϕ |11/D)|00.
    
    
    
    
    We can also re-write the state |ψ_0 in the concentric tadpole basis, where each ket represents the topological charge of the corresponding generalized tadpole,
    
     
    |ψ_0   = |inn. |mid. |out.
    
    where,
    
    
    |inn.   = _11
    
    |mid.   = 
    
    |out.   = _11.
    
    
    
    Here the first equality means that the plaquette represented by the innermost tadpole (p_1) and the plaquette represented by the annulus surrounding the innermost tadpole (p_2) each have a charge of _11. The second equality indicates that the total charge of plaquettes p_1 ∪ p_2, represented by the middle tadpole, is ; in other words, the pair of excitations on p_1 and p_2 are effectively pulled out of vacuum. Finally, the third equality implies that the plaquette represented by the annulus surrounding the middle tadpole (p_3) and the outside plaquette representing the remainder of the sphere (p_4) each also have a charge of _11. Due to spherical boundary conditions, the total charge of these two plaquettes is also , thus the excitations on these plaquettes are also pulled out of vacuum. 
    
    To return to the original lattice basis (tetrahedron) we apply three F-moves to qubits q_3, q_5 and q_1, as shown in Fig. <ref>(b), resulting in the state |ψ_1. The circuit definitions of the F-moves can be found in Fig. <ref>. At this point, we have created two pairs of  excitations on plaquettes p_1, p_2 and p_3, p_4 such that each pair fuse to vacuum. 
    
    
    
    Measurement: To measure the fusion coefficients resulting from cross-fusing excitations from opposite pairs, e.g., p_2 ∪ p_3 and p_1 ∪ p_4, we need to switch to a concentric tadpole basis in which two plaquettes from opposite pairs are enclosed by the middle tadpole. One such basis is shown in Fig. <ref>(c), in which the middle tadpole encloses the charge of p_2∪ p_3. In this basis, the innermost tadpole contains the charge of plaquette p_1, and the outermost tadpole corresponds to the total charge of plaquettes p_1 ∪ p_2 ∪ p_3 or, equivalently, plaquette p_4. This basis change can be achieved by applying three additional F-moves acting on qubits q_4, q_3 and q_5,  resulting in the state |ψ_2 in Fig. <ref>. In the concentric tadpole notation, this state has the form,
    
    
    |ψ_2   = 1/ϕ^2_11_11
       + 1/ϕ√(ϕ)_1τ(1 + t) _τ1
       +1/ϕ[
    1/ϕ_11_11_11 
    -1/ϕ√(ϕ)( _τ1_1τ_11 + _11_1τ_τ1) 
    + 1/ϕ^2_τ1_ττ_1τ].
    
    
    
    As was noted above, the information on the fusion channels of the total charge of p_2∪ p_3 is now represented by the state of the middle tadpole in |ψ_2. To identify the components of the fusion channels, we can use Eq. (<ref>)-(<ref>) to re-write |mid. = × as follows,
    
    
    |mid.   =   1/ϕ^2
       +   1/ϕ√(ϕ)(1  + t)  
       +   ,
    
    
    where,
    
       =   _11
       -   1/ϕ√(ϕ)( _1τ + _τ 1)
       +   1/ϕ^2_ττ.
    
    
    This is consistent with our expectation that in general the two chiral sectors of the  excitation fuse independently, resulting in four fusion channels, as given in Eq. (<ref>).
    
    The charges of individual plaquettes p_3 (represented by the innermost tadpole |inn.) and p_4 (represented by the outermost tadpole |out.) remain at . However, as we see in Eq. (<ref>), for these states different components of the  sector appear in different terms. 
    
    Similar to the previous case, it is possible to design a quantum circuit to simplify the state |ψ_1 such that the expected configuration will be mapped to an un-entangled state which can then be measured for verification.
    
    
    
    
    
     §.§ Fusion and Braiding with Nine Qubits: Tailed Θ Lattice
    
    
    To detect non-Abelian braiding phases in the DFib model, we need to create the chiral excitations 1 or t. To realize these excitations without violating the vertex operators Q_v, we employ the extended tailed lattice <cit.>. The simplest nontrivial tailed lattice for this purpose would be a tailed Θ lattice, consisting of nine qubits, as shown in Fig. <ref>(g). In what follows we describe circuits to create these excitations, change the basis to measure fusion channels, then braid the excitations and finally detect the braiding phases. 
    
    
    
    
    
    
      §.§.§ Fusion
    
    
    Here we focus on creating two pairs of 1 excitations on three plaquettes, aiming at detecting the following fusion relation,
    
    1 ×1 =  + 1,
    
    
    which results from fusing excitations from opposite pairs, created out of vacuum. The general procedure resembles the case of creating  excitations in the Θ lattice described above, but the initialization and measurement processes involve details that are specific to the chiral excitations, which we describe bellow. The full circuit is depicted in Fig. <ref>.
    
    Initialization: We initialize the tailed Θ lattice in a basis in which plaquettes p_1 and p_2 are enclosed by separate (generalized) tadpoles, resulting in |ψ_0, as shown in Fig. <ref>(c). The general form of initialization circuit for  1 anyons is given in Fig. <ref>. Here we apply these circuits independently to the two tadpoles, but initialize the tail of plaquette p_3, with an F-gate. This can be understood by first imagining two tails for plaquette p_3 by using two extra qubits q_10 and q_11, then initialize each tadpole independently according to Fig. <ref>. We then combine these tails by applying an F-move to q_9 and through away the extra qubits  q_10 and q_11, leaving us with |q_9=|0 + |1. Since everything about this process is deterministic, we choose to do away with the extra qubits and directly initialize the tail qubit q_9 with an F gate. 
    
    We then apply three F-moves to |ψ_0 to return the state to the Θ lattice basis, resulting in |ψ_1. This process is depicted in Fig. <ref>(c) where the effects of the F-moves on the lattice are shown. Here the colored solid lines are merely guides to the eye, to keep track of the locations of pairs of excitations. At this point we have created two pairs of 1 excitations on plaquettes p_1, p_3 and p_2, p_3, such that each of these pairs would fuse to the vacuum state .
    
    Measurement: To measure the  amplitudes of the superposition state |ψ_1, resulting from cross-fusing the charges of plaquettes p_1∪ p_2, we change into a concentric tadpole basis where plaquettes p_1 and  p_2 are enclosed by concentric tadpoles. This can be done with the aid of three F-moves, resulting in the state |ψ_2 as shown in Fig. <ref>(d). 
    
    In this state, the outer tadpole, consisting of qubits  q_5 and q_6, which form the head, and qubits q_2 and q_9, forming its tails, contains the information on the total charge content of plaquettes p_1 ∪ p_2, or equivalently, p_3. Likewise, qubits q_1, q_3, q_4 and q_7, which form the components of the inner tadpole, contain information on the charge content of plaquette p_1. Written in the concentric tadpole basis, |ψ_2 = |inn.|out., we have, 
     
    |ψ_2 = 
    1(1/ϕ +
    1/√(ϕ)t ).
    
    
    The components of the  and 1 sectors for the inner and outer tadpoles are shown in Fig. <ref>. In terms of these components, the inner tadpole consists of three terms (Fig. <ref>(c)) and the outer tadpole consists of two terms for the  sector (Fig. <ref>(a)) and three terms for the 1 sector (Fig. <ref>(b)), leading to fifteen terms for |ψ_2.
    
    To detect the fusion amplitudes, we  first simplify |ψ_2 using the  circuits given in Fig. <ref> applied independently to the inner and outer tadpoles. Switching to the qubit basis, these circuits first map the two tadpoles to the following:
    
    
    |q_4 q_8 q_7 q_1 q_3   =    |11110
    
    |q_9 q_2 q_6 q_5   =   |0000 +  |1110.
    
    
    To further simplify, we apply CNOT gates to q_2, q_6 and q_9 to un-entangle them, resulting in 
    
    
    |ψ_3   =    |q_1 q_2 q_3 q_4 q_5 q_6 q_7 q_8 q_9
       =     |1(|0 +  |1)|0100110.
    
    
    Thus, the information on the fusion channels of the combined plaquettes p_1 ∪ p_2 is now stored in the state of a single qubit, namely q_2 [By choosing different combinations of CNOTs we could choose any of q_2, q_6 or q_9 as our flag qubit.]. 
    
    To detect these amplitudes, as before, we either carry out statistical measurements of q_2 and measure the probability distribution of each fusion channel, or apply an F gate to q_2 as a “filter,” mapping it to |0, then use the result to control a CNOT acting on ancilla qubit A. If the fusion channels are as expected, the ancilla qubit should not change. 
    
    
    
    
      §.§.§ Braiding
    
    
    Staying within the tailed Θ lattice, here we discuss how to create and braid two 1 excitations and measure the resulting phases. This process is shown in Fig. <ref>.
    
    Initialization: We repeat the same initialization as the one we used in the previous section and create two pairs of 1 excitations, on plaquettes p_1, p_3 and p_2, p_3, resulting in the state  |ψ_1 in the Θ lattice basis. 
    
    
    
    
    Braiding: To braid the charge contents of plaquettes p_1 and p_2, we simply exchange the corresponding plaquettes in a counterclockwise manner using a series of F-moves <cit.> as shown in Fig. <ref>(d), resulting in the state |ψ̃_1. Note that, while we end up in the same lattice basis as the starting point, braiding effectively changes the arrangement of qubits on the lattice. We need to keep this in mind when preparing the state for read-out. 
    
    Measurement: To measure the braid phases, we again need to change the basis into one where plaquettes p_1 and p_2 form concentric tadpoles, so that the total charges of combined plaquettes p_1 ∪ p_2 will be enclosed by the outer tadpole. This process is the same as the one in the fusion case but with different qubits, as shown in Fig. <ref>(e). Using the concentric tadpole basis,   |ψ̃_2 = |inn.|out., we get,
    
    
    |ψ̃_2 = 1 
    ( e^-4π i/5/ϕ + e^+3π i/5/√(ϕ)1
    ).
     
    
    Thus the two states of the outer tadpole have acquired the phases corresponding to a counterclockwise exchange of two τ1 anyons. 
    
    
    To verify these phases with a quantum circuit, we begin by simplifying |ψ̃_2, using the detection circuits in Fig. <ref>. This simplification effectively maps the states of the two tadpoles to the following, 
    
    
    |q_1 q_7 q_8 q_5 q_4    =    |11110
    
    | q_3 q_9 q_2 q_6   =   e^-4π i/5/ϕ|0000 +e^+3π i/5/√(ϕ)|1110.
    
    
    To further simplify this state, we un-entangle qubits q_2, q_3 and q_9 by applying two CNOT gates, resulting in 
    
    
    |ψ̃_3   =    |q_1 q_2 q_3 q_4 q_5 q_6 q_7 q_8 q_9
       =    
    |1(e^-4π i/5/ϕ|0 + e^+3π i/5/√(ϕ) |1) |1010110.
    
    
    Thus, all the information on braiding phases of the two fusion channels is now stored in q_2. For the final detection, similar to the previous cases, we design another filter, which has the form,
    
    F̃ = [     ξ^2/ϕ    ξ/√(ϕ); ξ^-1/√(ϕ)  -ξ^-2/ϕ.;           ]
    
    
    
    We believe this is the simplest setting in which to demonstrate the braiding properties of the chiral excitations of the DFib model. One may also perform these exercises on a tailed tetrahedron (Fig. <ref>(h)) with fourteen qubits, such that each of the four plaquettes contains a single 1 excitation. The results are identical to the three plaquette lattice for both fusion and braiding. 
    
    Since this article is focused on minimal models, we do not provide that circuit here. 
    
    
    
    
    
    
     §.§ Twisting with Five Qubits: Tailed Tadpole
    
    
    A chiral excitation can be twisted within a plaquette and the resulting phase can be detected using additional tricks (see, e.g., <cit.>). The simplest lattice for this purpose consists of only four qubits, forming a single generalized tadpole, which can be thought as representing two plaquettes on a sphere, one shown and the other being the outside region representing the remainder of the sphere (see Fig. <ref>(f)). The circuit for the initialization, twisting and detection of the phase is shown in Fig. <ref> (with two additional qubits to simplify readout). 
    
    
    We start by initializing  the 1 state on the four qubits forming the generalized tadpole, using the initialization circuits in Fig. <ref>. We then apply a special case of an F-move, which effectively exchanges the two tails. This F-move is defined in Fig. <ref>. Written in the original basis, this is equivalent to twisting the 1 excitation in the front plaquette in a counterclockwise manner, resulting in,
    
    θ̂1 = e^+4π i/51.
     
    
    
    To detect the overall phase, we initialize an ancilla qubit |q_0 with a Hadamard gate. Re-writing the 1 state in qubit basis as, 
    
    1    =    |q_1 q_2 q_3 q_4
       =   1/D|00(|10 + ξ^2|01 +√(ϕ)ξ|11).
    
    
    results in the initial state, 
    
    
    |ψ_0   =    |q_0|q_1q_2q_3q_4
       =   1/√(2) (|0 + |1)1.
     
    
    We then then use |q_0 as a control qubit for the application of a controlled-F-move acting on q_3, resulting in
    
    
    |ψ_1   =    |q_0|q_1q_2q_3q_4
       =   1/√(2) (|0 + e^4π i/5|1)1.
    
    
    Thus, we use the phase kickback from twisting the 1 excitation in the tadpole consisting of  qubits q_1-q_4 to impart a phase shift on the ancilla qubit |q_0. We can then measure this phase shift by using the appropriate filter, which maps the state of q_0 to |0. The filter in this case is the following gate,
    
    M = [         1  e^4π i/5; e^-4π i/5        -1;           ].
    
    Finally we can use q_0 as a control for a CNOT acting on another ancilla qubit to verify the expected phase, as in the previous cases. Since the state of the ancilla qubit q_0 is un-entangled from the qubits forming the tadpole, we do not need to apply a simplification circuit to these qubits. However for ease of checking the results, simplifying gates can be adopted from Fig. <ref> to map the 1 state to |q_1q_2q_3q_4 = |1110. 
    
    
    We end by mentioning that this procedure can easily be generalized to larger plaquettes and be used to verify the relation between braiding and twisting, Eq. (<ref>), in the nine-qubit lattice. 
    
    
    
    
    § CONCLUSIONS AND OUTLOOK
    
    
    This work was inspired by a theoretical question: what are the smallest possible lattices in which the string-net rendition of the Doubled Fibonacci model can be realized and its non-Abelian nature revealed? We answered this question by designing quantum circuits that simulate the spectrum of this model and demonstrated their fusion, twisting and braiding properties, all on minimal lattices with the fewest number of qubits possible. 
    
    We argued that the ground state as well as a  pair of achiral  excitations can be realized on a lattice consisting of only one qubit. We then showed that with only three qubits, we can create two pairs of  excitations in a minimal three-plaquette lattice and demonstrate the amplitudes of their fusion channels. We further showed that by moving to a lattice with six qubits, additional fusion structure resulting from combining  excitations can be revealed. 
    
    We then created two pairs of chiral 1 excitations in a three-plaquette tailed lattice, consisting of nine qubits, braided them and measured the relative braiding phases. Finally, we showed that a total of four qubits is enough to create a pair of chiral excitations (1 or t) in two plaquettes, twist one of them and verify the resulting phase with the aid of a fifth (ancilla) qubit. 
    
    The Fibonacci anyon model exhibits one the simplest, yet most exotic, forms of topological order, with possible applications in active and passive fault-tolerant quantum computation. Our goal here was to find the minimal resources required to demonstrate the non-Abelian nature of Fibonacci anyons and we believe our circuits require sufficiently small quantum volume that they can be run on any modern qubit hardware with reasonable coherence times and gate fidelity, thus making this model more accessible to the wider physics community. 
    
    The methods discussed here can  be generalized to larger lattices, where more anyons can be created and quantum gates can be carried out topologically. An important future direction would be to quantify and measure the stability of these states in the presence of noise. Another interesting direction would be the design of minimal circuits that simulate other instances of the  string-net model, resulting in other forms of topological order, especially those that require multi-level qudits.
    
    
    
    
    § ACKNOWLEDGMENT
     
    
    We thank R. Koenig, M. Levin and A. Schotte for clarifying discussions at an earlier stage of this work, and F. Burnell for discussions and comments on an earlier draft of the manuscript. We also thank the Aspen Center for Physics
    for its hospitality during the completion of part of this
    work. This project was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. LH and JP acknowledge partial support from BNL LDRD 20-022. SB and RRA were supported by the U.S. Department of Energy, Office of Science’s SULI program.  SHS is supported by EPSRC Grants
    EP/S020527/1 and EP/X030881/1.
    
    
    Note Added: During the preparation of this work, we became aware of two other independently carried-out works on a related but different simulation of DFib anyons on specialized hardware <cit.> and <cit.>. Besides using different methods, the present work is distinct in that it focuses on designing minimal circuits suitable for realization on any quantum platform.


 Paper Title: Neuromorphic detection and cooling of microparticle arrays.



Article Title]Neuromorphic detection and cooling of microparticle arrays










1]Yugang Renyugang.ren@kcl.ac.uk

2]Benjamin Siegel

1]Ronghao Yin

1]Muddassar Rashid

1,3]James Millenjames.millen@kcl.ac.uk

[1]Department of Physics, King's College London, Strand, London, WC2R 2LS, United Kingdom

[2]Wright Laboratory, Department of Physics, Yale University, New Haven, Connecticut, 06520, USA

[3]London Centre for Nanotechnology, Department of Physics, King's College London, Strand, London, WC2R 2LS, United Kingdom





Micro-objects levitated in a vacuum are an exciting platform for precision sensing due to their low dissipation motion and the potential for control at the quantum level. Arrays of such sensors would allow noise cancellation, directionality, increased sensitivity and in the quantum regime the potential to exploit correlation and entanglement. We use neuromorphic detection via a single event-based camera to record the motion of an array of levitated microspheres. We present the first truly scalable method for multiparticle control by implementing real-time feedback to cool the motion of three objects simultaneously.


















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  §.§.§ Introduction


Modern technology relies on mechanical sensors, from accelerometers in mobile devices <cit.> to wearable health monitors <cit.>. The performance of a mechanical sensor is inversely proportional to it's volume due to dissipation through thermal contacts and surface strain <cit.>. By levitating nano- or micro-particles in optical, electrical or magnetic fields, one creates a mechanical oscillator with remarkably low dissipation <cit.>. Force sensitivities of zeptonewton <cit.> and torque sensitivities of (4.2 ± 1.2)× 10^-27 N m Hz^-1/2 <cit.> have been achieved, with levitating sensors achieving quality factors in excess of 10^10 <cit.>, motivating researchers to use these systems to search for dark matter <cit.> and gravitational waves <cit.>.

The control of levitated particles allows the exploration of a wide range of fundamental science <cit.>, and the demonstration of cooling to the ground state of an optical potential <cit.> opens the door to macroscopic quantum physics <cit.>. An emerging frontier in this field is the study of arrays of particles, which in the quantum regime would allow generation of entanglement <cit.> and tests of quantum gravity <cit.>. Interactions have been observed between pairs of levitated nanoparticles in optical <cit.>, electrodynamic (Paul) <cit.> and magnetic <cit.> traps. Detecting and controlling multiple particles in vacuum has so far involved single particle control with sympathetic cooling <cit.>, or entire optical systems for each individual particle <cit.>, which will not be scalable when working with arrays of tens, or even thousands of particles <cit.>.

We use neuromorphic detection in a truly scalable method for the control of particle arrays. Neuromorphic sensors are highly efficient detectors which mimic neurobiological vision <cit.>. Dynamic vision sensors mimic retinal response <cit.>, detecting changes across a threshold on each pixel in an array asynchronously to produce a stream of events <cit.>. Together with event-driven processing algorithms <cit.>, neuromorphic detection systems can achieve microsecond temporal resolution, sub-millisecond latency and high dynamic range (120 dB) detection with minimal data output at low power consumption <cit.>. Therefore, neuromorphic detection is highly suited to high-speed and real-time applications requiring low-power in environments with uncontrolled such as robotics <cit.>, autonomous driving <cit.> and space flight <cit.>, as well as finding applications in microscopy <cit.> and astronomy <cit.>. In this work, we use an event-based camera (EBC) with integrated neuromorphic sensor to monitor the motion of an array of levitated particles with a bandwidth high enough to demonstrate real-time simultaneous feedback control of multiple particles in an array.

We implement cold damping feedback to cool the motion of the levitated particles <cit.>, a technique with demonstrated ground state cooling capabilities <cit.>. We cool a single microsphere to sub-Kelvin temperatures and single degrees-of-freedom of multiple particles. In this work, the number of objects we can simultaneously control is only limited by the number of output channels in our feedback electronics. This single-device method for cooling particle arrays is readily scalable due to the low data output of neuromorphic detection. Such arrays of cooled micro-sensors will lead to enhanced sensitivity <cit.>, force gradient sensing <cit.>, and provide a larger interaction area without increasing the mass of the sensor <cit.>. Due to the low-power consumption of neuromoprhic detectors our presented methods are ideal for integration into chip-scale technology <cit.>.







  §.§.§ Neuromorphic imaging of levitated particle arrays



We levitate arrays of charged 5 μ m diameter silica microspheres in a linear Paul trap under vacuum conditions (see Methods) and record their motion using an event-based camers (EBC) <cit.>, see Fig. <ref>(a). The charged particles form a stable array due to the coulomb repulsion between them. The EBC uses a neuromorphic sensor, which is an array of independent pixels featuring contrast detectors which output an event in response to light levels on the pixel crossing a user-defined threshold. Pixels which do not experience the required level-change output no signal, removing the data-redundancy present in conventional cameras, while allowing the full sensor to be used at all times, sometimes referred to as a dynamic region-of-interest. The EBC hardware bundles asynchronous events into equal-length frames, and uses filters to identify objects within its field-of-view, afterwhich a propritetary generic tracking algorithm (GTA) tracks the motion of each object independently. For a more detailed explanation in the context of tracking single particles see <cit.>.

The EBC allows the tracking of multiple objects with a high bandwidth and a linear scaling in data output with the number of tracked objects, as compared to a rapid increase in data volume with increased region-of-interest in a conventional camera. Tracking of a single particle using the entire field-of-view of the EBC used ∼ 100 kB s^-1 as compared to 64,800 kB s^-1 using a standard CMOS camera with the same sensor size.  

In Fig. <ref>(b) we show an example of a single neuromorphic sensor being used to detect 10 levitated particles simultaneously. The EBC identifies each object and tracks it in 2D (illustrated by the coloured box around each image) and assigns each one a stable identification number, allowing us to process the positional data of each particle independently.

Particles levitated by a Paul trap undergo harmonic motion. In Fig. <ref>(c) we generate the power spectral density (PSD) for each particle in an array of four to analyze their motion in frequency space. Each particle has a different charge-to-mass ratio and is levitated in a different part of the confining field, meaning each particle has different frequency modes of oscillation, which are well separated under vacuum conditions. Below, we use this fact to independently cool the motion of multiple particles. Interactions between the particles are evident, as shown by the unlabelled peaks in Fig. <ref>(c). The signal-to-noise for the different particles varies due to non-uniform illumination and varying coupling to electronic noise. All data in this work is taken at gas pressures of 2.0-4.5× 10^-2mbar unless otherwise stated.




  §.§.§ Single particle cold damping using a neuromorphic detector




There are many reasons why it is desirable to control the energy of a levitated particle. Although the sensitivity of the levitated sensor does not change through cooling <cit.>, rapid damping of the motion increases the stability and measurement bandwidth of the system. By reducing the position fluctuations interactions between the particles can be minimized. Reduction of the particle energy to the ground-state of the levitating potential <cit.> opens up a toolbox of quantum control <cit.> and sensitivity enhancement <cit.>. 

Cold damping is a feedback method whereby a force proportional to the velocity of an oscillator is applied to its motion. Depending on the phase of the feedback signal, this method can cause damping or amplification of the particle motion without adding additional noise, hence the terminology cold damping. In reality, input and output noise of the feedback electronics still limits cooling.

We process the position data from the EBC using a Field Programmable-Gate-Array (FPGA) to generate a feedback signal proportional to velocity, with variable gain and phase. This signal is filtered and applied to an electrode of the levitating apparatus, see Fig. <ref>(a).

Figure <ref>(a) shows the PSD of a particle as a single degree-of-freedom (along the z-axis) is cooled via cold damping. The shape of the PSD is given by <cit.>:

    S_zz(ω)=2k_BT_CoMΓ_ t/m/(ω^2-ω_z^2)^2+Γ_ t^2ω^2,

where ω = 2π× f, m is the particle's mass, ω_z is the mode frequency, T_ CoM is the temperature associated with the mode being analyzed and Γ_ t = Γ_0 + Γ_ fb is the total momentum damping rate on the particle's motion, with Γ_0 being the calculable momentum damping rate due to the pressure of the surrounding gas <cit.> and Γ_ fb being the additional damping due to feedback. The parameters T_ CoM and Γ_ fb can be extracted from a measured PSD by fitting equation (<ref>) to the data. Due to voltage noise from the amplifiers driving our Paul trap, the equilibrium temperature of our particles without cooling is approximately T_0 = 900 K.

According to the equipartition theorem, the temperature of a levitated particle experiencing cold damping is <cit.>:

    T_CoM=T_0Γ_0/Γ_t+1/2mω_z^2/k_BΓ_fb^2/Γ_tS_nn,

where S_nn is noise due to the feedback defined by a constant spectral noise density. In  Fig. <ref>(b) we show the effect of increasing feedback gain on the temperature of a single mode of a single particle, and compare the data to the model in equation (<ref>), with only S_nn as a free parameter. 

The cooling depends on the phase ϕ between the feedback signal, which is proportional to velocity, and the detected motion of the particle, with the temperature varying as
<cit.>:

    T_CoM=Γ_0T_0/Γ_0+Γ_fbcos(ϕ+ϕ_0),

where ϕ_0 is the uncontrollable phase delay caused by the electronics. This model is fit to experimental data, Fig. <ref>(c), with Γ_fb and ϕ_0 as free parameters. The fitted value of ϕ_0 is 5^∘, noting that one- or two- periods of phase delay do not significantly effect the cooling for an underdamped oscillator as in our system <cit.>.

Finally, we combine the optimized Γ_fb and ϕ to push cooling to the limit by reducing Γ_0 through reducing the gas pressure in the experiment, as shown in Fig. <ref>(d). At low pressures we reach the noise floor of our system, indicated by the grey region, at a temperature corresponding to T_CoM=(0.5 ± 0.3) K, representing -30 dB of cooling. Further decreasing the pressure leads to noise squashing of our signals (not further cooling) due to our in-loop feedback loop <cit.>. To improve cooling further we could improve our particle illumination and imaging, decrease the noise in our levitation electronics, and replace the GTA on the EBC with an optimized tracking algorithm. 










  §.§.§ Simultaneous cooling of a microparticle array


Our neuromorphic imaging system gives us 2D readout of the motion of every object it identifies. We are able to process this signal, and make a feedback loop for each degree-of-freedom that is detected. Each feedback loop consists of a dedicated FPGA and set of analogue filters, and we are limited in our experiment to three loops in total. We stress that this is not a limitation of detection or processing power, simply the number of FPGA outputs available to us. For each degree-of-freedom the phase and gain of each feedback loop must be optimized, and filters must be set accordingly.

In Fig. <ref>(a) we cool two orthogonal degrees-of-freedom (the x- and z-axes) of a single particle. We are sensitive to all three degrees-of-freedom due to the angle our imaging system makes to the principal axes of the trap. The geometry of our Paul trap allows the control of all degrees-of-freedom with a single electrode. 

We optimize and fix the feedback parameters, then lower the background pressure to reduce Γ_0, hence lowering the temperature T_CoM. The temperature is limited not by our noise floor, but by imperfect filtering pumping energy from the feedback signal into the y-mode, since it is close in frequency to the z-mode. At low pressures this causes the particle to become unstable, preventing us from further lowering the pressure.



In Fig. <ref>(b) we extend our cooling to the z-mode of two separate particles, to the best of our knowledge the first direct cooling of two separate particles with a single detector. As we lower the pressure, the temperature of each mode drops, reaching ∼-10dB and ∼-15dB of cooling. Again, imperfect filtering leads to cross-talk between different modes, and at lower pressures this causes heating of the uncooled modes and particle instability. 

In Fig. <ref>(c) we cool the z-mode of three different particles, with the corresponding PSDs shown in Fig. <ref>(d). Different particles can have different noise floors, since their coupling to voltage noise depends on the particle charge. The issue of cross-talk is more pronounced when there are more modes of the system, yet we still achieve better than ∼-7dB of cooling.

The cross-talk between modes can be minimized through better filtering, use of phase-locked-loops <cit.>,  or further separation of the particle modes to enable more effective filtering. 



  §.§.§ Conclusions


We have presented a scalable method for the detection and control of arrays of levitated microparticles using a single neuromorphic detector. Neuromorphic imaging is ideally suited to this task, due to its natural affinity with detecting the motion of multiple objects and low data-transfer rate. The tracking speed is limited by the proprietary tracking algorithm of the EBC used in this work, but other work in our lab has achieved neuromorphic imaging at >100kHz frame-rates <cit.>.

The work presented here can straightforwardly be extended to arrays of 10s of microparticles, with multi-channel FPGA systems with high-quality digital filters being a common tool in research labs. Paul traps are stable at low pressures <cit.>, where the motional frequencies of levitated particles have sub-Hz linewidths <cit.>, meaning that the naturally varying charge-to-mass ratio of charged nano- and micro-particles will enable single-particle control and cooling even for large arrays. 

Since the motion of levitated sensors is well understood, simple machine-learning <cit.> could be used to optimize all of the feedback parameters in an array, and optimal tracking algorithms used which have been shown to enable quantum-level control <cit.>. When combined with the low power-consumption of neuromorphic imaging technology, and great progress in chip-scale particle levitation <cit.>, integrated devices containing arrays of quantum sensors are closer to being a reality.

Acknowledgements

We would like to thank Prof David Moore for useful discussions, Dr Ruvi Lecamwasam for assistance with writing FPGA cose, and Dr John Dale for bringing neuromorphic imaging to our attention. This work is supported by STFC Grant ST/Y004914/1, EPSRC Grant EP/S004777/1 and ERC Starting Grant 803277.







 §.§ Methods




  §.§.§ Experimental setup

Fig. <ref> illustrates the experimental setup surrounding our linear Paul trap. The trap consists of 4 parallel cylindrical trapping electrodes forming a square, with two coaxial cylindrical end-cap electrodes. The distance from trap centre to the surface of the 1 mm-diameter trapping electrodes is r=1.15  mm, and one opposing-pair are driven with 360 V amplitude at 1 kHz. The two endcaps are 300 μm-diameter and separated by 2z_0=800 μm. The proximity of the electrodes to the centre of the trap mean that a voltage applied to either one will create a field with a significant component in all axes, enabling 3D control with a single electrode. 

Laser-induced acoustic desorption (LIAD) <cit.> is used to launch micro-particles into the trap at a pressure of 1mbar and then lower the pressure to carry out the experiments presented in this manuscript. We typically trap particles of positive charge, ranging from 2×10^3 e to 2×10^4 e.

To image the particles, 18 mW of laser light of wavelength 520nm is weakly focused
onto the array. The scattered light is imaged onto an event-based camera (Prophesee EVK1 -Gen3.1 VGA (camera sensor: Prophesee PPS3MVCD, 640×480 pixels)) using a long-working distance microscope. The camera is precisely calibrated using a method outlined in detail in <cit.>. When dealing with multiple particles, calibration is performed via displacing a translation stage on which our imaging system is mounted by a known amount. 

Generation of feedback signals is described below, but a voltage is added to one endcap electrode to control the motion of the particles.































  §.§.§ Data processing

A data pipeline is shown in Fig. <ref>. The operation of the EBC is described in detail in <cit.>. The generic tracking algorithm (GTA) of the EBC outputs 2D position data for each object it detects. The EBC is communicated with via a Python script, which separates this 2D information into two 1D data streams for each object. The script passes each data stream to one of three FPGA systems (Red Pitaya STEMlab 125-14) to output the position of each particle which we can monitor. Code on the FPGA computes the velocity from the position data, and adds a variable gain and phase to the signal to generate the feedback signal. 

Each feedback signal is filtered with an anaolgue filter to isolate each frequency component of motion: a high-pass filter for the f_z signal and a low-pass filter for the f_x signal. The filtered feedback signals are combined with a summing amplifier, and sent to one endcap electrode.

Clock synchronization of the FPGA system and the EBC is achieved via enabling the camera clock only.
































































































 Paper Title: Non-Markovianity in Discrete-Time Open Quantum Random Walk on Arbitrary Graphs.


	
	Enabling Contextual Soft Moderation on Social Media
 through This paper is accepted for publication at the 2024 USENIX Security Symposium. Please cite accordingly.
    Pujan Paudel, Mohammad Hammas Saeed, Rebecca Auger, Chris Wells, and Gianluca Stringhini

Boston University

{ppaudel,hammas,raauger,cfwells,gian}@bu.edu

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

§ ABSTRACT 
		We present a new model of the Discrete-Time Open Quantum Walk (DTOQW) applicable to an arbitrary graph, thereby going beyond the case of quantum walks on regular graphs. The impact of noise is studied by constructing Kraus operators of different dimensions, making use of the Weyl operators. These Kraus operators are employed as the coin operators of the DTOQW. The walker dynamics is studied under the impact of non-Markovian amplitude damping, depolarizing and dephasing noise channels. In addition, the walk is implemented on arbitrary graphs, such as path graph, cycle graph, star graph, complete graph, and complete bipartite graph. The dynamics, due to the influence of noise, is gauged by computing the coherence and fidelity at different time steps. Further, the probability distribution, representing the availability of the quantum walker at different vertices of the graph, is computed at different time steps for the above noises.

  
		
		Keywords: Discrete-time open quantum walk, non-Markovian noise, probability distribution, coherence and fidelity. 
	
 
	
	
	

§ INTRODUCTION

	
    	The idea of quantum walk <cit.>, a quantum mechanical analogue of the classical random walk <cit.>, is a crucial tool for in quantum computation and information <cit.>. Different variants of quantum walks are applicable in quantum algorithms <cit.>, quantum simulation <cit.>, quantum communication <cit.>, quantum state transfer <cit.>, quantum search algorithms <cit.>, element distinctness <cit.> quantum routing <cit.>, ranking the vertices in a network <cit.>, among others. Different models of quantum walks include multiple variants of discrete-time quantum walk <cit.>, continuous-time quantum walk <cit.>, Szegedy quantum walk <cit.>, Fibonacci quantum walk <cit.>, open quantum walk <cit.>. The original ideas of discrete-time quantum walk, continuous-time quantum walk, Szegedy quantum walk and Fibonacci quantum walk are governed by a unitary operator that has no interaction with an external environment. Studying quantum walks in noisy environments is essential because real-world quantum systems are inevitably subject to various forms of noise <cit.>. It significantly affects the behaviour and performance of quantum walk <cit.>. Understanding the effect of these noises is crucial for designing robust quantum systems. The experimental implementations of quantum walk also analyze the effects of noise <cit.>.
     
        There are multiple proposals to incorporate the environmental noise in quantum walk <cit.>. A number of quantum operators are used as coin operators in discrete-time quantum walk, such as the Hadamard operator, Grover operator. Generally in traditional quantum walk models, noise and coin operators are treated separately <cit.>. Noise is usually applied as a separate process, mostly after the coin operation. The effects of noise on continuous-time quantum walk have also been discussed in the literature <cit.>.

        The open quantum walk <cit.> is a model of quantum walk which offers a platform to investigate the effect of noise on the dynamics of the quantum walk. The Discrete-Time Open Quantum Walk (DTOQW) is the discrete variant of an open quantum walk. In this work, we present a new model of DTOQW in a noisy environment on an arbitrary graph. In contrast to the other models of quantum walk in a noisy environment, our model integrates the noise directly into the coin operators, making the noise an intrinsic part of the quantum walk dynamics.  Understanding the properties of the associated noisy quantum channel is necessary to explore the effect of noise. We utilize a number of noisy quantum channels for our investigation, such as the non-Markovian Amplitude Damping Channel (ADC), non-Markovain dephasing (NMD) noise and non-Markovian depolarizing noise <cit.>.
    	 
    	The new model of DTOQW is defined on an arbitrary graph. This is another advantage of the new model, as the previous models of DTOQW were defined on regular graphs, where the degree of each vertex is the same, such as the infinite path graph, cycle graph <cit.>. In this article, we break this limitation. In our model, the vertices of the graph provide the location of the walker. The possible directions in which the walker can move are represented by the directed edges in the graph. 
    	
        To study the effect of noise we consider a number of quantum channels which were originally defined for qubit states. We can generalize these channels for qudits of arbitrary dimension using the Weyl operators <cit.>. This is essential as the degree of the vertices in an arbitrary graph is not unique. Corresponding to every outgoing edge from a vertex we assign a Kraus operator representing a noisy quantum channel. Our analysis involves the calculation of fidelity between the initial and final states of the walker at different time steps. It provides a quantitative measure of the impact of noise on quantum walker dynamics. Also, we work out the probability distribution for each vertex in different time steps. Coherence and fidelity are observed as a function of graph steps for various types of noise.
    	
        This paper is organized as follows. The preliminary ideas are elucidated in section 2. In section 3, we construct the coin operator of DTOQW on an arbitrary graph.  Here, we also generalize different noisy channels such as the non-Markovian amplitude damping channel, non-Markovian dephasing channel and non-Markovian depolarization channel, for arbitrary dimensions. In Section 4 we construct our DTOQW model on arbitrary graphs. In different subsections, we discuss DTOQW on different types of graphs, which are the path graph, cycle graph, star graph, complete graph, and complete bipartite graph. The probability distribution of the walker at different vertices of the graph, fidelity between the initial and final state as well as the coherence of the quantum state of the walker are calculated. We then conclude this article.
	
	
	

§ PRELIMINARIES

	
	A graph G = (V(G), E(G)) is a triple consisting of a vertex set V(G), a set of edges E(G) and a relationship that associates two vertices with each edge <cit.>. The number of vertices in a graph is called the order of the graph, which is denoted by n, throughout this article. A vertex u is said to be adjacent to a vertex v if (u, v) ∈ E(G). Also, the edge (u, v) is incident to u and v. A loop is an edge of the form (u, u) that connects a vertex u with itself. The neighbourhood of u is denoted by (u) which is the set of all vertices adjacent to it. The number of elements in (u) is the degree of vertex u, which we denote as d_u. We draw a vertex by a marked circle and an edge by a straight line joining two circles. 
	
	An edge with a direction is called a directed edge. A directed edge from the vertex u to the vertex v is denoted by (u, v). We say that the directed edge (u, v) is outgoing from u and incoming to v. The number of directed edges coming into a vertex v is called the indegree of v. Similarly, the number of edges going out from a vertex u is called the outdegree of u. We draw a directed edge by an arc with an arrowhead. A directed graph is a graph whose all edges are directed. A simple graph has no loop and directed edges. Given a simple graph G we can construct a directed graph G by assigning two opposite directions on every undirected edge. Therefore, if (u, v) be an undirected edge in a simple graph G, we construct two directed edges (u, v) and (v, u) in G.
	
	In this work, we utilize several simple graphs, such as path graphs, cycle graphs, star graphs, complete graphs, and complete bipartite graphs, depicted in figure <ref>. If a graph has n vertices we label them v_0, v_1, … v_n - 1. A path graph P_n has edges (v_0, v_1), (v_1, v_2), … (v_(n - 2), v_(n - 1)). Therefore, in P_n degree of v_0 and v_(n - 1) is 1 and all other vertices have degree 2. A cycle graph C_n has edges (u_0, u_1), (u_1, u_2), … (u_n-1, u_0). Therefore, every vertex in C_n has degree two. In a star graph S_n v_1, v_2, … v_n - 1 are adjacent only to the central vertex v_0. Therefore, there are (n - 1) vertices having degree 1 and v_0 has degree (n - 1). A complete graph K_n is a graph in which every vertex is adjacent to all other vertices. Therefore, every vertex in K_n has degree (n - 1). The vertex set of a complete bipartite graph K_m,n is partitioned into subsets V_1 and V_2, such that, there is no edge with both ends in the same partition.
	 
	
	Corresponding to a graph G with n vertices, we choose an n-dimensional Hilbert space ℋ^(n). A quantum state is a normalized vector |ψ⟩ belonging to a Hilbert space, with one such vector associated with each vertex. The space ℋ^(n) is spanned by the basis vectors |0⟩, |1⟩, …|(n - 1)⟩. Recall that, a quantum state of dimension 2 is called qubit. A quantum state is also represented by a density matrix. For a pure state |ψ⟩, the density matrix is ρ = |ψ⟩⟨ψ|. In general, the density matrix of a quantum state is a positive semidefinite, Hermitian matrix with unit trace. 
	
	A quantum channel is used to transmit either classical or quantum information and can be represented mathematically by a set of Kraus operators <cit.>. Any set of Kraus operators {B_k} representing a quantum channel must satisfy
	
    ∑_k B_k^† B_k = I,

	where I is the identity operator. The action of the quantum channel on the state ρ can be expressed as 
	
    ρ'=Λ(ρ) = ∑_k B_k ρ B_k^†,

	where ρ' is the resulting state after the evolution. 
	
	The qubit Hilbert space is spanned by the Pauli operators. For qudits, this is generalized by the Weyl operators <cit.>,which find a number of applications in quantum mechanics, particularly in the context of quantum optics <cit.>, quantum computation and information theory <cit.>, as well as in quantum teleportation <cit.>. The Weyl operators of order n are defined by 
	
    U_u,v= ∑_k=0^n-1 e^(2π i/n)ku|k⟩⟨(k+v)  n|,  where  0 ≤ v,u ≤ (n-1).

	It can be proved that U_u,v is unitary for all v and u, that is U_u,v^† U_u,v =  U_u,v U_u,v^† = I_n. Hence, they preserve the inner product structure of quantum states. When u = v = 0, we have U_0, 0 = I_n. If n = 2 for u = 1 and v = 0, we have U_1,0 = σ_z, which is the Pauli Z matrix. Also, we obtain σ_x  and i σ_y for U_01 and U_11, respectively.
	
	In this article, we study non-Markovian behaviour in a general open quantum walk. One of the most common signatures of non-Markovian behaviour is characterized by a flow of information from the environment back to the open quantum system, which suggests the presence of memory effects <cit.>. However, non-Markovian nature can be more complex in general <cit.>. We use a number of measures of non-Markovianity to study the open system under our consideration. The fidelity between two quantum states <cit.> represented by density matrices ρ and σ quantifies the distinguishability between them, and  is defined by
	
    F(ρ , σ) = (√(√(ρ)σ√(ρ)))^2,
       
	where 0 ≤ F(ρ, σ) ≤ 1. Also, F(ρ , σ) = 1 when ρ = σ. The dynamics of decoherence is one of the fundamental concerns in the study of open quantum systems. The quantum decoherence is the loss of quantum coherence in a state. Here, we use the l_1 norm of coherence to measure the quantum coherence of a state  <cit.> which is given by 
    C_l_1(ρ) = ∑_i≠ j|ρ_ij|.


	
 

§ CONSTRUCTION OF NON-MARKOVIAN COIN OPERATORS


    Since we are interested in studying the impact of noise in DTQWs on various graphs, we introduce a coin operation that directly accommodates the relevant noisy channel. 
    
	

 §.§ Amplitude Damping Channel (non-Markovian)

	
	The Amplitude Damping Channel (ADC) <cit.>, a non-unital process, describes the process by which a quantum system loses energy to its surroundings, that is energy dissipation of a quantum system to the environment. 
  The non-Markovian ADC on an arbitrary d-dimensional system can be defined as follows <cit.>
	
    C_0 = |0⟩⟨0| + √(1 - λ(t))∑_u = 1^d - 1|u⟩⟨u|,  and  C_u = √(λ(t))|0⟩⟨u|;

 Also, λ(t) = 1 - e^-gt[ g/lsinh(lt/2) + cosh(lt/2) ]^2,
	where l = √(g^2 - 2 γ g) and g is the spectral width of the system-environment coupling and γ is the spontaneous emission rate. Also, |0⟩ and |u⟩: u = 1, 2, … (d - 1) are the d dimensional state vectors in the computational basis of ℋ^(d). Note that, C_0 and C_u satisfy the conditions to be Kraus operators, as stated in equation (<ref>). 
	
	These operators are accommodated into a graph theoretic setting by applying them as the coin operators on the vertices of a graph. Let u be a vertex with outdegree d_u. The outgoing edges from u are u_1, u_2, … u_d_u. Also, we consider a loop (u, u) at the vertex u. Therefore, we need (d_u + 1) Kraus operators at u each of dimension n, which are as follows
	
    C_(u, u)    = |u⟩⟨u| + √(1 - λ(t))∑_(u, v)∈ E(G)|v⟩⟨v| + ∑_(u, w)∉ 	E(G)|w⟩⟨w|,
    
    	C_(u, v)    = √(λ(t))|u⟩⟨v|.

	Here, |u⟩, |v⟩, and |w⟩ are state vectors in ℋ^n. We apply C_(u, u) for the loop (u, u) and C_(u, v) for the edge (u, v). Note that,
	
    C_(u, u)^† C_(u, u) + ∑_(u, v)∈ E(G) C_(u, v)^† C_(u, v)
    
    	=    [|u⟩⟨u| + √(1 - λ(t))∑_(u, v)∈ E(G)|v⟩⟨v| + ∑_(u, w)∉ 	E(G)|w⟩⟨w|]^†
       ×[|u⟩⟨u| + √(1 - λ(t))∑_(u, v)∈ E(G)|v⟩⟨v| + ∑_(u, w)∉ E(G)|w⟩⟨w|] + ∑_(u, v)∈ E(G)[√(λ(t))|u⟩⟨v|]^†[√(λ(t))|u⟩⟨v|]
    
    	=    |u⟩⟨u | u|⟨%s|⟩u + (1 - λ(t)) ∑_(u, v)∈ E(G)|v⟩⟨v | v|⟨%s|⟩v + ∑_(u, w)∉ E(G)|w⟩⟨w | w|⟨%s|⟩w + λ(t) ∑_(u, v)∈ E(G)|v⟩⟨u | v|⟨%s|⟩v
    
    	=    |u⟩⟨u| + (1 - λ(t)) ∑_(u, v)∈ E(G)|v⟩⟨v| + ∑_(u, w)∉ E(G)|w⟩⟨w| + λ(t) ∑_(u, v)∈ E(G)|v⟩⟨v|
    
    	=    |u⟩⟨u| + ∑_(u, v)∈ E(G)|v⟩⟨v| + ∑_(u, w)∉ E(G)|w⟩⟨w| = I_n.

	Here, the parameter λ(t) is a function of time. Therefore, the coin operators are changed at every time step of the walker. Note that, at every step, the coin operators fulfil the condition for being Kraus operators.
	
	

 §.§ Non-Markovian Dephasing Channel

	
	The Kraus operators representing the Non-Markovian Dephasing (NMD) noise, an unital process, for qudits <cit.> are as follows,
	
    C_u,v = √(1 - κ(p)) U_00    when  u = 0, v = 0; 
    √(κ(p)/d^2 - 1) U_u, v    for  0 ≤ u, v ≤ (d - 1)  and  	(u, v) ≠ (0, 0).
 
	where
	
    κ(p) = p 1 + η (1 - 2p) sin(ω p)/1 + η (1 - 2p),

	and 0 ≤ p ≤1/2 as well as η and ω are two constants characterizing channel strength and frequency. 
	
	In equation (<ref>) there are (d^2 - 1) operators of the form U_u, v. We want to fit these operators with the outgoing edges from a vertex. Let u be a vertex with outdegree d_u. Thus, we need (d_u +1) coin operators associated to the outgoing edges from u. Hence, we modify the Kraus operators as follows:
	
    C_(u, u) = √(1 - κ(p)) U_0,0,  and  C_(u, v) = √(κ(p)/d_u)  U_u, v;

	Here, C_(u,u) is the coin operator associated to the loop (u,u) and C(u, v) are the coin operators associated to the directed edges (u, v). Note that,
	
    C_(u, u)^† C_(u, u) + ∑_(u, v)∈ E(G)  C_(u, v)^† C_(u, v)
    
    	=    [√(1 - κ(p))U_0,0]^†[√(1 - κ(p))U_0,0] +  ∑_(u, v)∈ E(G)[√(κ(p)/d_u) U_u,v]^†[√(κ(p)/d_u) U_u,v]
    
    	=     (1-κ(p))U_0,0^† U_0,0 + κ(p)/d_u∑_(u, v)∈ E(G) U_u,v^† U_u,v = (1  -κ(p))I_n + κ(p)/d_u d_u I_n = I_n.

	Here, the coin operators satisfy the Kraus operator condition for every value of p.	
	
	

 §.§ Non-Markovian Depolarization Channel

	
	Depolarization noise, unital in nature, also affects the quantum systems by causing the loss of information about their quantum states. For the qudits, this noise is characterized by the following Kraus operators <cit.>,
	
    C_u,u =    √(1 + (N^2 - 1)(1 - p)/N^2Λ_1)  I_N   for  u=0; 
    
    	C_u,v=    √(%s/%s)p Λ_2N^2  U_u, v for  0 ≤ u, v ≤ (N - 1),  and  (u, v) ≠ (0, 0).

	Also, Λ_1 = -α p, Λ_2 = α(1 - p) and (1-p)Λ_1 + pΛ_2 = 0.
	
	In order to utilize the non-Markovian depolarization channel as a coin operator on graph vertices, we need to further generalize it. Hence, the Kraus operators for this channel are of the form, 
	
    C_u,u = √(1 + d_u(1-p)Λ_1/d_u+1)  U_0,0,  and  C_u,v = √(pΛ_2/d_u+1)  U_u,v,

	where d_u is the outdegree of u. We apply C_(u, u) for the loop (u, u) and C_(u, v) for the edge (u, v). Note that
	
    C_(u, u)^† C_(u, u) + ∑_(u, v)∈ E(G) C_(u, v)^† C_(u, v)
    
    	=   [√(1 + d_u(1-p)Λ_1/d_u+1) U_0,0]^†[√(1 + d_u(1-p)Λ_1/d_u+1) U_0,0] + ∑_(u, v)∈ E(G)[√(pΛ_2/d_u+1)U_u, v]^†[√(pΛ_2/d_u+1)U_u,v]
    
    	=    (1 + d_u(1-p)Λ_1/(d_u+1)) U_0,0^† U_0,0 + pΛ_2/(d_u+1)∑_(u, v)∈ E(G) U_u,v^† U_u,v
    
    	=    (1+d_u(1-p)Λ_1/d_u+1)I_n + (pΛ_2/(d_u+1))(d_u) I_n
    
    	=    (1 + d_u(1-p)Λ_1/(d_u+1) + d_u pΛ_2/d_u+1)I_n = (1 + d_u/d_u+1((1-p)Λ_1 + d_uΛ_2))I_n = I_n.

	Here, also the coin operators satisfy the Kraus operator condition for every value of Λ_1 and Λ_2.
	
	
	
	

§ DTOQW ON ARBITRARY GRAPHS

	
	To describe the dynamics of the walker on any simple graph G, we convert it to a directed graph G by assigning two opposite directions on every undirected edge. Also, we add loops to all the nodes in G. Further, every directed edge acts as a channel. Let u be any vertex with degree d_u in G. For simplicity, we can assume that there are directed edges (u, u_1), (u, u_2), …(u, u_d) and a loop (u, u) incident to u in the graph G. We depict a graph G, and G in figure <ref>. 
	
	
	The discrete-time quantum walk is implemented by the coin and shift operators. The coin operators were discussed above in section <ref>. For every directed edge (u, v) we associate a coin operator C_(u,v) and a shift operator S_(u,v) = |v⟩⟨u|. The transition operator for each edge (u,u_l) for l = 1, 2, … d_u is given by B_(u,u_l) = C_(u,v)⊗ S_(u,v). Also, for the loop (u, u) we can define the transition operator as B_(u, u) = C_(u,u)⊗ S_(u,u). Here, ⊗ denotes the tensor product between two operators. In figure <ref>, we indicate the transition operators on a digraph G. Now, we find a set of operators at each node u which can lead a walker through the edges going out from it.
	As we assume S_(u,v) = |v⟩⟨u|. Therefore, S_(u,v)^† S_(u,v) = |u⟩⟨v|v|⟨%s|⟩u = |u⟩⟨u|.
	
	For the transition operators associated with the outgoing edges from a vertex u, we find that
	
    B_(u, u)^† B_(u, u) + ∑_(u, v)∈ E(G) B_(u, v)^† B_(u, v) = C_(u,u)^† C_(u,u)⊗|u⟩⟨u| + ∑_(u, v)∈ E(G) C_(u,v)^† C_(u,v)⊗|u⟩⟨u|
        = (C_(u,u)^† C_(u,u) +  ∑_(u, v)∈ E(G) C_(u,v)^† C_(u,v)) ⊗|u⟩⟨u| = I_n ⊗|u⟩⟨u|.

	Now, considering all the vertices in G we observe that
	
    ∑_u ∈ V(G)[B_(u, u)^† B_(u, u) + ∑_(u, v)∈ E(G) B_(u, v)^† B_(u, v)] 
        = ∑_u ∈ V(G) I_n ⊗|u⟩⟨u| = I_n ⊗∑_u ∈ V(G)|u⟩⟨u| = I_n ⊗ I_n = I_n^2.
    or    ∑_u ∈ V(G)∑_(u, v)∈ E(G)∪{(u,u)} B_(u,v)^† B_(u,v) = I_n^2.

	Therefore, the set of all transition operators B_(u,v) forms a set of Kraus operators on G. The total number of B_(u,v) operators is (2m+n), where m is the number of edges in G and n is the number of vertices in G. 
	
	Let the initial state of the walker be given by the density matrix.
	
    ρ^(0) = ∑_u ∈ V(G)ρ_u^(0)⊗|u⟩⟨u| + ∑_v ≠ u ∈ V(G)ρ_v^(0)⊗|v⟩⟨v|.
 
	The walker can initiate walking from an arbitrarily chosen vertex. Hence the probability of selecting the initial vertex is uniform which is 1/n. Therefore, here we consider ρ_u^(0) = J_n/n and ρ_v^(0) = J'_n/n, where J_n is all-one matrix of order n and J'_n are all-zero matrices of order n. The probability of getting the walker at vertex u at time t = 0 is (ρ_u^(0)). Now, the density matrix of the walker after (k + 1)-th step is given by
	
    ρ^(k+1)    =  Λ(ρ^(k)) = ∑_u ∈ V(G)∑_(u, v)∈ E(G)∪{(u,u)} B_(u,v)ρ_u^(k)⊗|u⟩⟨u| B_(u,v)^†
        = ∑_u ∈ V(G)∑_(u, v)∈ E(G)∪{(u,u)} C_(u,v)ρ_u^(k) C_(u,v)^†⊗|v⟩⟨u|u|⟨%s|%s⟩⟩u|u⟨v|
        = ∑_u ∈ V(G)[∑_(u, v)∈ E(G)∪{(u,u)} C_(u,v)ρ_u^(k) C_(u,v)^†] ⊗|v⟩⟨v|
        = ∑_u ∈ V(G)[∑_(v, u)∈ E(G)∪{(v, v)} C_(v,u)ρ_v^(k) C_(v, u)^†] ⊗|u⟩⟨u|.  [By changing the index u and v.]

	Comparing with equation (<ref>) we find that 
	
    ρ_u^(k + 1) = ∑_(v, u)∈ E(G)∪{(v, v)} C_(v,u)ρ_v^(k) C_(v, u)^†.

	The probability of getting the walker at vertex u at time-step (k + 1) is (ρ_u^(k + 1)). Below we discuss the quantum walk and its non-Markovian properties, characterized by coherence and fidelity, on the graphs depicted in figure <ref>. We assume that the walker starts to move from the node u_0 in all the considered graphs. Further, we take different types of graphs like path, cyclic, star, complete and complete bipartite graphs under the non-Markovian ADC, NMD and non-Markovian depolarization noise channels, respectively. These coin operators are applied to the state of the walker at different time steps and the probability distribution at different vertices, fidelity and coherence of the state with steps and channel parameters are calculated and discussed below.
	
	

 §.§ DTOQW on path graphs
 
	
	Using equation (<ref>), we can derive the coin operators corresponding to the non-Markovian ADC channel for the path graph with n vertices P_n. They are as follows:
	
    C_(u,u) = |u⟩⟨u| + √(1-λ)|u+1⟩⟨u+1| + ∑_w ≠ u|w⟩⟨w|    if  u = 0,
    |u⟩⟨u| + √(1-λ)|u-1⟩⟨u-1| + ∑_w ≠ u|w⟩⟨w|    if  u = (n - 1)
    |u⟩⟨u| + √(1-λ)(|u-1⟩⟨u-1| + |u+1⟩⟨u+1|) 
     + ∑_w ≠ u|w⟩⟨w|    if  u ≠ 0  and  u ≠ (n - 1)
       C_(u,v) = √(λ)|u⟩⟨v|.

	In matrix form C_(u,u) is a diagonal matrix of order n and C_(u,v) is a matrix of same order n with √(λ) in (u,v)-th position with other zero entries. For simplicity, we consider a path graph P_5 with five vertices which is depicted in figure <ref>. Also, the shift operators will be S_u,v = |v⟩⟨u| for u, v = 0, 1, … (n - 1).
	
	S_(0,0) = |0⟩⟨0| , S_(1,1) = |1⟩⟨1|, S_(2,2) = |2⟩⟨2|, S_(3,3) = |3⟩⟨3|, S_(4,4) = |4⟩⟨4|, S_(0,1) = |1⟩⟨0|, S_(1,0) = |0⟩⟨1|, S_(1,2) = |2⟩⟨1|, S_(2,1) = |1⟩⟨2|, S_(2,3) = |3⟩⟨2|, S_(3,4) = |4⟩⟨3|, S_(4,3) = |3⟩⟨4|.
	We apply the DTOQW operators on the initial state discussed in equation (<ref>). Figure <ref> shows that only the walker's probability of being at different vertices changes as the step count rises for nodes 0 and 1 under non-Markovian ADC but the probability for nodes 2, 3 and 4 remains constant throughout the steps, showing that the walker is not visiting these nodes. This is because the connectivity and structure of the graphs significantly influence the impact of noises on the quantum walker. Also, ADC causes a loss of energy from the system to its environment and energy loss translates into a reduction in the probability of the walker's state. Path graph, as described above, has limited connectivity due to which the walker can move only to its neighbour vertex, figure <ref>. The figure <ref> shows the coherence and the fidelity of a quantum walk on a path graph under non-Markovian ADC. Coherence decreases with increasing steps, indicating a steady loss of coherence over time. Lower γ values result in a slower decline in coherence, whereas higher γ values cause faster decoherence. The fidelity graph fluctuates significantly in the initial steps before gradually stabilizing under the given parameter conditions.
	
	 
	
	
	
	The coin operators for the path graph under the NMD channel are as follows,
	
    C_(u,u) = √(1-κ) I_n,  C_(0,1) = √(κ) U_(0,1)
       C_(n-1,n-2) = √(κ) U_(n-1,n-2),  C_(u,v) = √(κ/2) U_(u,v) for  u ≠ 0  and  u ≠ (n-1).

	The operators C_(u,u) form a diagonal matrix with all the same elements √(1-κ) on the diagonal. The effects of these coin operators on the probability distribution of the walker, coherence and fidelity under NMD noise are shown in figures <ref> and <ref> respectively. The probability of the walker decreases at node 0 while increasing at all other nodes with steps and noise parameter p. The coherence and fidelity decrease with steps and channel parameters. 

	 
	
	 
	
	The coin operators for the non-Markovian depolarization channel are defined as,
    
    C_u,u = √(1 + 2(1-p)Λ_1/3) U_0,0,  for  u ≠ 0  and  u ≠ (n-1)  
       C_(u,u) = √(1 + (1-p)Λ_1/2) U_0,0,  for  u = 0 = (n-1)  
       C_u,v = √(pΛ_2/3) U_(u,v),  for  u ≠ 0  and  u ≠ (n-1)
       C_u,v = √(pΛ_2/2) U_(u,v),  for  u= 0  and  u = (n-1).

	
	Again, we apply these coin and shift operators on the initial density matrix (<ref>) of the walker and see the effect of these operators on the probability distribution, coherence and fidelity under non-Markovian depolarization noise, as depicted in figures <ref> and <ref>, respectively.  Here the probability distribution for the path graph shows non-zero probability at all nodes because depolarization noise primarily affects the coherence and randomness, allowing the walker to maintain its ability to explore the graph and is shown in figure <ref>. The coherence and the fidelity decrease with the steps and p while α is constant.  We observe that the coherence and the fidelity patterns for all the other considered graphs are qualitatively the same under the respective non-Markovian noises.
	
	 
	
	
	
	

 §.§ DTOQW on cycle graphs

	
	In this type of graph, the walker can move to any vertices through any edge. A cycle graph with three vertices is shown in figure <ref>.  The coin operators of this cycle graph for the non-Markovian ADC are given as
	
    C_(u,u) = |u⟩⟨u| + √(1-λ) (|u-1⟩⟨u-1| + |u+2⟩⟨u+2|) + ∑_w ≠ u|w⟩⟨w|, 
       C_(u,v) = √(λ)|u⟩⟨v|.
 
	In matrix form C_(u,u) is a diagonal matrix of order n and C_(u,v) is a matrix of same order n with √(λ) in (u,v)-th position with other entries zero. Also, the shift operators will be of the form
	
	S_(0,0) = |0⟩⟨0| , S_(1,1) = |1⟩⟨1|, S_(2,2) = |2⟩⟨2|,  S_(0,1) = |1⟩⟨0|, S_(1,0) = |0⟩⟨1|, S_(0,2) = |2⟩⟨0|, S_(2,0) = |0⟩⟨2|, S_(1,2) = |2⟩⟨1|, S_(2,1) = |1⟩⟨2|. The probability distribution for the cycle graph <ref> are shown in figures <ref>. It can be observed from figure <ref> that due to connectivity and multiple paths, the walker can move to all the nodes, despite the effect of damping. We observe that the probability of the walker becomes stationary after around 25 steps. Thereafter, there is no effect of the noise on the walker's probability with time-steps and an approximately stationary probability distribution is achieved. 
	
	
	
	The coin operators for non-Markovian dephasing channels can be defined as
	
    C_(u,u) = √(1-κ) I_n,
       C_(u,v) = √(κ/2) U_(u,v).

	Here also, the operators C_(u,u) form a diagonal matrix with the same elements √(1-κ) on the diagonal. We apply these coin and shift operators on the DTOQW of the walker. The probability distribution of the walker are shown in figures <ref>.  The probability of the walker at node 0 decreases while at the other nodes of the cyclic graph increases with steps and p. 
	
	
 	
	The coin operators for the non-Markovian depolarization channel can be defined for the cyclic graph as follows,
	
    C_u,u = √(1 + 2(1-p)Λ_1/3) U_0,0,
       C_u,v = √(pΛ_2/3) U_(u,v).

	We apply these coin and shift operators on the DTOQW of the walker. In the figure <ref>, we observe the probability of the walker on all vertices due to the connectivity on the cyclic graph <ref>. We get the same probability distribution on the adjacent vertices due to the same path from the starting vertex.
	
	 
	
	

 §.§ DTOQW on star graphs

	
	A star figure is shown in figure <ref>. Here the walker can jump only between adjacent vertices. The coin operators for the star graph under the non-Markovian ADC noise are 
	
    C_(0,0) = |0⟩⟨0| + √(1-λ) (∑_r ≠ 0^n-1|r⟩⟨r|), 
        C_(u,u) = |u⟩⟨u| + √(1-λ)|0⟩⟨0| + ∑_w ≠ u|w⟩⟨w|,
        C_(u,v) = √(λ)|u⟩⟨v|.

	
	In matrix form C_(u,u) is a diagonal matrix of order n and C_(u,v) is a matrix of same order n with √(λ) in (u,v)-th position with other entries zero. Also, the shift operators are,
	
	S_(0,0) = |0⟩⟨0| , S_(1,1) = |1⟩⟨1|, S_(2,2) = |2⟩⟨2|, S_(3,3) = |3⟩⟨3|, S_(4,4) = |4⟩⟨4|, S_(0,1) = |1⟩⟨0|, S_(0,2) = |2⟩⟨0|, S_(0,3) = |3⟩⟨0|, S_(0,4) = |4⟩⟨0|, S_(1,0) = |0⟩⟨1|, S_(2,0) = |0⟩⟨2|, S_(3,0) = |0⟩⟨3|, S_(4,0) = |0⟩⟨4|. Here non-Markovian ADC noise shows different behaviour as for the star graph, the central vertex is connected to all other vertices thereby enhancing the connectivity. The probability at the node 0 starts decreasing and at the other nodes starts increasing with steps due to this type of structure and the probability distribution becomes stationary after some steps, as can be seen from figure <ref>.
	 
	
	The coin operators for the star graph under the NMD channel are given by 
	
	
    C_(u,u) = √(1-κ) I_n,
       C_(u,v) = √(κ/n-1) U_(u,v), for  u=0
       C_(u,v) = √(κ) U_(u,v) for  u ≠ 0.
	
	
	From equation <ref>, we can see the operators C_(u,u) form a diagonal matrix with the same elements √(1-κ) on the diagonal.  The observations for the probability distribution under NMD noise for the star graph with 5 vertices are shown in figures <ref>. The probability of the walker at the node 0 on the star graph decreases while increasing at the other nodes with steps and p.
	
	 
		

	The coin operators for the star graph under a non-Markovian depolarization channel can be described as
	
    C_0,0 = √(1 + (n-1)(1-p)Λ_1/n) U_0,0, 
       C_(u,u) = √(1 + (1-p)Λ_1/2) U_0,0,  for  u = 0  
       C_(0,u) = √(pΛ_2/n) U_(0,u)
       C_u,0 = √(pΛ_2/2) U_(u,0),  for  u =0.

	The observed probability distribution on the star graph under non-Markovian depolarization noise is shown in figure <ref>. As can be seen from figure  <ref>, depolarization noise causes the probability at vertex 0 to decrease while increasing the probability at peripheral vertices (vertices 1 to 4). Over time, the probabilities stabilize, reflecting a stationary distribution across all nodes due to the randomizing effect of non-Markovian depolarization noise. Due to the structure of the star graph, the probability for the other nodes in the graph increases up to 10 steps and then becomes stationary.
	
	
	

 §.§ DTOQW on complete graphs

	
	A complete graph K_5 with five vertices is shown in figure <ref>. The non-Markovian ADC coin operators for this graph are,
	
    C_(u,u) = |u⟩⟨u| + √(1-λ)∑_r ≠ u^n-1|u⟩⟨u|,  and  C_(u,v) = √(λ)|u⟩⟨v|.
 			  
	
	In matrix form C_(u,u) is a diagonal matrix of order n and C_(u,v) is a matrix of same order n with √(λ) in (u,v)-th position with other zero entries. Also the shift operators for the graph <ref> are of the form, 
	
	S_(0,0) = |0⟩⟨0| , S_(1,1) = |1⟩⟨1|, S_(2,2) = |2⟩⟨2|, S_(3,3) = |3⟩⟨3|, S_(4,4) = |4⟩⟨4|, S_(0,1) = |1⟩⟨0|, S_(0,2) = |2⟩⟨0|, S_(0,3) = |3⟩⟨0|, S_(0,4) = |4⟩⟨0|, S_(1,0) = |0⟩⟨1|, S_(1,2) = |2⟩⟨1|, S_(1,3) = |3⟩⟨1|, S_(1,4) = |4⟩⟨1|, S_(2,1) = |1⟩⟨2|, S_(2,3) = |3⟩⟨2|, S_(2,4) = |4⟩⟨2|, S_(3,1) = |1⟩⟨3|, S_(3,2) = |2⟩⟨3|, S_(3,4) = |4⟩⟨3|, S_(4,1) = |1⟩⟨4| S_(4,2) = |2⟩⟨4|, S_(4,3) = |3⟩⟨4|. 
	
	Probability distribution plots are shown in figures <ref> for the complete graph <ref> under non-Markovian ADC. In figure <ref>, initial oscillations in the probabilities are due to high connectivity, gradually dampening over time and becoming stationary after some steps.
	
	
	The non-Markovian dephasing coin operators for the complete graph can be described as follows,
	
    C_(u,u) = √(1-κ) I_n,  C_(u,v) = √(κ/n-1) U_(u,v).

	Here, the operators C_(u,u) form a diagonal matrix with all the same entries √(1-κ). The plots for the probability distribution of the walker on the graph under NMD are shown in figures <ref>. The probability plots show that the probability decreases over steps with oscillations due to memory effects at vertex 0 while the probability of a quantum walker increases over steps at vertices 1, 2, 3 and 4. 
	 
	
	The coin operators for the complete graph under non-Markovian depolarization channel can be described as,
	
    C_u,u = √(1 + (n-1)(1-p)Λ_1/n) U_0,0, C_u,v = √(pΛ_2/n) U_(u,v).

	Here the coin operators for loops C_(u,u) forms a diagonal matrix with all entries equal to √(1 + (n-1)(1-p)Λ_1/n).
	
	The observations for the probability distribution of the walker for the complete graph <ref> under non-Markovian depolarization noise can be seen in figures <ref>. The probability plots show that the probability of the walker at the node 0 is decreasing while increasing with equal distribution at the other nodes with the steps due to the structure of the complete graph.
	 
	

	

 §.§ DTOQW on complete bipartite graphs

	
	A complete bipartite graph is shown in figure <ref> in which one subset belongs to the set (u_0, u_1) and the another subset to  (u_2, u_2, u_3, u_4).
	
	The general form of coin operators for the complete bipartite graph under non-Markovian ADC are,
	
    C_(u,u) = ∑_u=0^m |u⟩⟨u| + √(1-λ)∑_r ≠ u^n-1|r⟩⟨r|, 
       C_(u,u) = √(1-λ)∑_u=0^m |u⟩⟨u| + √(1-λ)∑_r ≠ u^n-1|r⟩⟨r|, 
       C_(u,v) = √(λ)|u⟩⟨v|.

	
    The shift operators are of the form defined as,
	
	S_(0,0) = |0⟩⟨0| , S_(1,1) = |1⟩⟨1|, S_(2,2) = |2⟩⟨2|, S_(3,3) = |3⟩⟨3|, S_(4,4) = |4⟩⟨4|, S_(0,2) = |2⟩⟨0|, S_(0,3) = |3⟩⟨0|, S_(0,4) = |4⟩⟨0|, S_(1,2) = |2⟩⟨1|, S_(1,3) = |3⟩⟨1|, S_(1,4) = |4⟩⟨1|,  S_(2,0) = |0⟩⟨2|, S_(2,1) = |1⟩⟨2|, S_(3,0) = |0⟩⟨3|, S_(3,1) = |1⟩⟨3|, S_(4,0) = |0⟩⟨4|, S_(4,1) = |1⟩⟨4|.
	The probability distribution for the complete bipartite graph <ref> under non-Markovian ADC is depicted in figures <ref>. The probability distribution of the walker on the complete bipartite graph decreases at vertex 0 with steps and then becomes stationary. Vertices 2, 3, and 4 show increasing probabilities with oscillatory patterns initially, thereafter it becomes stationary with steps. The probability at vertex 1 remains negligible due to its initial disconnected state.
	 
		
	The coin operators for the general complete bipartite graph under a non-Markovian dephasing channel are defined as,
	
    C_(u,u) = √(1-κ) I_n,
       C_(u,v) = √(κ/n-1) U_(u,v), 
       C_(u,v) = √(κ/n-1) U_(u,v).

	Here the operators C_(u,u) form a diagonal matrix with all entries equal to √(1-κ). The probability distribution of the walker on the complete bipartite graph under NMD noise are shown in the figures <ref>. The probability of the walker decreases at vertex 0 while increasing at the other nodes with steps and becomes stationary after 20 steps.
	 
	

	The coin operators for the complete bipartite graph in general, can be characterized as,
	
    C_u,u = √(1 + (n-1)(1-p)Λ_1/n) U_0,0,
       C_u,v = √(pΛ_2/n) U_(u,v), 
       C_u,v = √(pΛ_2/n) U_(u,v).

	Here, the coin operators for loops C_(u,u) form a diagonal matrix with equal entries √(1 + (n-1)(1-p)Λ_1/n). The probability distribution of the walker on the complete bipartite graph under non-Markovian depolarization noise is depicted in the figure <ref>. The probability of the walker decreases at vertex 0 while increasing at the other nodes with steps and becomes stationary after some steps. 
	 
	

	
	

§ CONCLUSION


        We focus on the effect of noise in discrete-time quantum walks on arbitrary graphs, going beyond the case of regular graphs. A new model is proposed to investigate the effect of noise in the Discrete-Time Open Quantum Walk (DTOQW). To study the impact of noise in discrete-time quantum walks, the coin and noise operators are usually applied separately. In contrast, in this work, we replace the coin operators with the Kraus operators of the corresponding noisy channels. Weyl operators are used to develop the Kraus operators in arbitrary dimensions.	
 
        Noise from the surrounding environment inevitably affects a quantum walk. We have taken into consideration three physically relevant noise models: non-Markovian amplitude damping, non-Markovian dephasing, and non-Markovian depolarization. The Kraus operators corresponding to each considered non-Markovian noise was generalized for the qudit system in order to make them compatible for application on an arbitrary graph. The specific examples of arbitrary graphs chosen for illustration were the path, cyclic, star, complete, and completely bipartite graphs. The probability distribution of the walker at different vertices of the considered graph were calculated. This shed light on the type of graph on which the DTOQW was implemented. Further, the fidelity between the two states of the walker at different time steps and the coherence of the state were also studied. We observed that under the considered non-Markovian noise channels, the coherence and fidelity patterns are qualitatively similar for all the considered graphs bringing out their dependence on the nature of noise rather than the structure of the graphs.
	
        DTOQW would help in modelling the behaviour of complex networks, including social, biological, and computer networks. Understanding their dynamics could provide insight into various real-world phenomena. In this setting, our findings would be useful.	


    

§ FUNDING


        This work is partially supported by the project “Transmission of quantum information using perfect state transfer" (CRG/2021/001834) funded by the Science and Engineering Research Board, Government of India. 

        



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 Paper Title: SQUIDs for detection of potential dark matter candidates.


    
    
    

SQUIDs for detection of potential dark matter candidates
    Siddarth Sivakumar^1, Manan Agarwal^2, Hannah Rana^1

    ^1Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA, USA

   ^2Anton Pannekoek Institute/GRAPPA, University of Amsterdam, Amsterdam, The Netherlands
    
    
====================================================================================================================================================================================================================================

§ ABSTRACT 
Superconducting QUantum Interference Devices (SQUIDs) are extremely sensitive magnetic flux sensors which render them useful in a wide array of instrumentation. SQUIDs are often paired with other detectors as a readout mechanism to obtain quantitative insight. SQUIDs have impacted many fields but much less addressed is its impact on the field of fundamental physics, particularly in the search for dark matter. Dark matter is believed to make up around 27% of all mass-energy content of the universe and will provide critical insight into understanding large-scale structures of the universe. Axions and WIMPs are the prominent two dark matter candidates whose search has been fuelled by the usage of SQUID readouts. 



§ INTRODUCTION

Superconducting QUantum Interference Devices (SQUIDs), inherently flux-to-voltage transducers, can be used to measure any physical quantity convertible into magnetic flux, used in an array of instrumentation such as signal amplifiers, magnetometers and gradiometers. SQUIDs are considered vital in instrumentation due to their quantum sensitivity (field resolution of 10 -17 T <cit.>) and enormous operational bandwidth. SQUIDs employ Josephson junctions in a superconducting loop; two in direct current (DC) SQUIDs and one in radio frequency (RF) SQUIDs. Due to their dependence on superconductors, they require cryogenic temperatures around 9 K. Although, recent advancements have brought the advent of high-T_C SQUIDs which have a relatively high operational temperature of 77 K. SQUID-based applications <cit.> extend to various fields; recently even concepts of military applications have been conceived <cit.>. SQUIDs are used as magnetometers in biology instrumentation: magnetoencephalography, magnetocardiography and gastro-magnetometry. Geophysical applications include the liquid-nitrogen-cooled SQUIDs that provide insight on low conductivity anomalies at depths from a few metres to tens of kilometres, allowing us to predict the presence and depth of oil and mineral ores. High-TC SQUIDs have piqued the interest in Non-Destructive Evaluation (NDE) of materials used to locate flaws that may compromise the integrity of structures later on; it can be used to test aircraft structures, steel-reinforced concrete, and more. Our focus in this review, however, will be their extended usage in particle detection for the detection of dark-matter candidates <cit.>, an emerging approach in the search for dark matter in fundamental physics. SQUIDs are used as readout amplifiers in the detection of axions and Weakly Interacting Massive Particles (WIMPs), potential dark-matter candidates, as they offer better sensitivity than their traditional counterparts.




§ SEARCH FOR DARK MATTER

Dark matter has been theorized to help us explain the formation and evolution of galaxies and large-scale structures in the universe; the existence of dark matter is inferred from various astrophysical observations, such as the rotation curves of galaxies, gravitational lensing, and the cosmic microwave background <cit.>. It is believed to constitute about 27% of the universe's mass-energy content. Despite no interaction with light and being detectable only through its gravitational effects, our search for dark matter continues through multiple experiments pushing the boundaries. Researchers have proposed several hypothetical candidates for dark matter particles, with WIMPs and axions being two prominent possibilities.


    
  * WIMPs interact only through the weak nuclear force and gravity, making them difficult to detect. WIMPs are predicted by extensions of the Standard Model of particle physics, such as supersymmetry <cit.>. Their mass is expected to be in the range of 10 to 1000 times that of a proton (10 to 1000 GeV/c^2), and interact with normal matter at a rate much lower than that of photons or other particles.
    
  * Axions, proposed as a solution to the strong CP problem in quantum chromodynamics, interact extremely weakly with normal matter and could contribute to the dark matter density in the universe. They are expected to have masses much lower than WIMPs, potentially in the microelectronvolt (μ eV) range <cit.>.




§ ADMX - AXION DETECTION



 §.§ Background

The Axion Dark Matter eXperiment (ADMX) at Lawrence Livermore National Laboratory has been searching for axions since 1996 <cit.>. The primary challenge in axion detection is their extremely weak interaction with ordinary matter, requiring highly sensitive detection methods. Traditional detection systems, such as those using the GaAs field-effect transistor (HFET) amplifiers, have inherent noise limitations. The novel approach of using SQUID amplifiers, with their near quantum-limited noise performance, significantly enhances the sensitivity and scan rate of the search. 



 §.§ Experimental Setup



The experimental setup includes:


    
  * A 7.6 Tesla superconducting solenoid with a 0.5-meter diameter bore.
    
  * A cylindrical, copper-plated microwave cavity inside the magnet bore.
    
  * A DC SQUID amplifier placed above the magnet to minimize interference from the magnetic field.

Dark matter axions passing through the cavity are expected to convert into microwave photons, detectable by the resonant cavity. The experiment scanned for axion signals by tuning the cavity frequency and integrating the signal over time to improve the signal-to-noise ratio. The cavity's frequency is tunable by moving copper-plated axial rods, allowing the experiment to scan different axion masses. The expected power from axion-photon conversion, given by the inverse Primakoff effect, is extremely low, requiring ultra-sensitive amplification. The power generated by axion-photon conversion depends on several factors, including the magnetic field strength, cavity volume, axion density, and the quality factors of the cavity and axion signal.



 §.§ Role of SQUIDs




In the ADMX experiment, SQUID amplifiers replace conventional HFET amplifiers, due to several key advantages:


    
  * Noise Temperature: The SQUID amplifier operates with a noise temperature close to the quantum limit, significantly lower than that of HFET amplifiers, allowing for better sensitivity and faster scan rates. The SQUID amplifier has achieved noise temperatures as low as 47 mK, close to the quantum noise limit of 33 mK at 700 MHz.
    
  * Thermal Stability: SQUIDs maintain their low noise temperature down to very low physical temperatures, unlike HFET amplifiers which plateau at a few Kelvin.
    
  * Gain Performance: The specific design of the SQUIDs used in ADMX includes a resonant microstrip input coil, which mitigates the gain roll-off issue at microwave frequencies.

 


 §.§ Results and Analysis


    
  * Noise Reduction: In the ADMX experiment, the SQUID amplifiers demonstrated a noise temperature of approximately 50 mK at 1 GHz, which is close to the quantum limit of 33 mK. This low noise temperature is crucial for enhancing the signal-to-noise ratio (SNR) and, consequently, the sensitivity of the axion search. 
    
  * Sensitivity: The experiment's sensitivity is quantified by the Dicke radiometer equation, which links SNR to system noise temperature, integration time, and bandwidth. With the introduction of SQUID amplifiers, the system noise temperature is minimized, leading to a higher SNR and allowing for a more efficient scan of axion masses.
    
  * Improvement: The use of the SQUID amplifier significantly improved the scan rate compared to previous runs using HFET amplifiers.
    
  * Findings: The ADMX experiment was able to rule out certain models of dark matter axions with masses between 3.3 and 3.53 μeV, with 90% confidence, assuming a local dark matter density of 0.45 GeV/cm^3. This new result extends previous research that had only ruled out axions up to 3.3 μeV. While the hunt for dark matter candidates and axions in particular will continue, current findings suggest that they are less likely to have masses in the 3.3 to 3.53 μeV range. 




 §.§ Future Directions and Conclusion

Future experiments will use dilution refrigeration to further lower the temperature of the SQUID and cavity, potentially increasing the scan rate and sensitivity. The goal for the future is to cover the entire plausible axion mass range with higher sensitivity, even for the most pessimistic axion-photon couplings. The application of SQUIDs in the ADMX experiment showcases its potential in particle detection instrumentation. The improvements in sensitivity and scan rate afforded by SQUID amplifiers represent a critical step forward in the ongoing quest to detect dark matter axions. 



§ CRESST - WIMP DETECTION




 §.§ Background

The Cryogenic Rare Event Search with Superconducting Thermometers (CRESST) experiment <cit.> uses cryogenic detectors that leverage SQUIDs, which play a huge role in enhancing the sensitivity and precision of WIMP detection. The experiment is set up in the Laboratori Nazionali del Gran Sasso (LNGS) underground laboratory in Italy, which provides a shielded environment to minimize background noise from cosmic rays and other sources. 



 §.§ Experimental Setup

The core idea of the 2004 data run of the CRESST-II phase of the experiment is to identify WIMP interactions by detecting the tiny amounts of energy deposited by the phonon (heat) signal and scintillation light, both of which need to be detected simultaneously and with high precision by sensitive tungsten phase-transition thermometers, when a WIMP collides with a nucleus in the scintillating CaWO4 crystals used as the detector material. The SQUID readout system subsequently measures resistance changes that allow us to infer changes in temperature. The setup includes:


    
  * Detectors: Two 300 g CaWO4 scintillating crystals were used as targets in their respective modules with reflecting housings, with associated cryogenic light detectors that use the phonon-light technique, which measures both phonon and light signals to identify nuclear recoils indicative of WIMP interactions. Using both channels help distinguish nuclear recoils from background events.
    



    
  * Phonon Channel: The phonon signal is the energy transfer to a nucleus from a WIMP-nucleus elastic scattering event, which is measured using a cryogenic calorimeter read out by a SQUID. The amount of energy deposited by a WIMP is expected to be very low and SQUID's sensitivity allows it to detect the minute temperature changes resulting from the phonon signal with high accuracy and precision.
    
  * Light Channel: Alongside the phonon channel, a cryogenic light detector measures the scintillation light. This dual measurement system—phonon and light—discriminates between nuclear recoils (potential WIMP signals) and electron recoils (background noise) and enables the identification and suppression of background events, improving the signal-to-noise ratio crucial for WIMP detection. 




    
  * Readout: The system uses SQUID readout electronics to measure resistance changes in the tungsten thermometers attached to the detectors.
    
  * Stability: Temperature regulation and periodic calibration pulses ensure stability and accurate energy measurements throughout the experiment.



 §.§ Role of SQUIDs

    
  * Sensitivity: SQUIDs provide unmatched sensitivity in detecting the minute signals produced by WIMP interactions.
    
  * Noise Reduction: The high precision of SQUIDs helps in significantly reducing background noise, which is crucial for identifying rare events.
    
  * Dual-channel Detection: By enabling simultaneous detection of phonons and scintillation light, SQUIDs enhance the ability to discriminate between different types of interactions, improving overall detection accuracy. 

    





 §.§ Results and Analysis




    
  * Data Collection: Data was collected from January 31 to March 23, 2004, with a net exposure of 20.5 kg days.
    
  * Nuclear Recoil Events: A total of 16 events in the nuclear recoil acceptance region (12-40 keV) were observed, corresponding to a rate of (0.87 ± 0.22) events/(kg day), which is consistent with expected neutron background. This demonstrates the effectiveness of the SQUID-based detection system in distinguishing between neutron-induced recoils and potential WIMP signals. 
    
  * Background Suppression: The phonon-light technique provides strong suppression of non-nuclear recoil backgrounds, enhancing the experiment's sensitivity to WIMPs. The combination of phonon and light signals from the scintillating cryogenic detector allowed for a strong suppression of non-nuclear recoil backgrounds, enabling accurate identification of potential WIMP interactions.
    
  * Quenching Factor: It is the measure of the reduction in light yield for nuclear recoils compared to electron/gamma events. For CaWO4, the quenching factor was found to be 7.4 for neutrons and 40 for tungsten recoils.
    
  * Findings: The experiment set significant limits on the WIMP-nucleon cross-section. The experiment set exclusion limits on the same, with the best module showing no events in the region corresponding to tungsten recoils. Hence, no events could be conclusively attributed to WIMPs and the search for dark matter candidates continues.




 §.§ Future Directions and Conclusion

Challenges remain, such as the need for extremely low temperatures and the complexity of the SQUID system. Future upgrades will include additional shielding and more detector modules to further reduce background and improve sensitivity, with a focus on enhancing the resolution of the light detectors and studying quenching factors to better understand WIMP interactions. The application of SQUIDs in the CRESST-II phase of the experiment showcases its potential in particle detection instrumentation. The improvements in sensitivity and background suppression due to SQUIDs represent a critical step forward in the ongoing quest to detect WIMPs.














§ CONCLUSION

The integration of SQUIDs has significantly advanced dark matter research by enhancing the detection of weakly interacting massive particles (WIMPs) and axions. In the ADMX experiment, SQUID amplifiers achieved near quantum-limited noise temperatures, improving sensitivity and scan rates, and ruling out axion models with masses between 3.3 and 3.53 μeV. In the CRESST experiment, SQUID readouts enabled precise dual-channel detection of phonon and scintillation light signals, improving background suppression and setting exclusion limits on the WIMP-nucleon cross-section. These advancements underscore SQUIDs' critical role in reducing noise and increasing detection accuracy, showing great promise for facilitating future discoveries in the search for dark matter and bringing us closer to understanding the universe's unseen mass.


§ ACKNOWLEDGEMENTS

Special thanks goes to our supervisors H. Rana and M. Agarwal for their incredible mentorship and moral support, and to the CCIR team for giving me an opportunity to connect with incredible people and provide extensive support.

 unsrt   
 



 Paper Title: Time-resolved Stochastic Dynamics of Quantum Thermal Machines.





ahegde4@rochester.edu

Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA



patrick.potts@unibas.ch
Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82 CH-4056, Switzerland


glandi@ur.rochester.edu

Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA


§ ABSTRACT 
Steady-state quantum thermal machines are typically characterized by a continuous flow of heat between different reservoirs.
However, at the level of discrete stochastic realizations, heat flow is unraveled as a series of abrupt quantum jumps, each representing the exchange of finite quanta with the environment.
In this work, we present a framework that resolves the dynamics of quantum thermal machines into cycles that are classified as engine-like, cooling-like, or idle.
We explore the statistics of each cycle type and its duration, enabling us to determine both the fraction of cycles useful for thermodynamic tasks and the average waiting time between cycles of the same type.
Our framework presents a novel approach in characterizing thermal machines, with significant relevance to modern experiments, such as mesoscopic transport using quantum dots.


Time-resolved Stochastic Dynamics of Quantum Thermal Machines
    Gabriel T. Landi
    Received ...; accepted ...
=============================================================



Introduction.—
A typical quantum thermal machine consists of a system situated between hot and cold thermal baths, extracting or absorbing energy in the form of work, as depicted in Fig. <ref> <cit.>.
As an engine, it extracts work while transferring heat from hot to cold; as a refrigerator, it absorbs work to move heat from cold to hot.
In autonomous machines, this is usually pictured as a continuous process, where heat and work constantly flow through the system <cit.>.
However, within the microscopic domain, the stochastic nature of system and bath interactions endows an alternative perspective where energy is exchanged with the baths in the form of abrupt jumps. 
This is the basis for stochastic thermodynamics in classical (Pauli) rate equations <cit.>, as well as quantum models in the quantum jumps formalism <cit.>.

The jumps occur at random times and in random “channels.”
Let us broadly classify these channels as either an injection (I) or an extraction (E) of energy into or out of the hot (h) or cold (c) baths, resulting in four distinct types of monitored channels 𝕄 = {I_h, E_h, I_c, E_c}. 
Generalizing to multiple injection and extraction channels per bath is straightforward.
The quantum trajectory of such a machine, in the quantum jump unraveling, appears as a random string of the form

    I_h E_c I_c I_h E_h E_c I_h I_c E_h I_c …,

along with their timestamps t_1, t_2, …, indicating when each jump occurred. This representation of the dynamics is grounded in experimental observations <cit.>.
Note that (<ref>) accounts solely for events representing heat exchange with the environment. 
We assume work-based events are undetectable, as they are often associated with unitary drives in many models <cit.>.



The central question we address in this study is can specific thermodynamic cycles be identified solely from strings like (<ref>) so that their statistics can be explored?
For instance, one might intuitively characterize the sequence I_c E_h as a refrigeration cycle, since an energy quanta was injected from c to the system, and subsequently extracted to h suggesting work consumption.
Similarly, I_h E_c could be seen as an engine-like process (or accelerator <cit.>).
These are both examples of “useful cycles.”
Conversely, pairs such as I_h E_h and I_c E_c are events that fail to peddle quanta of energy overall, and incur no entropy production. We refer to these as “idle cycles".
While a machine might operate as an engine on average, the stochastic nature of these processes manifests in individual realizations yielding different cycles.

Classifying cycles raises several meaningful questions, such as
-0.1cm
    
  * What is the probability of each type of cycle?
    
  * How are cycles related to steady-state currents?
    
  * What is the time required to complete each cycle?
    
  * How many idle cycles precede a useful one?

These questions relate to the extensive literature on Full Counting Statistics (FCS) <cit.> and Fluctuation Theorems <cit.>.
Addressing them involves exploring time-resolved and cycle-resolved quantities, offering a nuanced understanding of the dynamics.

In attempting to classify cycles this way, a challenge arises when the system can withhold multiple excitations at once. 
For instance, in the string (<ref>), what meaning should be ascribed to the substring I_c I_h E_h E_c?
Because excitations are indistinguishable, it is impossible to infer if this was 
1̧ I_c 2̧ I_h 2̧ E_h 1̧ E_c (two idles) or
1̧ I_c 2̧ I_h 1̧ E_h 2̧ E_c (a refrigeration followed by an engine cycle).
While this is not an issue as far as the average heat and work currents are concerned, it does cause ambiguity in defining time-resolved quantities.
In this letter, we focus on systems that can retain only one excitation at a time; i.e., when injections and extractions alternate (I_∙ E_∙ I_∙ E_∙…) in the trajectory.
This assumption is common in experiments involving single <cit.> or double <cit.> quantum dots in the Coulomb blockade regime, as well as realizations of quantum heat engines <cit.>.

First, we establish the restrictions imposed by the single-excitation hypothesis on a quantum Markovian master equation.
Then, we employ the tools of waiting time distributions of the quantum jump unraveling <cit.> to fully characterize the statistics of cycles. 
Finally, we illustrate our results with a three-level maser example. 

Theory.—
We consider a finite-dimensional system weakly coupled to hot and cold baths.
Work may be performed either by a driven Hamiltonian H(t) or by additional work reservoirs. 
It is assumed that the dynamics can be described by a quantum master equation <cit.> (ħ = k_B = 1 throughout),

    dρ/dt = ℒ_tρ≡    -i[H(t), ρ] + ∑_n  𝒟[K_n] ρ
       +∑_α∈{h,c}, j(γ_α j^-𝒟[L_α j] + γ_α j^+𝒟[L_α j^†])ρ ,

where 𝒟[L]ρ = L ρ L^† - 1/2{L^† L, ρ}.
Here, {L_α j} are jump operators for the hot (α = h) and cold (α = c) baths, with 
L_α j denoting extractions and L_α j^† denoting injections, each occurring at rates γ_α j^∓, respectively.
Finally, K_n are jump operators of work reservoirs, which are often used in describing absorption refrigerators <cit.>.

We assume one can only monitor whether energy is injected (extracted) to (from) the hot or cold baths without identifying the specific jump operator (indexed by j) responsible. Therefore, the four corresponding jump superoperators are

    𝒥_E_αρ = ∑_j γ_α j^- L_α j^ρ L_α j^†,
          𝒥_I_αρ = ∑_j γ_α j^+ L_α j^†ρ L_α j^.


As our first result, we prove in the Supplemental Material <cit.> that the condition for the quantum trajectory to have alternating injections and extractions (i.e. at most a single excitation)  is achieved, if and only if, there exist two subspaces  and  spanning the system Hilbert space ℋ, such that: 



    L_α j = 𝒫_E L_α j𝒫_I   ∀ α,j; 
       𝒫_E H(t) 𝒫_I = 𝒫_I H(t) 𝒫_E =0; 
       𝒫_E K_n 𝒫_I = 𝒫_I K_n 𝒫_E=0,


where 𝒫_E/I are projection operators onto ℋ_E/I, satisfying 𝒫_E + 𝒫_I = 1.
Thus, the jump operators of the baths must be block upper-triangular, while those of the work reservoir, and Hamiltonian must be block diagonal in the joint basis of ℋ_E and ℋ_I.

Consequently, L_α j takes the system to  by extracting energy, while L_α j^† directs it to  by injecting.
We refer to  and  as post-extraction and post-injection subspaces. 
While the unitary dynamics and work reservoirs can inject (extract) work into (out of) the system, this result implies that such processes must occur within each subspace.
Transitions between these subspaces are only feasible through interactions with the baths.
An example is the three-level maser [see Fig. <ref> (a)]; other examples are hinted in <cit.>.

We henceforth assume, as is often the case, that there exists a rotating frame where H(t) is time-independent, and that the steady-state (ℒ=0) in this frame is unique.
The single-excitation hypothesis implies a conservation law for the average excitation current exchanged with the baths,

    := ( - )  = - ( - ) ,

which is deduced by noting that ddt𝒫_E ρ(t) 𝒫_E → 0 as the system approaches the steady-state. 
Equation (<ref>) does not imply that the heat currents to both baths are equal, as jumps to each bath generally involve different energies.
Indeed, their mismatch accounts for the work exchanged.

Statistics of cycles.—
Under the single-excitation assumption, the trajectory (<ref>) can be characterized in terms of the statistics of four possible pairs: I_h E_c, I_c E_h, I_h E_h, and I_c E_c.
We refer to each pair I_∙ E_∙ as a “cycle” and label them as X=1,2,3,4, respectively (see Fig. <ref>).
X=1 is a work extraction cycle [Alternatively, it could also represent an accelerator, wherein the direction of heat flow remains the same, but work is injected instead of extracted. For simplicity, we will continue to refer to this as a work cycle. This scenario always holds when the energy exchanged with the cold bath is smaller than that with the hot bath, an assumption we make implicitly.] and X=2 a refrigeration cycle, while X=3, 4 are idle cycles.

We are interested in the long-time steady-state behavior of strings of the form X_1X_2… = I_∙ E_∙ I_∙ E_∙…, adopting the convention that strings always begin with an injection. 
Then, as explained in <cit.>, the probability of observing a specific sequence X_1,…,X_n, with durations τ_1,…,τ_n, is given by

    p_X_1,…,X_n^(τ_1,…,τ_n) = 𝒪_X_n,τ_n…𝒪_X_1,τ_1π_E ,

where

    𝒪_X,τ≡∫_0^τ dt  e^ℒ_0(τ-t) e^ℒ_0t,

and ℒ_0 =  - ∑_α (𝒥_E_α + 𝒥_I_α) is the no-jump superoperator. 
In Eq. (<ref>), we have introduced the jump steady-state <cit.> 

    π_E = ( + )/( + )∈,

to ensure the jump sequence is stationary.

Marginalizing Eq. (<ref>) over all (X_i,τ_i) except one yields the probability that a single cycle is of type X and duration τ,

    p_X^(τ) = 𝒪_X,τ^π_E .

Integrating over τ yields the  probability that the cycle is of type X:

    p_X^ = ∫_0^∞ dτ p_X^(τ) = 𝒪_X π_E,

where 𝒪_X = ∫_0^∞ dτ 𝒪_X,τ =.

The average cycle time given it is of type X reads

    E(τ| X) = 1/p_X^∫_0^∞ dτ τ p_X^(τ).

In <cit.>, we show

    E(τ) = ∑_X=1^4 E(τ| X) p_X^ = 2/,

where  is the dynamical activity of the baths representing the average number of jumps per unit time in the steady-state.

The p_X^ in Eq. (<ref>) represent the relative occurrence of each cycle type over many trajectories, regardless of their duration.
The proportion of p_1/2 over p_3/4 yields information on the intermittency of the machine, revealing how often useful cycles occur over idle ones overall.
In <cit.>, we prove that p_1/2 and the excitation current from Eq. (<ref>) are related by 

    = p_1 - p_2/E(τ),

which provides a fundamental connection between usual steady-state currents and our results: 
the system functions as an engine when p_1 > p_2, and as a refrigerator when p_1 < p_2.

Statistics of useful cycles.— 
Drawing on the concept of intermittency, we can further address the distribution of idle cycles and the intervals between useful ones.
Let 𝒪_ u = 𝒪_1 + 𝒪_2 and 
𝒪_ id = 𝒪_3 + 𝒪_4 denote the superoperators associated with useful and idle cycles, respectively (and similarly for 𝒪_ u,τ and 𝒪_ id,τ).
We find the probability of n idles between two useful cycles to be <cit.>

    ℙ_u(n) = 𝒪_ u𝒪_ id^n𝒪_ uπ_E /𝒪_ uπ_E ,
             n = 0,1,2,…

and, similarly, the time interval t between them to be <cit.>,

    ℙ_u(t) = ∫_-∞^∞dz/2π𝒪_ u,z (1-𝒪_ id,z)^-1𝒪_ uπ_E /𝒪_ uπ_E e^ z t,

where 𝒪_X,z = ∫_0^∞ dτ e^- z τ𝒪_X,τ.
These probabilities provide a refined overview on the performance of the machine.
Remarkably, both these equations fully account for the statistical dependence between cycles.
This discussion concludes our characterization of the questions posed in the beginning, taking into account not only the behavior of individual cycles but also their correlations in sequences of arbitrary length.



Example: Three-level system.—
We apply our results to a three-level maser <cit.> whose schematic is depicted in Fig. <ref>. It is coupled to hot and cold baths at energy ω_α and temperature T_α with their populations following a Bose-Einstein distribution given by n̅_α=[exp(ω_α/T_α)-1]^-1. The maser is driven by a Hamiltonian H(t) = (ω_h-ω_c)σ_11 + ω_h σ_22 + ϵ(e^iω_d tσ_01 + e^-iω_d tσ_10) with a Rabi drive of strength ϵ and frequency ω_d. 
The jump operators are 
L_h = σ_02, L_c = σ_12 (and K_n = 0) with rates 
γ_α^- = γ_α(n̅_α + 1) and γ_α^+ = γ_αn̅_α.
Here, σ_ij = |i⟩⟨ j| are the transition operators.

The post-extraction subspace is spanned by {|0⟩,|1⟩}, and the post-injection by {|2⟩}. As anticipated, the Hamiltonian is block diagonal in the joint basis of these subspaces.
In this model, we find that the excitation current is directly related to the steady-state heat and work currents as
Q̇_h = ω_h, Q̇_c = - ω_c, and Ẇ = ω_d <cit.>.



Explicit formulas for the the cycle probabilities of this model are provided in <cit.>.
Fig. <ref> (b) illustrates p_X(τ) [Eq. (<ref>)] as a function of time.
The wiggles are reminiscent of coherent Rabi oscillations between |0⟩ and |1⟩. For large τ, all probabilities scale as p_X^(τ) ∼ e^-γτcos(ϵτ) where γ = (γ_h+γ_c)/2.

The marginals p_X [Eq. (<ref>)] are shown in Fig. <ref> (c) as a function of ratio of bath temperatures.
The plot highlights the different regimes of operation, which changes from refrigeration to engine at T_h/T_c = ω_h/ω_c.
It is noteworthy that

    p_1 - p_2 ∝n̅_h - n̅_c,     p_3/p_1 = p_2/p_4 = (n̅_h + 1) γ_h/(n̅_c + 1) γ_c,

implying that across all the parameter ranges, the probabilities of idle cycles bound those of useful ones <cit.>.
As a result, it is always more likely to observe the machine undergoing a cycle with no net heat transfer.

For small ϵ, the probability of a useful cycle p_u := p_1+p_2 scales as
ϵ^2/(Γ^2 + Δ^2), where Γ = (n̅_h γ_h + n̅_c γ_c)/2 
is the net decoherence rate
and
Δ = (ω_h-ω_c) - ω_d is the detuning.
This provides insights on maximizing the occurrence of useful cycles: stronger pumps, more resonant drives and lower damping. 
For this model Eq. (<ref>) simplifies to the geometric distribution, 
ℙ_ u(n) = p_u(1-p_u)^n. 
The distribution of time between two useful cycles [Eq. (<ref>)] is analyzed in <cit.>.

The average duration of cycles [Eqs. (<ref>), (<ref>)] are plotted in Fig. <ref> (d); for this model, it turns out that E(τ| 1) = E(τ| 3) and E(τ| 2)=E(τ| 4). 
Noticeably, the cycles tend to take much longer in the refrigeration regime.  
Moreover, at resonance (Δ = 0), all conditional averages tend to become very close (although not strictly equal), as shown in the inset of Fig. <ref> (d).

Conclusions.— We showed how to unravel the time-dependent statistics of quantum thermal machines, which allowed us to classify the stochastic dynamics in terms of distinct types of cycles, describe how often each cycle occurs, and how long they take.
Our results encompass all statistical correlations between cycles, and also connect with known results in FCS for the average excitation current and dynamical activity. 
This framework opens up a new avenue for characterizing quantum thermal machines, based entirely on experimentally observable data.
In particular, our formalism could be readily employed to analyse, e.g., mesoscopic transport in quantum dot experiments, shedding light on the underlying thermodynamics.

Of particular importance to this analysis is the notion of intermittency — i.e., how often a machine is actually performing a thermodynamically useful task. 
Since our approach enables the identification of both useful and idle cycles, we now have the tools to optimize the intermittency for fixed efficiency and output power.
Our results [cf. Eq. (<ref>)] also allow us to examine cycle “bunching”, specifically how the occurrence of one useful cycle influences the probability of observing another immediately after.
These insights have the potential to significantly deepen our understanding and interpretation of quantum stochastic processes.


A.S.H. expresses thanks to Joey Smiga, Felix Binder, and Guilherme Fiusa for fruitful discussions.







Supplemental Material for “Time-Resolved Stochastic Dynamics of Quantum Thermal Machines"
















Sections S1 to S5 prove various results mentioned in the main text, and the last section S6 pertains to the technical details of three-level maser example.



§ S1. PROOF OF THE SINGLE-EXCITATION CONSTRAINTS ON A MASTER EQUATION

In this section, we prove the necessary and sufficient conditions given in Eq. (<ref>) on the master equation from  Eq. (<ref>) to achieve quantum trajectories that have alternating injections and extractions (I_∙E_∙I_∙E_∙…) of excitations in a system.
Specifically, we comment on the structure of Hamiltonian, jump operators, and work reservoirs.
In what follows, without loss of generality, we shall drop the index referring to the bath α for jump operators L_α j as it is not relevant for this proof in particular. We first assume that there are no work reservoirs. We also assume that the Hamiltonian is time independent, even though the proof is unaltered otherwise.

We begin by imposing that any two successive extractions out of the system cannot occur.
For any k, q and for all times t_1, t_2, we demand

    𝒥_E_k e^ℒ_0 t_2𝒥_E_q e^ℒ_0 t_1ρ = 0,

and likewise for two consecutive injections. 
Then, the action of a no-jump operator ℒ_0 is equivalent to an effective non-Hermitian operator, ℒ_0 ρ = -i [H_e ρ - ρ H_e^†], where H_e = H - i∑_jL_j^†L_j^†/2.
Thus, we can express e^ℒ_0tρ = e^-iH_e tρ e^iH_e^† t.
Expanding the extraction superoperators, we obtain

    γ_kγ_q L_k e^-i H_e t_2 L_q e^-i H_e t_1ρ e^i H_e^† t_1 L_q^† e^i H_e^† t_2 L_k^† = 0,

for γ_k, γ_q > 0. 
Since we want this to be true for all times t_1 and t_2, Taylor expanding e^-i H_e t = ∑_n=0^∞(-i H_e t)^n/n! then allows us to conclude that this will be possible if and only if all powers of H_e satisfy the condition,

    L_k H_e^n L_q = 0,      n = 1,2,…

as well as 

    L_k L_q = 0   ∀  k, q.




 §.§ Constraints on each jump operator

Setting k = q in Eq. (<ref>) renders L_k^2 = 0.
Let us temporarily ignore the index for the jump operator.
The condition L^2=0 means the jump operators must be nilpotent matrices of order two.
From Ref. <cit.> (p.no. 157) one can show that for any such matrix, it is always possible to find a unitary matrix V such that L = V Λ V^†, where

    Λ = σ_1 [ 0 1; 0 0 ]⊕⋯⊕σ_r [ 0 1; 0 0 ]⊕ 0_d - 2r,

with σ_1 ≥⋯≥σ_r > 0 being the positive singular values of L and ⊕ denoting direct sum. 
Here, d= dim(L) and r =  rank(L).
It follows from the rank-nullity theorem <cit.> that r ≤⌊d2⌋.
Thus, we can choose d-r orthonormal vectors {|μ_i⟩}_i=1^d-r 
satisfying Λ |μ_i⟩ = 0, as well as r vectors 
{|ν_j⟩}_j=1^r satisfying Λ |ν_j⟩ = σ_j |μ_i⟩. 

We now introduce a rotated basis defined by |ϵ_i⟩ = V |μ_i⟩ and |ι_j⟩ = V |ν_j⟩, so that the jump operator L acts in the following way:

    L|ϵ_i⟩   = 0, 
    
    L|ι_j⟩   = ∑_i=1^r c_i |ϵ_i⟩,    c_i ∈ℝ.

From this, we can form orthogonal projectors, 𝒫_E =  span{|ϵ⟩}
and 𝒫_I =  span{|ι⟩}
that obey 𝒫_E + 𝒫_I = 1.
This reformulation allows us to express the operators as L = 𝒫_E L 𝒫_I.
Thus, we conclude that there is a basis of the Hilbert space in which each jump operator is written as a strictly upper-triangular matrix.

To further clarify the procedure, 
let us consider an explicit example for d = 5 with r = 2 case. According to our proof, such a jump operator L must be unitarily similar to

    Λ = 
    [   0 σ_1   0   0   0;   0   0   0   0   0;   0   0   0 σ_2   0;   0   0   0   0   0;   0   0   0   0   0 ],

where σ_1 ≥σ_2 > 0 are positive singular values of L. As one may verify, we can find the orthonormal set of vectors and the action of Λ on them as,

    .
    |μ_1⟩   = (1, 0, 0, 0, 0)^T
     
    |μ_2⟩   = (0, 0, 1, 0, 0)^T
    
    |μ_3⟩   = (0, 0, 0, 0, 1)^T}Λ |μ_i ⟩   = 0  for i = 1, 2, 3,
    .
    |ν_1⟩   = (0, 1, 0, 0, 0)^T
     
    |ν_2⟩   = (0, 0, 0, 1, 0)^T}Λ |ν_i ⟩   = σ_i |μ_i⟩ for i = 1, 2.

Now, the exact forms of unitary matrices V are dependent on L.



 §.§ Joint injection/extraction subspaces for the entire set of jump operators


We now return to the full set of jump operators L_k.
As just shown, to each one we can associate two orthonormal projectors 𝒫_E_k and 𝒫_I_k such that L_k = 𝒫_E_k L_k 𝒫_I_k, where each pair satisfies 𝒫_E_k + 𝒫_I_k = 1.
Next, we show that all 𝒫_E_k must coincide. 
This follows from the additional condition supplied by Eq. (<ref>).

Since for any pair L_k L_q = (𝒫_E_k L_k 𝒫_I_k)(𝒫_E_q L_q 𝒫_I_q) = 0, this must imply that 𝒫_I_k𝒫_E_q = (1 - 𝒫_E_k)𝒫_E_q = 0.
Thus 𝒫_E_q = 𝒫_E_k𝒫_E_q, allowing us to conclude that 𝒫_E_q = 𝒫_E_k. 
This also implies that 𝒫_I_q = 𝒫_I_k leading to only two unique projectors  𝒫_E and 𝒫_I which must be the same for all jump operators associated to the hot and cold baths.



 §.§ Hamiltonian and work reservoir operators


From Eq. (<ref>), setting n = 1  we have that L_k H(t) L_q = 0. This implies (𝒫_E L_k 𝒫_I) H(t) (𝒫_E L_q 𝒫_I) = 0 which reduces to 𝒫_I H(t) 𝒫_E = 0 = 𝒫_E H(t) 𝒫_I. Since the work reservoir jump operators also do not induce any transitions between the subspaces  and , it follows that 𝒫_I K_n 𝒫_E = 0 = 𝒫_E K_n 𝒫_I.

The converse holds true trivially; i.e., given the above forms for the jump operators and the Hamiltonian, it is straightforward that conditions listed in Eq. (<ref>) are satisfied.



 §.§ Other schematic examples of thermal machines satisfying our condition

A key insight to take away is that a Rabi drive cannot connect post-injection and post-extraction subspaces for the single-excitation assumption to hold.
To clarify the restrictions imposed by the assumption from a physical point of view, we show in Fig. <ref> a schematic of a four-level system for which the restriction is applicable and another example for which it is not. 






§ S2. CYCLE STATISTICS IN THE LONG-TIME LIMIT

For a given initial state ρ, the probability of observing a trajectory of the sort given in Eq. (<ref>) takes the form <cit.>,

    𝒫(τ_1, k_1, …, τ_n, k_n) = 𝒥_k_ne^ℒ_0τ_n⋯𝒥_k_1e^ℒ_0τ_1ρ,

for all k_i ∈{I_h, E_h, I_c, E_c} representing the monitored channels and τ_i ≥ 0 referring to the time interval between the jumps.

For the case of single-excitations, we shall replace k_i's with alternating injections and extractions.
As in the main text, we adopt the convention that the cycle always begins with an injection. 
For each pair (τ_1,τ_2), (τ_3,τ_4), … we change variables to e.g. (τ_1,τ_2)→ (t_1,s_1-t_1), and then integrate over s_i. 
In this way, we bundle each pair I_∙ E_∙ together, thus making a cycle. 
Carrying out this operation in Eq. (<ref>) yields

    p_X_1,…,X_n^(τ_1,…,τ_n) = 𝒪_X_n,τ_n…𝒪_X_1,τ_1ρ,

with the cycle superoperators 𝒪_X,τ as defined in Eq. (<ref>).
This result differs from Eq. (<ref>) of the main text only in the choice of initial state, which is left arbitrary here. 
Next, we turn to the justification for beginning in a specific initial state π_E.

We are interested in the distribution (<ref>) in the long-time limit of operations of the machine. 
Specifically, we consider the machine to be in operation for an extended period before we begin counting cycles.
We therefore need to choose the initial state ρ which makes Eq. (<ref>) stationary.
Interestingly, it is not the steady-state of the master equation that possesses this property, but rather the so-called jump steady-state (JSS) <cit.>.
The steady-state captures how long the system stays in a given corner of Hilbert space, while the JSS captures how often it visits that region, an intuition borrowed from classical stochastic processes <cit.>.

Given the unique steady-state = 0, we define the JSS via 

    π = /,

where 

    = ∑_α∈{h, c} (𝒥_E_α + 𝒥_I_α),

is the sum of all jump operators of hot and cold baths, and 

    = ,

is the dynamical activity of the hot and cold baths together – i.e., the average number of jumps per unit time in the steady-state.

Because of the single-excitation hypothesis, the JSS in our case has to be block diagonal in the post-injection and post-extraction subspaces 

    π = p_E^π_E^  + p_I^π_I^.

To see this, we can expand Eq. (<ref>) to

    π   = 1/∑_α∈{h, c} (𝒥_E_α + 𝒥_I_α) 
       = 1/( ∑_α, jγ_α j^- L_α j^ L_α j^†)
    +  1/( ∑_α, jγ_α j^+ L_α j^† L_α j^).

Now, we define the (properly normalized) post-extracted and post-injected parts of the JSS, along with their weights as follows:

    π_E    = ∑_α, jγ_α j^- L_α j^ L_α j^†/∑_α, jγ_α j^-L_α j^† L_α j^,
         p_E^ = 1/(∑_α, jγ_α j^-L_α j^† L_α j^) ≡𝒦_E/, 
    π_I    = ∑_α, jγ_α j^+ L_α j^† L_α j^/∑_α, jγ_α j^+L_α j^ L_α j^†,
         p_I^ = 1/(∑_α, jγ_α j^+L_α j^ L_α j^†) ≡𝒦_I/,

leading to the desired form.

We can think of 𝒦_E and 𝒦_I as the dynamical activities of the  and  subspaces. 
As proved in Sec. S1, in case of single-excitation machines, the states are partitioned into mutually exclusive subspaces. 
Consequently, the state has an equal probability of visiting either subspace, leading to,

    p_E^ = p_I^ = 1/2,    𝒦_E = 𝒦_I = /2,    π = π_E + π_I/2.


We shall now prove that the distribution (<ref>) is stationary when the initial state ρ is the post-extraction JSS π_E. 
The key point is that since injections and extractions alternate, the state just before an injection must reside in the post-extraction subspace.



 §.§ Proof of stationarity


To prove stationarity it suffices to consider a sequence of two cycles of types X_1 and X_2, each with durations τ_1 and τ_2, respectively. The corresponding trajectory probability (<ref>) for the initial state as π_E is 

    p_X_1,X_2^(τ_1, τ_2) = 
    ∫_0^τ_2 dt_2
    ∫_0^τ_1 dt_1
    𝒥_E_X_2 e^ℒ_0(τ_2-t_2)𝒥_I_X_2 e^ℒ_0 t_2𝒥_E_X_1 e^ℒ_0(τ_1-t_1)𝒥_I_X_1 e^ℒ_0 t_1π_E .

Marginalizing over all quantities of the first cycle amounts to the following substitution:

    ∑_X_1∫_0^∞ dτ∫_0^τ dt 𝒥_E_X_1 e^ℒ_0(τ_1-t_1)𝒥_I_X_1 e^ℒ_0 t_1 = ℳ_E ℳ_I,

where ℳ_k = -∑_α∈{h, c}𝒥_k_αℒ_0^-1 for k ∈{I, E} such that ℳ_I and ℳ_E lie in  and , respectively. Combining these two operators gives

    ℳ = ℳ_E + ℳ_I = - ,

where the final equality follows from Eq. (<ref>). Notably, this operator has a special property that only the jump steady-state satisfies <cit.>: 

    ℳπ = -𝒥ℒ_0^-1𝒥/𝒥 = -𝒥ℒ_0^-1(ℒ-ℒ_0)/𝒥 = π.

Since ℳ_I π_I = 0 = ℳ_E π_E, it follows that


    ℳ_I π_E    = π_I,
    ℳ_E π_I    = π_E.


Thus, marginalizing Eq. (<ref>) over the first cycle leaves us with 

    ℳ_E ℳ_I π_E = ℳ_E π_I = π_E,
 
i.e., cycling back to where we started.

As a result, we obtain the following identities which in turn prove stationarity of Eq. (<ref>):

    ∑_X ∫_0^∞ dτ 𝒪_X,τπ_E    = π_E
        
    ∑_X ∫_0^∞ dτ 𝒪_X,τ∙   = ∙.

Thus, marginalizing Eq. (<ref>) from either side yields the same distribution, proving stationarity.
The above properties also ensure the proper normalization in Eq. (<ref>),

    ∑_X ∫_0^∞ dτ p_X(τ) = ∑_X p_X = 1.




§ S3. MOMENTS OF WAITING TIME DURATION OF A CYCLE

The n-th moment of the waiting time, given a cycle of type X is a generalization of  Eq. (<ref>), and reads

    𝔼(τ^n | X) = 1/p_X^∫_0^∞ dτ τ^n p_X^(τ).

Substituting for the probability from Eq. (<ref>) and changing the integration variable to u = τ - t results in the following:

    𝔼(τ^n | X)    = 1/p_X^∫_0^∞ dτ τ^n ∫_0^τ dt  e^ℒ_0(τ - t) e^ℒ_0 tπ_E 
       = 1/p_X^∫_0^∞ dτ τ^n ∫_0^∞ du  e^ℒ_0u e^ℒ_0(τ - u)π_E Θ(τ - u) 
       = 1/p_X^∫_0^∞ du  e^ℒ_0u(∫_0^∞ dτ τ^n e^ℒ_0τΘ(τ - u)) e^-ℒ_0uπ_E ,

where we introduced Heaviside step function Θ(τ - u) in the second equality.
Let us first solve the integral in the parentheses,

    I_n    ≡∫_0^∞ dτ τ^n e^ℒ_0τΘ(τ - u)
       = τ^n (-e^ℒ_0τ)Θ(τ - u)|_τ = 0^∞ + [n ∫_0^∞dτ τ^n-1e^ℒ_0τΘ(τ-u) + ∫_0^∞dτ τ^n e^ℒ_0 τδ(τ - u)]
       = n  I_n - 1 + u^n  e^ℒ_0 u

where the first term in the second equality vanishes at both the limits τ = 0 and τ = ∞ (the latter is because of the assumption that there are no dark subspaces). Solving this recursion relation yields,

    I_n = [∑_i=0^n-1n!/(n-i)!u^n-iℒ_0^-(i+1) - n! ℒ_0^-(n+1)]e^ℒ_0u,

Plugging this back into Eq. (<ref>), we find

    E(τ^n | X) = 1/p_X^(∑_i=0^n-1n!/(n-i)!ℒ_0^-(n-i+1)ℒ_0^-(i+1)π_E - n! ℒ_0^-1ℒ_0^-(n+1)π_E).

For example, if n = 1, we obtain the expectation value for the duration of a given cycle X,

    E(τ| X) = - 1/p_X^(ℒ_0^-2π_E  + ℒ_0^-2π_E ).

On the other hand, the unconditional waiting time is 

    E(τ) = ∑_X=1^4 p_X E(τ| X) = - ∑_X = 1^4 (ℒ_0^-2π_E  + ℒ_0^-2π_E).


We can relate E(τ) in Eq. (<ref>) to the dynamical activity defined in Eq. (<ref>) through the notion of residence times.
We refer P_ no(τ|ρ) = e^ℒ_0 τρ to denote the survival probability that the system remains in a given state ρ for a time t.
Since -dP_ no/dτ is then the waiting-time distribution, we can determine the average time the system will remain in state ρ before jumping as 

    E(τ|ρ) = -∫_0^∞ dτ τ dP_ no(τ|ρ)/dτ = -ℒ_0^-1ρ.

A consequence of assuming at most a single-excitation in the system is that the no-jump superoperator ℒ_0 cannot produce transitions between subspaces. Accordingly,


    -ℒ_0^-1π_I    = E(τ|π_I) π_I,
    
    
    -ℒ_0^-1π_E    = E(τ|π_E) π_E.


Specifically, starting in the jump steady-state yields the well-known result that the dynamical activity is the inverse of the average waiting time in the jump steady-state,

    E(τ|π) = - / = -  ( - ℒ_0) / = 1/,

where we have used =0 and 
(·) = 0.
This relation can be extended to the unconditional expectations as well:

    E(τ)    = ∑_X ∫_0^∞ dτ τ∫_0^∞ dt  e^ℒ_0 (τ-t) e^ℒ_0 tπ_E 
       = ∑_X ∫_0^∞ du ∫_0^∞ dt (u+t)  e^ℒ_0u e^ℒ_0 tπ_E 
       = -ℳ_E ℒ_0^-1ℳ_I π_E + ℳ_E ℳ_I ℒ_0^-1π_E 
       = E(τ|π_I)π_I  + E(τ|π_E)π_E 
       = 2E(τ|π)  
       = 2/,

where the second equality comes from a substitution u = τ - t and the last equality follows from Eq. (<ref>).



§ S4. CONNECTION WITH THE AVERAGE EXCITATION CURRENT


Eq. (<ref>) of the main text provides a fundamental connection between the statistics of cycle and the average excitation current, a much more standard measure to characterize the operation of an engine. 
In order to prove that relation  we shall first recast the probabilities from Eq. (<ref>) as,

    p_X^   = π_E
       = 2 π
       = 2/,

where we have used Eq. (<ref>) in going to the second equality, and Eq. (<ref>) in the third.
Since E(τ) = 2/ from Eq. (<ref>), we have 

    p_X^/E(τ)   = 
       =  ( - ℒ_0) 
       = -,

where made use of (·) = 0 in the last step. Thus, we can express

    p_1 - p_2/E(τ)   =  - 
       = ( - ℒ_0 -  -  - )  -  ( - ℒ_0 -  -  - ) 
       =  - 
       ≡,

which proves Eq. (<ref>). In simplifying the above, we repeatedly made use of (·) = 0 and ℒ = ℒ_0 +  +  +  +.




§ S5. PROBABILITIES FOR OBSERVING INTERMEDIATE CYCLES BETWEEN TWO OF THE SAME CYCLES


Eqs. (<ref>) to (<ref>) in the main text refer to the statistics of individual cycles. 
However, armed with the full trajectory distribution (<ref>), we can take this a step further and ask questions pertaining to the occurrence of multiple cycles in specific orders, and how correlations between  outcomes might affect that. 
In particular, here we prove the following two results from the main text: 

    
  * Eq. (<ref>): the probability that there are n idle cycles between two useful ones;
    
  * Eq. (<ref>): the probability that it takes a time t between the conclusion of two useful cycles. 

Recall that the useful cycles are X=1,2 and the idle cycles are X=3,4. 
Starting with the cycle superoperators in Eq. (<ref>), we define

    𝒪_ u = 𝒪_1 + 𝒪_2, 
            𝒪_ id = 𝒪_3 + 𝒪_4,

and similarly for 𝒪_ u,τ and 𝒪_ id,τ.


The proof of Eq. (<ref>) is straightforward.
We are seeking the probability of observing n idle cycles until a next useful one given that we have already observed a useful cycle. Post the first useful cycle, the state is updated to 𝒪_ uπ_E/𝒪_ uπ_E. We then readily obtain,

    ℙ_u(n) = 𝒪_ u𝒪_ id^n𝒪_ uπ_E/𝒪_ uπ_E.


To prove Eq. (<ref>) we proceed as follows. 
Suppose we are interested in observing a specific cycle X = x by time t such that we implicitly assume only other cycles X x occur until then.
This cumulative probability ℙ_X = x, t is then an infinite sum where each term defaults to observing X in the last trial while assigning the probability for observing X until the last but one, 

    ℙ_X = x(t) =    p_X = x(t) 
       + ∑_X_1∫_0^t dt_1 p_X_1, X_2 = x(t_1, t-t_1) 
       + ∑_X_1, X_2∫_0^tdt_2∫_0^t_2dt_1 p_X_1, X_2, X_3 = x(t_1, t_2 - t_1, t - t_2) 
       + ⋯,

where we have made use of p_X(t) from Eq. (<ref>). 
Written in terms of the cycle operators, this becomes

    ℙ_X = x(t) =    𝒪_X = x, tπ_E 
       +∑_X_1∫_0^t dt_1 𝒪_X_2 = x, t - t_1𝒪_X_1, t_1π_E 
       +∑_X_1, X_2∫_0^t dt_2∫_0^t_2 dt_1 𝒪_X_3 = x, t-t_2𝒪_X_2, t_2 - t_1𝒪_X_1, t_1π_E 
       +⋯.

We can now recast it in terms of Eq. (<ref>) after replacing for the duration of cycles, τ_i = t_i - t_i-1, for i = 1, 2, … with τ_1 = t_1 and τ_n = t - t_n-1,

    ℙ_X = x(t)    = ∑_n=0^∞∑_X_1, ⋯, X_n-1∫_0^∞ dτ_n ⋯ dτ_1 𝒪_X_n = x, τ_n^𝒪_X_n-1, τ_n-1⋯𝒪_X_1, τ_1π_E  δ(t - ∑_n=1^∞τ_i).

Here we shall make use of the Fourier expansion of delta function, δ(t) = ∫_-∞^∞ dz e^izt/2π, and also introduce the Fourier transformed version of the cycle operator,

    𝒪_X,z = ∫_0^∞ dt 𝒪_X, te^-izt,

so that we can simplify the above to,

    ℙ_X = x(t)
       = ∑_n=0^∞ ∑_X_1, …, X_n-1 ∫_-∞^∞dz/2π𝒪_X_n = x, z^𝒪_X_n - 1, z⋯𝒪_X_1, zπ_E  e^izt.

Since all X_k for k = 1, ⋯, n-1 are identical so that X_k ≡X, we can further reduce the above equation to,

    ℙ_X = x(t)   = ∑_X∫_-∞^∞dz/2π𝒪_X_n = x, z(∑_n = 0^∞𝒪_X, z^n - 1)π_E e^ zt, 
       = ∑_X∫_-∞^∞dz/2π𝒪_X_n = x, z(1 - 𝒪_X, z)^-1π_E e^ zt,

where we have made use of the summation of infinite geometric series: ∑_n=0^∞ x^n = 1/(1-x) for x < 1.

Given that we begin with a useful cycle, we can update the initial state to π_E →𝒪_ uπ_E/𝒪_ uπ_E.
Summing over the idle cycles X = {3, 4} in Eq. (<ref>), we obtain the result from Eq. (<ref>),

    ℙ_u(t) = 1/𝒪_ uπ_E ∑_X = {3, 4} ∫_-∞^∞dz/2π 𝒪_X, z(1 - 𝒪_X, z)^-1𝒪_ uπ_E e^izt.




§ S6. EXPRESSIONS FOR STATISTICAL QUANTITIES IN THREE-LEVEL MASER


In this section, we provide the equations specific to three-level masers for the statistical measures introduced in the main text. We begin with evaluating the essential operators involved in obtaining those measures.
The computational basis is spanned by |0⟩, |1⟩, |2⟩.
We use the notation σ_ij = |i⟩⟨ j| throughout.



 §.§ Time-independent Hamiltonian

The maser Hamiltonian is

    H(t) = (ω_h-ω_c)σ_11 + ω_h σ_22 + ϵ(e^iω_d tσ_01 + e^-iω_d tσ_10).

Choosing an appropriate rotating frame of reference X = ω_d σ_11 + ω_h σ_22 and rotate according to H_ rot = U H U^† where U = e^ X t as in Ref. <cit.>, we obtain a time-independent Hamiltonian

    H = H_ rot - X = Δσ_11 + ϵ(σ_01 + σ_10),

where Δ = (ω_h - ω_c) - ω_d is the detuning parameter. We use this Hamiltonian for all further evaluations.



 §.§ Jump operators and steady-state

According to Eq. (<ref>) the jump operators for the three-level maser appear as,

    2(ρ)    = γ_h(n̅_h+1)σ_02ρσ_02^†,           (ρ) = γ_h n̅_h σ_20ρσ_20^†,
    (ρ)    = γ_c(n̅_c+1) σ_12ρσ_12^†,           (ρ) = γ_c n̅_cσ_21ρσ_21^†.

In this case, the jump steady-state along with its post-extracted and post-injected parts are,


    π_E    = ( γ_c(n̅_c + 1)γ_h(n̅_h + 1) + 1 )^-1 |0⟩⟨ 0| + ( γ_h(n̅_h + 1)γ_c(n̅_c + 1) + 1 )^-1 |1⟩⟨ 1|, 
    π_I    = |2⟩⟨ 2|,
    π   = π_E + π_I/2,


Because there are no work reservoirs, the action of the no-jump superoperator can be described solely in terms of the non-Hermitian Hamiltonian H_e = H - i/2 ∑_j L_j^† L_j^. 
For the maser, this reads

    H_e = 1/2[                      n̅_h γ_h                             ϵ                             0;                             ϵ                 n̅_c γ_c + 2Δ                             0;                             0                             0 γ_c (1+n̅_c) + γ_h (1 + n̅_h) ].




 §.§ Time-resolved probability of completing a cycle

Plugging the above operator forms into Eq. (<ref>) describing time-resolved cycle probabilities, we obtain

    p_1^(τ)    = ∫_0^τ dt 𝒥_E_c e^ℒ_0 (τ-t) 𝒥_I_h e^ℒ_0 tπ_E 
    
       = γ_h^2 n̅_h(n̅_h + 1) γ_c (n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|0⟩|^2 
           + γ_h n̅_h γ_c^2 (n̅_c + 1)^2/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|1⟩|^2.


Likewise, we arrive at similar expressions for the other cycles:

    p_2^(τ)     = γ_h^2 (n̅_h + 1)^2 γ_c n̅_c/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|0⟩|^2 
           + γ_h (n̅_h + 1) γ_c^2 n̅_c(n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|1⟩|^2, 
    
     p_3^(τ)    = γ_h^2  γ_c n̅_h(n̅_h + 1)^2/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|0⟩|^2 
           + γ_h n̅_h (n̅_h + 1) γ_c^2 (n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 0| e^-i H_e t|1⟩|^2,
    
     p_4^(τ)     = γ_h^2 (n̅_h + 1) γ_c n̅_c (n̅_c + 1)/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|0⟩|^2 
           + γ_h γ_c^2 n̅_c(n̅_c + 1)^2/γ_h(n̅_h + 1) + γ_c (n̅_c + 1)∫_0^τ dt  |⟨ 2| e^-iH_e(τ - t)|2⟩⟨ 1| e^-i H_e t|1⟩|^2.

These are plotted in Fig. <ref> (a) of the main text.



 §.§ Probability of observing a cycle of each kind irrespective of waiting times


Marginalizing the previous results over τ yield the probabilities p_X of observing a cycle of type X [Eq. (<ref>)]. 
Let us introduce the simplifying variables:


    Γ   = 12(n̅_h γ_h + n̅_c γ_c),
        
    γ   = 12(γ_h + γ_c),
        
    χ   = n̅_h n̅_c γ_h γ_c,
        
    n̅_ ave   = Γγ = n̅_h γ_h + n̅_c γ_cγ_h + γ_c,
        
    κ   = χ2Γ = n̅_h n̅_c γ_h γ_cn̅_h γ_h + n̅_c γ_c,
        
    γ_ cl   = 2ϵ^2 ΓΔ^2 + Γ^2.


The last quantity is the classical transition rate between |0⟩ and |1⟩ derived by assuming Lorentzian broadening in Fermi's golden rule, as explained in Ref. <cit.>.
With these definitions, we find

    p_1^   = 𝒥_E_cℒ_0^-1𝒥_I_hℒ_0^-1π_E , 
       =
    κ/2Γ(1 + 1/n̅_c)/(1+1/n̅_ ave)[(1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_h)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1],
    
    
    p_2^   = 𝒥_E_hℒ_0^-1𝒥_I_cℒ_0^-1π_E 
       = 
    κ/2Γ(1 + 1/n̅_h)/(1+1/n̅_ ave)[(1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_c)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1],
    
    
    p_3^   = 𝒥_E_hℒ_0^-1𝒥_I_hℒ_0^-1π_E 
       =
    κ/2Γn̅_h γ_h/n̅_c γ_c(1 + 1/n̅_h)/(1+1/n̅_ ave)[
    (1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_h)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1],
    
    
    p_4^   = 𝒥_E_cℒ_0^-1𝒥_I_cℒ_0^-1π_E 
       = 
    κ/2Γn̅_c γ_c/n̅_h γ_h(1 + 1/n̅_c)/(1+1/n̅_ ave)[
    (1 + κ/γ_ cl)^-1
    +
    (1 + 1/n̅_c)/(1+1/n̅_ ave)(1 + γ_ cl/κ)^-1].

It is straightforward to verify that p_1 + p_2 + p_3 + p_4 = 1.
Remarkably,

    p_1 - p_2    = 1/2Γ(1/κ + 1/γ_ cl)^-1(1 + 1/n̅_ ave)^-1(1/n̅_c - 1/n̅_h), 
    p_3/p_1 = p_2/p_4   = (n̅_h + 1) γ_h/(n̅_c + 1) γ_c.

As expected from Eq. (<ref>), p_1 - p_2 relates to the particle current. In this model, this is indeed evidenced by its dependence on the difference in Bose occupations, n̅_h - n̅_c.
These expressions are plotted in Fig. <ref> (b) of the main text, where we also set γ_h = γ_c leading to the following conclusions:

    
  * In the refrigeration regime (n̅_h < n̅_c), we have p_3 < p_1 < p_2 < p_4. 
    
  * In the engine regime (n̅_h > n̅_c), we have  p_4 < p_2 < p_1 < p_3.

Thus, the idle cycle probabilities bound those of useful cycles across all parameter ranges.



 §.§ Useful cycle probabilities

The probability of observing a useful cycle is  p_u := p_1 + p_2. 
Because the post-injection subspace in this case is one-dimensional, it turns out that the cycles are independent of each other. 
As a consequence, the probability of observing n idle cycles between two useful ones [Eq. (<ref>)] will be given simply by the geometric distribution 

    ℙ_ u(n) = p_u(1-p_u)^n.




Figure <ref> (a) shows the mean and variance: ⟨ n ⟩ and Var(n) of this distribution.
We find the optimal temperature ratio T_h/T_c = ω_h/ω_c minimizes the mean number of idle cycles, thereby enhancing resource efficiency.
However, a steeper temperature gradient between the baths does not necessarily result in efficient heat transport, as it also promotes the participation of idle cycles in system dynamics.
To quantify this, we turn to the time between two useful cycles given in Eq. (<ref>) and computed numerically in this case.
The results plotted in Fig. <ref> (b) show that a steeper temperature gradient facilitates faster heat transport.


 Paper Title: Quantum Non-Demolition Measurements and Leggett-Garg inequality.






Dipartimento di Fisica, Università di Genova, via Dodecaneso 33, I-16146, Genova, Italy.
INFN - Sezione di Genova, via Dodecaneso 33, I-16146, Genova, Italy.


Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), Largo Enrico Fermi 6, I-50125 Firenze, Italy.
European Laboratory for Non-linear Spectroscopy, Universitá di Firenze, I-50019 Sesto Fiorentino, Italy.

§ ABSTRACT 
Quantum non-demolition measurements define a non-invasive protocol to extract information from a quantum system that we aim to monitor.
They exploit an additional quantum system that is sequentially coupled to the system. 
Eventually, by measuring the additional system, we can extract information about temporal correlations developed by the quantum system dynamics with respect to a given observable.
This protocol leads to a quasi-probability distribution for the measured observable outcomes, which can be negative.
We prove that the presence of these negative regions is a necessary and sufficient condition for the violation of macrorealism.
This is a much stronger condition than the violation of the Leggett-Garg inequalities commonly used for the same task.
Indeed, we show that there are situations in which Leggett-Garg inequalities are satisfied even if the macrorealism condition is violated.
As a consequence, the quantum non-demolition protocol is a privileged tool to identify with certainty the quantum behavior of a system. As such, it has a vast number of applications to different fields from the certification of quantumness to the study of the quantum-to-classical transition.


Quantum Non-Demolition Measurements and Leggett-Garg inequality
    Stefano Gherardini
    August 5, 2024
===============================================================






§ INTRODUCTION


How can we distinguish a quantum system from a classical one?
What are the distinctive features of a quantum system? 
These are two of the questions physicists have asked from the dawn of quantum theory. 
To give a clear and definite answer is difficult because of the smooth boundary between the classical and quantum worlds, and due to our inability to directly assess quantum features without altering them.
For example, when a quantum system interacts with an environment, it quickly loses its quantum features and starts to behave classically.
Still, the identification of protocols to spot and quantify a system's quantum features is of paramount importance and it would have a deep impact on both the foundation of quantum mechanics and quantum technologies.


The milestones in this field are the papers by John Bell in 1964 <cit.> and by Leggett and Garg <cit.> in 1985.
They both try to answer our initial questions, identifying and giving some criteria to highlight two quantum features.
The famous Bell's inequalities <cit.> focus on the non-locality and the quantum correlations related to the entanglement while the Leggett-Garg's inequalities (LGIs), which formally have a similar structure, discuss the violation of the macrorealism (MR) <cit.>.


The concept of MR was introduced to identify the features that a classical system should have and then test if they are present at a quantum level. If not, one would conclude that a quantum system behaves in a non-classical way.
In the following, for the sake of continuity with the current literature, we first use the original terminology of MR <cit.>, which is still adopted in several contributions, but then we will give a modern and precise definition <cit.>.
The Leggett and Garg's assumptions to test the MR are 1) Macrorealism per se (MRps), whereby the system is in one of the states available to it at each moment, and 2) Noninvasive measurability (NIM) that identifies the conditions under which one can determine the state of a system without disturbing the subsequent dynamics.


Later, an additional condition (implicit in the original article <cit.>) was identified <cit.> as 3) Induction: future measurements cannot affect the present state. All these assumptions are surely satisfied by a classical system, while they might be violated for quantum systems.
Therefore, they give a testable way to distinguish between quantum and classical behavior. In this regard, it would be thus desirable to identify a condition that certifies the violation of macrorealism, eventually with certainty.


The LGIs are usually stated in terms of quantum correlators of sequential measurements performed on a single quantum system.
The violation (resp. validation) of the LGIs is a signature of the quantum (resp. classical) behavior of the system of interest.
However, the violation of LGIs is only a sufficient condition for the violation of MR <cit.>, which is equivalent to stating that the validity of LGIs is a necessary condition such that even the MR is fulfilled.
Practically, this means that there could be situations in which the MR is violated but the LGIs are satisfied.


In this paper, we focus on the MRps condition, comprising the MR, by giving a new general protocol guaranteeing both a necessary and sufficient condition to identify its violation. 
The violation of the MRps implies the violation of the MR's assumption that, in turn, identifies the presence of quantum features. Such a protocol is based on quantum non-demolition (ND) measurements <cit.> and it was proposed in Refs. <cit.>. 
To highlight the practical implications and experimental feasibility of the protocol, we will start by describing its implementation which makes use of a quantum detector to store the desired information. In this way, by measuring the state of the detector, one can construct a quasi-probability distribution (QPD) of the corresponding measurement outcomes. 
As clarified in Refs. <cit.> other quasi-probability distributions have been defined in the literature, ranging from Kirkwood-Dirac quasi-probabilities <cit.> to the full counting statistics <cit.> and Keldysh quasi-probabilities <cit.>. 
At the same time, alternative protocols to determine the violation of the LGIs with weak measurements have been proposed <cit.>.
Nevertheless, in this paper, our focus goes to quasiprobabilities based on ND measurements, as we are going to prove that the corresponding distribution has negative regions if and only if the MRps condition is violated. 


In order to show the effectiveness of our results, we compare the ability of our protocol to reveal a violation of MR with the LGIs, representing so far the proper tool for such a task. For the sake of a fair comparison, we derive, from the full QPD, the multi-time quantum correlators leading to the corresponding LGIs.
With an in-depth analysis, we identify the contributions of the QPD that lead to the violation of the LGIs and we show that only in some particular cases the LGIs spot the violation of the MR, thus confirming that such a circumstance is only a sufficient condition.
To further stress this point and test on a concrete case study, we consider a specific example taken from Ref. <cit.>.
We show how, by using the negative regions of the QPD, we are always able to identify any violation of MRps and thus of MR, while LGIs statistically fail in half of the cases.


To the best of our knowledge, the protocol based on ND presented here is the first allowing for the certain identification of purely quantum behaviors associated with the violation of MR.
This feature joined with the fact that it finds application beyond binary observables, for which usually the LGIs are discussed, has implications for the foundation of quantum mechanics and quantum technologies. For example, it can be used (i) to characterize the quantum-to-classical transition <cit.>, (ii) to monitor the stability and robustness against noise of quantum devices and computers subject to decoherence process <cit.>, or (iii) to certify quantum random number generator <cit.>.
Such a reliable and quantitative tool would not only enhance our comprehension of quantum phenomena but also inspire innovative methods for preserving the quantum properties essential to technological advancements.




§ MACROREALISM CONDITIONS



The original definition of MRps given by Leggett and Garg  <cit.> was ambiguous and prone to different interpretations.
During the years, this led to confusion about its meanings and implications. Only recently some authors have clarified that it was intended to understand if a quantum system can be described in terms of classical random variables <cit.>.
More precisely, once the observable to be measured is identified, the MRps assumption is satisfied if quantum superpositions are not allowed, at any time, within the Hilbert space spanned by the eigenbasis of the observable or by a statistical mixture of them <cit.>.
In other words, the MRps assumption is satisfied if there cannot be coherent superpositions and the state of the system is always diagonal in the basis that diagonalizes the observable to be measured <cit.>.


The NIM assumption, entering MR, is equally difficult to formalize. For this reason, it is usually flanked by the no-signaling in time (NSIT) assumption <cit.>, which is regarded as a statistical (and slightly less strong) version of NIM.
As extensively discussed by Halliwell <cit.>, this allows us to relate the NIM with the possibility of attaining the marginal probability of single events from the joint probability of the sequence of them.


That is, as an example, if p(s_1,s_2,s_3) is a generic joint probability of recording s_1, s_2 and s_3, the NSIT condition over, say, the outcome s_3, reads: ∑_s_1, s_2  p(s_1, s_2, s_3) = P(s_3), where P(s_3) is the probability to measure s_3 in correspondence of the state of the system at the time the measurement is performed, ensuring that no previous measurements in the sequence have affected it. For the sake of clarity, throughout the manuscript, the symbol P will always denote probabilities at single times, given by Born's rule, and probabilities at multi-times respecting the theory of classical probability.


Below, we show that the ND quasi-probability always satisfies the NSIT assumption so that the violation of MR can be reduced only to the violation of the MRps' assumption, for which we give both a necessary and sufficient condition.




§ THREE MEASUREMENT QUASI-PROBABILITY DISTRIBUTION



We consider the following scheme for sequential non-demolition measurements. Suppose we have a quantum system 𝒮 evolving under a unitary transformation in a time interval 0 ≤ t ≤. At times t_0≤ t_1 ≤ t_2=, we measure a generic observable Â (Hermitian operator).


In general, any projective measurement perturbs a quantum system inducing the collapse of its wave function, thus destroying the quantum coherence between eigenstates of the measured observable. To avoid this drawback in a scheme with sequential measurements, according to the ND approach, we use an auxiliary quantum detector to store the desired information in the phase of the detector's state, which is eventually measured. This scheme allows us to preserve the quantum coherence in the initial density operator and obtain the average value of observables at multi-times, not perturbed by the interaction with the detector <cit.>.


Sequential non-demolition measurements rely on a sequence of fast (with respect to the timescale of the system evolution) system-detector couplings. 
The unitary transformations describing the coupling processes are û_k  = exp{ i (λ/2) Â⊗p̂}, which occurs at any time t_k with k=0, 1, 2 (see Refs. <cit.> for more details). Here, λ is an effective coupling, and p̂ is an operator acting on the detector's degrees of freedom.


 


We denote the unitary transformation from time t_j-1 to t_j acting on the system only as _j ⊗ = ( t_j, t_j-1 ) ⊗ (with j=1,2) so that the total unitary transformation acting on both the system and detector is _tot = û_2 _2 û_1 _1 û_0.
In the following, to simplify the notation we will denote _i ≡_i ⊗. Explicitly, the total evolution operator _tot reads as

    _tot = e^ i λ/2Â⊗p̂_2   e^ i λ/2Â⊗p̂_1   e^ i λ/2Â⊗p̂.

Notice that, since in general [Û_j, Â] ≠ 0, this scheme can also be applied to describe the sequential measurement of three distinct non-commuting operators Â, B̂ and Ĉ. Observe also that this ND protocol can be implemented in any quantum platform that allows for a tunable interaction between the quantum system of interest and an auxiliary system acting as the detector. For example, in Ref. <cit.>, the scheme was implemented using several qubits to describe the system and one qubit for the detector.


We decompose the initial state of the system in the basis {|i⟩} in which the observable Â is diagonal, i.e. Â|i⟩ = a_i |i⟩ with a_i scalars, so that |ψ_0⟩ = ∑_i  ψ^0_i |i⟩. 
Throughout the paper, we assume that |ψ_0⟩ is not an eigenstate of Â.
On the other hand, the state of the detector is taken to be |ϕ_0⟩ = (1/√(M)) ∑_p |p⟩, where p̂|p⟩ = p |p⟩. The calculation of the evolution of the system and detector together is a straightforward extension of the one presented in Refs. <cit.>, and the final total state is 

    |Ψ⟩ = 1/√(M)∑_p ∑_i, j, k e^i λ p/2(a_k + a_j + a_i)     U_2,kj U_1,jiψ^0_i|k⟩|p⟩

where U_2,kj = k_2j and U_1,ji = j_1i.


The information on the evolution of the monitored system is extracted by measuring the phase accumulated in the detector between two generic states |± p⟩.
This is obtained by taking the off-diagonal element of the detector's density operator and tracing out the degrees of freedom of the system. 
Formally, writing the final total density operator as R̂ = |Ψ⟩⟨Ψ| and the density operator of the detector as r̂^0 = |ϕ_0⟩⟨ϕ_0|, this is 𝒢_λ = 
[ (⊗⟨ p|)R(⊗ |-p⟩) ] / pr^0-p.


Normalizing p to one (in some opportune unit), 𝒢_λ explicitly reads as

    𝒢_λ = ∑_i, l, j, m, k e^i λ (a_k + a_j+a_m+a_i+a_l/2)   U_2,kj U_1,jiρ^0_il U^*_1,lm U^*_2,mk

where ρ̂^0 = |ψ_0⟩⟨ψ_0| is the density operator of the system at the initial time t_0=0, and ρ^0_il = iρ̂^0l.
The function 𝒢_λ is also called quasi-characteristic function <cit.>.


As usual, the ND quasi-probability distribution is obtained by calculating the inverse Fourier transform of 𝒢_λ, i.e.,  _ND(Δ) = (2 π)^-1∫exp{ -i λΔ}𝒢_λ d λ. If we introduce

    Δ_k, j, m, i, l = a_k +  a_j+a_m+a_i+a_l /2

and we use the projectors Π̂_k = |k⟩⟨k| (with Π̂_k^2 = Π̂_k, ∑_k Π̂_k = and Π̂_k Π̂_j = δ_kjΠ̂_k), _ND(Δ) is

    _ND(Δ) = ∑_i, l, j, m, k P_ND(k, j, m, i, l) δ[ Δ - Δ_k, j, m, i, l]

where P_ND(k, j, m, i, l) = U_2,kj U_1,jiρ^0_il U^*_1,lm U^*_2,mk = [ Π̂_k _2 Π̂_j _1 Π̂_i ρ̂^0 Π̂_l ^†_1 Π̂_m ^†_2 ], and δ[·] denotes the Dirac's delta. Thus, P_ND(k, j, m, i, l) is the probability amplitude to arrive in the state |k⟩ following the superposition of paths i → j → k and l → m → k (see Fig. <ref>).


The ND quasi-probability distribution _ND(Δ) comprises both a classical and a quantum contribution, _ cl(Δ) and _ q(Δ), respectively. The former contribution is the one in which a well-defined value of Â is recorded at any measurement time by measuring the system. That is, for m=j and l=i, so that

    _ cl(Δ) = ∑_i, j, k P_ND(k, j, j, i, i) δ[ Δ - Δ_k, j, j, i, i].
 
These can be interpreted as probabilities obeying classical probability theory since P_ND(k, j, j, i, i) = P^(0)_i P^(1)_i → j P^(2)_j → k where P^(0)_i =  ρ^0_ii, P^(1)_i → j = |U_1,ji|^2 and P^(2)_j → k = |U_2,kj|^2 are the probabilities of finding the system initially in the state |i⟩, of having a transition from state i to j and from j to k, respectively. 


The quantum contributions in Eq. (<ref>), thus with m≠ j or l≠ i, are associated with quantum trajectories where the system is in a coherent superposition of paths defined by the measurement outcomes; see Fig. <ref>.
More specifically, at time t_0, the measurement outcomes might be a_i or a_l while, at time t_1, a_j or a_m. Formally, _ q(Δ) is

    _ q(Δ) = '∑_i, l, j, m, k P_ND(k, j, m, i, l) δ[ Δ - Δ_k, j, m, i, l]

where the sum '∑ does not include the terms for which m = j and l = i entering the classical distribution _ cl(Δ); this means that in _ q(Δ) are allowed contributions with (l = i, m ≠ j), (m = j, l ≠ i) and (m ≠ j, l ≠ i).


We stress that the violation of MRps in the _ q(Δ) shall be manifest also in the presence of classically forbidden values of Δ_k, j, m, i, l <cit.>, as given in (<ref>).
For example, suppose that a_i = ± 1. 
The possible classical (i.e., m = j and l = i) values of Δ are { -3, -1, 1, 3 }. On the contrary, the quasi-distribution _ q(Δ) also accounts for terms associated with Δ = ± 2 (for example, with a_k=a_j=a_m=a_i=1 and a_l=-1) or Δ = 0 (for example, with a_k=a_j=1 and a_m=a_i=a_l=-1).


The QPD _ND(Δ) in (<ref>) satisfies the noninvasive measurability condition, in terms of the no-signaling in time assumption <cit.>. This means that marginalizing _ND(Δ) over all the measurement outcomes but the ones at a single time returns the corresponding probability to measure the observable Â given by the Born's rule. 
For example, to obtain the probability of recording a final eigenvalue a_k, we must sum over all the intermediate outcomes a_i, a_l, a_j, a_m. It can be shown (see Methods) that  
P(a_k) = ∑_i, l, j, m P_ND(k, j, m, i, l) = [ Π̂_k Ûρ̂^0 Û^†].
Indeed, P(a_k) is the probability to measure Â at time , when the system has evolved under the unitary transformation Û <cit.>.
Similarly, the probability to record the eigenvalues a_i and a_l at time t=0, is obtained from (<ref>) by summing over the indices l, j, m, k and i, j, m, k respectively. Thus, we have P(a_i) = ∑_l, j, m, kP_ND(k, j, m, i, l) = [ Π̂_i ρ̂^0 ] as expected.
With similar calculations, it can be shown that also the intermediate measurements satisfy the NIM condition, such that P(a_j) = ∑_i, l, m, k P_ND(k, j, m, i, l) = [ Π̂_j Û_1 ρ̂^0 Û_1^†] and P(a_m) = ∑_i, l, j, k P_ND(k, j, m, i, l) = [ Π̂_m Û_1 ρ̂^0 Û_1^†]. 
It is also worth observing that summing P_ND(k, j, m, i, l) over three indices returns the Kirkwood-Dirac quasi-probabilities at two-times <cit.>, apart the 3-tuples (j,m,k) and (i,l,k) that give rise to a single-time probability. By construction, Kirkwood-Dirac quasiprobabilities respect the assumption of no-signaling in time.




§ NEGATIVE REGIONS ARE NECESSARY AND SUFFICIENT FOR VIOLATING MACROREALISM



We now prove that _ND(Δ) has negative contributions if and only if the macrorealism, i.e., MRps, condition is violated. The proof is composed of three steps.
First, we show that _ND(Δ) is real, i.e. given by a distribution of real numbers. Then, we prove that it is normalized to 1 and that this normalization comes uniquely from the classical contribution _ cl(Δ). This implies that the terms entering the quantum contribution _ q(Δ) cancel out each other and that some of them must be negative. The last part of the proof involves the connection between the macrorelism and the negativity of _ND(Δ).


To prove that _ND(Δ) is real, we group the terms in the distribution that are multiplied by the same δ-function δ[Δ - Δ_k, j, m, i, l]. There are indeed identical values of Δ_k, j, m, i, l because of the symmetries between the exchange of the indices (i,l) and (j,m). Formally, using the properties of the matrix elements, e.g., U_2,km = k_2 m =  ( m_2 k)^*, we have P_ND(k, m, j, i, l) = ( P_ND(k, j, m, i, l) )^* so that (<ref>) can be rewritten as 

    _ q(Δ) = 2 ”∑_i, l, j, m, k[P_ND(k, j, m, i, l) ]  δ[ Δ - Δ_k, j, m, i, l]

where ”∑ sum over half the terms than '∑ as it includes terms for which l > i or m > j.
Since the distribution _ cl(Δ) contains (positive) classical joint probabilities by construction and _ q(Δ) comprises real numbers, we conclude that _ND(Δ) is real.


The proof that _ cl(Δ) is normalized comes from noting that its contributions are positive joint probabilities obeying the classical probability theory. In fact, we have that ∫ d Δ_ cl(Δ) = ∑_i, j, k P^(0)_i  P^(1)_i → j P^(2)_j → k = ∑_i, j, k[ Π̂_k _2 Π̂_j _1 Π̂_i ρ̂^0 Π̂_i ^†_1 Π̂_j ^†_2 ] = 1.


Regarding the integration of the quantum contribution _ q(Δ) of the ND quasi-distribution _ND(Δ), it is worth distinguishing two distinct sums of terms: one with m ≠ j, and the other with (m = j, l ≠ i). Then, as above, we integrate both sums of terms in _ q(Δ) over Δ.
The integration over Δ of the sum with m ≠ j leads to

    2 ∑_k, i, l, j, m>j [ Π̂_k _2 Π̂_j _1 Π̂_i ρ̂^0 Π̂_l ^†_1 Π̂_m ^†_2 ] =0,

where we have used the projectors' properties Π̂_kΠ̂_j = δ_k,jΠ̂_k = 0 and ∑_k Π̂_k =.
An analogous result is obtained from integrating over Δ the sum of terms in _ q(Δ) with (m = j, l ≠ i). In fact, one can get that

    2  ∑_k, i, j, l>i [ Π̂_k _2 Π̂_j _1 Π̂_i  ρ̂^0 Π̂_l ^†_1 Π̂_j ^†_2 ] = 0  .


Therefore, the normalization of _ND(Δ) uniquely comes from the classical distribution _ cl(Δ). 
Given that the terms in _ q(Δ) are real numbers and cancel out upon integration, it follows that at least some of them must be negative. Hence, the negativity of _ND(Δ) derives from contributions in _ q(Δ) that are built with the coherent superposition of Â's eigenstates at times t_0, t_1, t_2. We can thus state that necessary condition for the negativity of _ND(Δ) is the presence of contributions in _ q(Δ) coming from the coherent superposition of eigenstates associated to an observable that is measured at multiple times.


Notably, using ND quasi-probabilities, the statement above is also a sufficient condition, since the negativity of _ND(Δ) can only derive from _ q(Δ).
Since the two summations in (<ref>)-(<ref>) must always be zero, whenever there is a coherent superposition of the measurement observable's eigenstates, i.e., _ q(Δ)≠ 0, _ND(Δ) must contain negative (real) terms. That is, a sufficient condition for the negativity of _ND(Δ) is the presence of the coherent superposition of eigenstates associated with the observable measured at multiple times.


We conclude by noting that if a system is in a coherent superposition of the eigenstates of an observable measured at distinct times, then the MRps assumption is violated. Accordingly, the MRps assumption is violated if and only if the ND quasi-probability distribution _ND(Δ) exhibits negativity.




§ LEGGETT-GARG INEQUALITIES



Having established that negative regions in the ND quasi-probability distribution are necessary and sufficient indicators of the violation of MRps, we now compare our results with the usual tool employed so far to identify such a quantum feature: the Leggett-Garg inequalities.


The LGIs are usually formulated for the so-called binary observables <cit.>, that is, for observables that can have only two outcomes.
Hence, while in the previous discussions, the observable Â had a generic discrete spectrum, here we assume that its eigenvalues can be only a_i = ± 1 for any time t_i. LGIs can be built using quantum correlators at two times, say t_i and t_j, which are defined as <cit.>
C_ij =  [  (Â(t_i) Â(t_j) + Â(t_j) Â(t_i) ) ρ̂^0]/2, where Â(t_i) = ^†(t_i, 0)Â(t_i, 0).
The expression of C_ij can be rewritten as <cit.>

    C_ij = ∑_i,j a_i a_j P(a_j,a_i)  = ∑_i, j, k a_i a_j P(a_k, a_j , a_i)

where, by construction, P(a_j, a_i) = ∑_k P(a_k, a_j, a_i) and, as above, P(a_k, a_j, a_i)=P^(0)_i P^(1)_i → j P^(2)_j → k is the joint probability to record the measurement outcomes a_k, a_j and a_i at times t_i,t_j,t_k by means of a sequential scheme.  The sequential measurements are performed at three times t_0=0, t_1 and t_2. 


Following Ref. <cit.>, we can calculate the Leggett-Garg (LG) parameter 

    K ≡ C_01 + C_12 - C_02.

Notably, for any classical system fulfilling MR, the LG parameter K reads as

    K = 1 - 4 [ P(1,-1,1) + P(-1, 1,-1) ]

that results in the LGI -3 ≤ K ≤ 1. That is, if the -3 ≤ K ≤ 1 is not satisfied, then at least a condition of MR is violated.


In this context, it would be desirable to find a decomposition of K made of two terms: one equivalent to the right-hand-side of (<ref>) (valid under MR), and the other different from zero whenever the LGI -3 ≤ K ≤ 1 is violated. For this purpose, we can still resort to the ND quasiprobabilities given that, for binary observables (here, the eigenvalues are ± 1), the correlators C_ij can be equivalently expressed as <cit.> 

    C_ij = ∑_i,j a_i a_j P_ND(j, i),

where P_ND(j, i) = ∑_l, m, k [ P_ND(k, j, m, i, l) +  P_ND(k, m, j, i, l) + P_ND(k, j, m, l, i) +  P_ND (k, m, j, l, i) ] with P_ND(j, i) given by Kirkwood-Dirac quasiprobabilities as commented before. Accordingly,

    C_01 = 1/4∑_i, j a_i a_j ∑_l, m, k[ P_ND (k, j, m, i, l) +  P_ND (k, m, j, i, l)    
       + P_ND (k, j, m, l, i) +  P_ND (k, m, j, l, i) ].

Writing the correlator at two-times C_01 as a function of the three-time ND quasiprobabilities brings the advantage of including also the superposition of the observable Â's eigenstates resulting in multiple different possible outcomes: a_i and a_j at times t_0=0, and a_j and a_m at time t_1.


The definition of the correlators C_12 and C_02 involving the (final) measurement at time t_2 are similar to that of C_01: C_12 = ∑_j, k a_k a_j P_ND(k, j) and C_02 = ∑_i, k a_k a_i P_ND(k, i), whose extended expression is in the Methods.


Since we are working with binary variables, each of them can take two values; in our case, a_i and -a_i. For simplicity, we refer to them with the indices i and i̅; for example, P_ND (k̅, j, m, i, i̅) shall stand for P_ND(-a_k, a_j, a_m, a_i, -a_l).
Using the correlators C_ij as defined above, the LG parameter K for the ND procedure reads as

    K    =   1/4∑_i, l, j, m, k f(k, j, m, i, l) P_ND(k, j, m, i, l)

with f(k, j, m, i, l) = (a_i +a_l)(a_j +a_m) + a_k (a_j +a_m-a_i -a_l); see Methods for details. 
It is convenient to separate in K the classical contributions when m=j and l=i, i.e.,

    _cl = 1/2∑_k, j, i[ a_i a_j  + a_k (a_j -a_i ) ] P_ND(k, j, j, i, i),

from the quantum ones when m=j and l=i̅≠ i 

    _q,1 = 1/2∑_i, j, k a_k a_j  P_ND(k, j, j, i, i̅)

and when m=j̅≠ j and l= i 

    _q,2 = - 1/2∑_i, j, k  a_k a_i P_ND(k, j, j̅, i, i).



First, we now show that from the classical contributions in (<ref>) we recover the classical results in the right-hand-side of (<ref>).
To do this, we recall that P_ND(k, j, j, i, i) = P^(0)_i  P^(1)_i → j P^(2)_j → k with ∑_i, j, kP^(0)_i  P^(1)_i → j P^(2)_j → k=1, namely the total probability over a complete set of classical paths must sum to 1. The latter can be written explicitly as 
∑_i, k P_ND(k, i, i, i, i) + ∑_k P_ND(k, k, k, k̅, k̅) +∑_k  P_ND(k,  k̅, k̅, k, k) = 1 where we have split the contributions for j=i (and any k), j=k and i = k̅≠ k, and j = k̅≠ k and i = k. Using this separation criterion to (<ref>), the classical contribution _cl can be written as

    _cl = 1 - 4 ∑_k P_ND(k, k̅, k̅, k, k).

Expanding the sum for k = ± 1 leads to the result in (<ref>) so that _cl satisfies the LGI, i.e., -3 ≤_cl≤ 1.


 



As shown in the Methods, the remaining terms in (<ref>) can be simplified by noting that _q,1 = 0. Therefore, when present, the violation of the LGI is due to the second quantum contribution _q,2 in (<ref>).
_q,2 can be further simplified by observing that ∑_k P_ND(k, j,j̅, i, i) = 0 so that P_ND(k̅, j, j̅, i, i) = - P_ND(k, j, j̅, i, i). In this way, setting a_k=1 without loss of generality, (<ref>) equals to

    _q,2 =-4 ∑_k[ P_ND(k, j, j̅, k, k) ].

The derivation of (<ref>) is in the Methods.


Some remarks are due to this point.
Using the ND quasi-probability distribution highlights the connection between the violation of macrorealism and LGIs.
In particular, the contributions with m = j̅≠ j and l=i, denoting the superposition between eigenstates of Â, are the ones responsible for the violation of the LGI. However, there are situations where the MRps condition is violated but the LGI is satisfied.
This occurs for any initial condition for which _q,2 vanishes. In such a case, the total quantum contribution _q vanishes as well, _q = 0, with the result that = _cl and the LGI is satisfied. Nevertheless, we can have that l ≠ i, which means that the system is initially in a superposition of eigenstates of Â and this fact naturally violates the MRps.
On the contrary, if we use the ND quasi-probability _ND(Δ), the terms P_ND(k, j, j, i, i̅) in (<ref>) always contribute to the negative regions of the quasi-probability distribution. This is a confirmation that the LGIs give only a sufficient condition for the violation of the MRps and thus of the MR's assumption, while the ND quasi-probability _ND(Δ) gives both a sufficient and necessary condition.




§ EXAMPLE


Following Ref. <cit.>, we discuss a specific example that clarifies the results obtained.
Let us consider a two-level quantum system that is initially in the state |ψ_0⟩ = (|↑⟩ + i |↓⟩)/√(2) where |↑⟩ and |↓⟩ are the eigenstates of the Pauli operator σ̂_z.
The dynamics is generated by the Hamiltonian Ĥ = ωσ̂_x/2 and we measure the operator Â=σ̂_z at times t_0=0, t_1=τ and t_2=2τ.


The correlators C_ij [see (<ref>)] and the corresponding LG parameter K can be calculated analytically:

    K = C_01 + C_12 - C_02 = 2  cos(ωτ) - cos(2 ωτ).

As discussed above, for a classical system -3 ≤ K ≤ 1. Hence, any value of K outside this range implies the violation of such an inequality. If we consider the ND protocol, the whole dynamics of the system and the detector is given by (<ref>), which leads to the ND quasi-probability distribution in (<ref>).


The numerical results for the example are shown in Fig. <ref>.
The top-left panel shows the value of K as a function of ωτ with the shaded regions representing the situations in which the LGI -3 ≤ K ≤ 1 is violated.
The colored shapes (dot, square, triangle) give the value of K for three specific values of ωτ. The other panels of the figure show the ND quasi-probability distribution [as obtained from (<ref>)] that are associated with each colored shape. As we can see, the LGI is violated only for some parameter choices, e.g., the green dot in the top-left panel.
However, for intermediate values of ωτ (the orange square and purple triangle), the LGI is satisfied but the dynamics exhibit quantum features given by the presence of quantum coherence along the eigenbases of Â. As proved above, this circumstance is identified by determining negative regions of the corresponding distribution _ND(Δ).
As expected, the ND quasi-probability distribution is always able to identify the quantum features of the system even when the LGI fails.


From (<ref>), the violation of the LGI -3 ≤ K ≤ 1 occurs for 0 ≤ωτ≤π/2 and 3 π/2 ≤ωτ≤ 2 π. This means that the LGI correctly identifies the violation of MRps in only half of the cases, while the ND protocol always succeeds.




§ CONCLUSIONS


We have presented a protocol that allows us to identify the presence of genuinely quantum behaviors unambiguously, in terms of the violation of macrorealism per se (MRps).
This protocol is based on performing sequential quantum non-demolition measurements of a given observable Â, and this can be attained using a quantum detector. Using the corresponding measurement outcomes recorded at multiple times, we can construct a quasi-probability distribution. We have proven that the presence of negative regions in such a quasi-probability distribution is a necessary and sufficient condition for the violation of the MRps (and macrorealism) that outlines the presence of quantum behaviors.


The non-demolition protocol has some additional features that make it interesting for more practical applications.
First, it can provide results overcoming some limitations imposed by the LGIs, which are usually employed for binary observables, as the protocol with ND measurements can be implemented for observables with arbitrary discrete spectra. Secondly, being based on an experimental protocol, the ND protocol gives an operational procedure that, in principle, could be implementable on any quantum platform <cit.>.


The advantage of the ND protocol over the LGIs ultimately is a consequence of the more information we extract from the system and its dynamics.
While using LGIs we get information only about correlators at two times, with the ND protocol we have access to a full quasi-probability distribution and, therefore, to all their moments <cit.>.
While it is natural to think that the ND protocol gains more information about non-classicality and, thus, needs more resources than other methods as LGIs, the comparison in terms of resources is not straightforward.
Regarding the ND quasi-probability distribution, it is obtained by a Fourier transform of the measured data. Therefore, the resources needed to implement the ND protocol depend critically on the requested precision in performing the Fourier transform. To evaluate this a more detailed analysis is needed.


The research fields that can benefit mostly from the implementation of the ND protocol as a tool to identify quantum behavior are the foundation of quantum mechanics and quantum technologies.
A few examples are the generation of certified random numbers <cit.>, the study of quantum gravity in mesoscopic systems, and the cooling of optomechanics systems up to a scale where quantum effects play a role <cit.>.
In all these cases, the question of when and how the quantum-to-classical transition occurs is of critical importance, and it might help to improve already-performed experimental results <cit.> or to better identify causes leading to decoherence in open quantum systems, quantum computers or quantum annealers <cit.>.
This could be possible by showing that a given observed quantum-to-classical transition corresponds to a reduction of negative regions in an ND quasi-probability distribution that is built over a properly chosen observable Â. The results in Refs. <cit.> seem to suggest that such a circumstance could be in principle attainable. 







§ NON-INVASIVE MEASUREMENT


To compute the probability of measuring an outcome a_k (eigenvalue of Â) from the measurement at time t_2, we sum over all the intermediate outcomes a_i, a_l, a_j, a_m. In this way, using the properties of the projectors (Π̂_k^2 = Π̂_k, ∑_k Π̂_k = and Π̂_k Π̂_j = δ_kjΠ̂_k), we get 

    P(a_k)    =   ∑_i, l, j, m P_ND(k, j, m, i, l) 
       =   ∑_i, l, j, m[ Π̂_k _2 Π̂_j _1 Π̂_i ρ̂^0 Π̂_l ^†_1 Π̂_m ^†_2 ] 
       =   [ Π̂_k ρ̂^0 ^†]  = [ Π̂_k ρ̂() ]

with Û = _2 _1 and ρ̂() = ρ̂^0 ^†. 
P(a_k) is the probability to record a_k at time  as given by the Born's rule, consistently with Ref. <cit.>.


Concerning the measurement at time t=0 of the ND procedure, we have to sum over two possible realizations built with the indices i and l. Thus, renaming the indices, the probability to measure the eigenvalue a_i at time t=0, can be written as   

    P(a_i)    =   1/2∑_l, j, m, k{ P_ND(k, j, m, i, l) + P_ND(k, j, m, l, i) }
       =   1/2∑_l, j, m, k{[ Π̂_k _2 Π̂_j _1 Π̂_i ρ̂^0 Π̂_l ^†_1 Π̂_m ^†_2 ] + 
       +   [ Π̂_k _2 Π̂_j _1 Π̂_l ρ̂^0 Π̂_i ^†_1 Π̂_m ^†_2 ] }
       =   [ Π̂_i ρ̂^0 ].

In a similar fashion, we obtain the probability P(a_l) = 1/2∑_i, j, m, k{ P_ND(k, j, m, i, l) + P_ND(k, j, m, l, i) }.


The calculation for the measurement at time t_1 of the ND procedure follows a common reasoning than the one used to get P(a_i). In this case, indeed, there are two possible realizations over the indices j and m, such that 

    P(a_j)    =   1/2∑_i, l, m, k{  P_ND(k, j, m, i, l) + P_ND(k, j, m, l, i) }
       =   [ Π̂_j _1 ρ̂^0  ^†_1 ] = [ Π̂_j ρ̂^0(t_1) ]

that, as expected, corresponds to the probability of recording the outcome a_j after the system has evolved to the state ρ̂^0(t_1).
Observing that in general the sum of P_ND(k, j, m, i, l) over three indices returns the Kirkwood-Dirac quasiprobabilities at two times, we can conclude that the quasi-probability distribution _ND(Δ) satisfies the non-invasive measurability condition in terms of the assumption of no-signaling in time.




§ CORRELATORS AND LG PARAMETER


The two-time quantum correlators C_12 and C_02, defined in the main text, explicitly read as

    C_12   =   ∑_j, k a_k a_j ∑_i, l, m[  P_ND (k, j, m, i, l) + P_ND (k, m, j, i, l) ] 
    
    	C_02   =   ∑_i, k a_k a_i ∑_l, j, m[  P_ND (k, j, m, i, l) + P_ND (k,  j, m, l,i) ].

In the expressions of C_12 and C_02, we can rearrange the indices to factorize a common P_ND(k, j, m, i, l). For example, we can rewrite C_12 by changing the indices j ↔ m in P_ND(k, m, j, i, l), so that 

    C_12 = ∑_i, l, j, m, k (a_k a_j + a_k a_m) P_ND (k, j, m, i, l).
 
By repeating this rearrangement of terms also for C_01 and C_02, we can obtain (<ref>) in the main text. 


Now we are going to show that _q,1 = 0. To do this, we have to derive some properties of the ND quasi-probability distribution. 
Let us consider the case in which m = j and l ≠ i.
We want to show that P_ND(k, k, k, i, i̅) = - P_ND(k̅, k̅, k̅, i, i̅). For a two-level system and binary observables, we have that Π̂_k =  - Π̂_k̅. Using this relation, by direct calculation, it holds that

    U_2,kk U^*_2,kk   =   [ Π̂_k _2 Π̂_k _2^†] = [ Π̂_k̅_2 Π̂_k̅_2^†] 
       =    U_2,k̅k̅ U^*_2,k̅k̅ .

Analogously, we get that 

    U_1,ki U^*_1,  i̅ k   =   ⟨i̅|_1^† (  - Π̂_k̅) _1 |i⟩ = 
         - U^*_1,i̅k̅ U_1,k̅ i.

Hence, 

    P_ND(k, k, k, i, i̅)    =     U_2,kk U_1,kiρ^0_i i̅ U^*_1, i̅ k U^*_2,kk
       =   	 - U_2,k̅k̅ U_1,k̅ iρ^0_i i̅ U^*_1,i̅k̅ U^*_2, k̅k̅
       =    - P_ND(k̅, k̅, k̅, i, i̅).

In a similar way, it can be proven that P_ND(k, k̅, k̅, i, i̅) = -P_ND(k̅, k, k, i, i̅).
At this point, let us take the expression of _q,1 [(<ref>) in the main text]. Separating the contribution for a_j=a_k and a_j ≠ a_k and recalling that a_k^2=1, _q,1 can be equivalently written as

    _q,1 = ∑_i{∑_k P_ND(k, k, k, i, i̅) -  ∑_k P_ND(k, k̅, k̅, i, i̅) }.

Using the properties of the ND quasi-probability derived above, we have that both the sums vanish with the result that _q,1 = 0.


Let us now derive the simplified expression of _q,2 as given by (<ref>) in the main text. We are going to use the relation ∑_k P_ND(k, j, j̅, i, i) = 0, whose proof is in the following 1-line calculation:

    ∑_k P_ND(k, j, j̅, i, i)    =   ∑_k[ Π̂_k _2 Π̂_j _1 Π̂_i ρ̂^0 Π̂_i _1^†Π̂_j̅_2^†] 
       =   ∑_k [ Π̂_j _1 Π̂_i ρ̂^0 Π̂_i _1^†Π̂_j̅] = 0.

From this equality, we get P_ND(k, j, j̅, i, i) = -P_ND(k̅, j, j̅, i, i).
Moreover, recalling that P_ND(k, m, j, i, l) = ( P_ND(k, j, m, i, l) )^*, we also have that ∑_j P_ND(k, j, j̅, i, i) = 2  [P_ND(k, j, j̅, i, i) ]. As a result, the contribution _q,2 in (<ref>) (with constraints m=j̅ and l=1) simplifies to

    _q,2   =    - ∑_i, j, k a_k a_i   P_ND(k, j, j̅, i, i) 
       =   
        - ∑_j, k{ P_ND(k, j, j̅, k, k) - P_ND(k, j, j̅, k̅, k̅) }
       =    -2 ∑_j{   P_ND(k, j, j̅, k, k) + P_ND(k̅, j, j̅, k̅, k̅)  }
       =    -4 {[ P_ND(k, j, j̅, k, k) ] + [ P_ND(k̅, j, j̅, k̅, k̅) ]    }
       =    -4 ∑_k[ P_ND(k, j, j̅, k, k) ]

that corresponds to (<ref>) in the main text. 




 §.§ Acknowledgments


The authors acknowledge the fruitful discussion with Andrea Smirne.
P.S. acknowledges financial support from INFN. S.G. would like to thank the PRIN project 2022FEXLYB Quantum Reservoir Computing (QuReCo), the PNRR MUR project PE0000023-NQSTI funded by the European Union–Next Generation EU, and the MISTI Global Seed Funds MIT-FVG Collaboration Grant “Revealing and exploiting quantumness via quasiprobabilities: from quantum thermodynamics to quantum sensing”.




 §.§ Authors' contributions


P.S. designed and performed the research. P.S. and S.G. discussed the results and wrote the paper.






 Paper Title: Robust Implementation of Discrete-time Quantum Walks in Any Finite-dimensional Quantum System.



APS/123-QED








abamitsaha@gmail.com



^1A. K. Choudhury School of Information Technology, University of Calcutta, India

^2Institute of Radiophysics and Electronics, University of Calcutta, Kolkata, India
^3Institut National de Recherche en Informatique et en Automatique (Inria), Paris, France
^4École Normale Supérieure (ENS), PSL University, Paris, France



             

§ ABSTRACT 
Research has shown that quantum walks can accelerate certain quantum algorithms and act as a universal paradigm for quantum processing. The discrete-time quantum walk (DTQW) model, owing to its discrete nature, stands out as one of the most suitable choices for circuit implementation. Nevertheless, most current implementations are characterized by extensive, multi-layered quantum circuits, leading to higher computational expenses and a notable decrease in the number of confidently executable time steps on current quantum computers. Since quantum computers are not scalable enough in this NISQ era, we also must confine ourselves to the ancilla-free frontier zone. Therefore, in this paper, we have successfully cut down the circuit cost concerning gate count and circuit depth by half through our proposed methodology in qubit systems as compared to the state-of-the-art increment-decrement approach. Furthermore, for the engineering excellence of our proposed approach, we implement DTQW in any finite-dimensional quantum system with akin efficiency. To ensure an efficient implementation of quantum walks without requiring ancilla, we have incorporated an intermediate qudit technique for decomposing multi-qubit gates. Experimental outcomes hold significance far beyond the realm of just a few time steps, laying the groundwork for dependable implementation and utilization on quantum computers.



                              
Robust Implementation of Discrete-time Quantum Walks in Any Finite-dimensional Quantum System
    Amlan Chakrabarti^1
    August 5, 2024
=============================================================================================







§ INTRODUCTION

    The field of quantum computing <cit.> is in rapid development in the current era. Quantum computers are far more efficient in doing certain tasks than our best state-of-the-art classical computing algorithms. They can leverage quantum mechanical phenomena like superposition, entanglement, and measurement giving them a significant edge over classical computers. For example, Grover's search <cit.> can search in O(√(N)) time where N is the size of the input space. In 1994, Peter Shor showed that we can prime factor a number much more efficiently using quantum computers <cit.>. Like these quantum algorithms, quantum walk algorithms have also applications in various fields such as quantum simulation, quantum search algorithms, and quantum cryptography. They also offer potential advantages over classical algorithms in certain scenarios, particularly in tasks that involve searching through large databases or graphs.
    
    Quantum walks <cit.> is the quantum mechanical counterpart of classical random walks. Random walks have been used to simulate the motion of molecules in gas or liquid, also known as Brownian motion <cit.>, unpredictable fluctuating stock prices <cit.>, PageRank <cit.> algorithms in search engines like Google, and recommendation algorithms also make use of it. Therefore, there is a significant amount of use cases for random walk algorithms but how is quantum walk different from classical random walk? We get a normal distribution of the probability of finding the walker in different states of the space. Therefore, around the state from which the walker starts walking has the highest probability, or in other words, the walker is most likely to return to where it started. In contrast, a quantum walker is the exact opposite. In a quantum walk, the probability of finding the walker around the state it started is the lowest. It is highest on the states farthest from the starting position. Therefore, we can say that, quantum walks move faster than their classical counterparts <cit.>, which can be useful for certain applications such as in developing cryptographic applications <cit.>. 

    Two types of quantum walks are proposed in the literature: discrete-time quantum walks (DTQW) <cit.> and continuous-time quantum walks (CTQW) <cit.>. In the first model, we have a walker, a coin, and an evolution operator which is applied to both the walker and the coin in discrete time steps. On the other hand, in the second model, we have a walker and an evolution that can be applied to the walker with no timing restrictions. In this article, we have only worked with the first model i.e., DTQW. Before we discuss our motivation for implementing a scalable generalized quantum walk algorithm, we address why one should study quantum walks in the first place i.e., its applications. Childs in his work showed that quantum walks can be considered a universal computational model <cit.>. Ambainis showed that quantum walks can be used to efficiently solve the element distinctness problem which is a problem of finding two distinct items among N given ones <cit.>. The application domain of quantum walks is not bounded by theoretical computer science only. We can use quantum walks to simulate natural processes like photosynthesis <cit.>. The primary challenge in quantum computing lies in optimizing qubit utilization and circuit depth to maximize the implementation of Discrete-Time Quantum Walks (DTQW) within the constraints of qubit quantity and coherence time for its applications.

    Motivation: In the initial development of quantum walk algorithms <cit.> and in recent developments <cit.> also, for a general quantum walk implementation, without considering special cases such as quantum walks on complete graphs <cit.>, the naive approach <cit.>  is used in binary quantum systems. This is a generalized approach for implementing n-qubit quantum walks, albeit, this is inefficient concerning gate count and circuit depth. The noise level is also colossal because of its inefficient circuit structure. Our main motivation is to develop an efficient generalized approach to robust circuit implementation of DTQW that can also be scaled up to any finite-dimensional quantum system to make it noise-resilient. In <cit.>, the authors have proposed a generalized quantum walk implementation for 5-qubit quantum hardware, but not scalable to the n-qubit systems. Another efficient quantum circuit for the DTQW on the cycle has been presented as a notable advancement in <cit.>. By employing just one QFT and one IQFT, the approach in <cit.> markedly enhances the most efficient state-of-the-art QFT-based implementation. Albeit, this QFT-based implementation is not efficient in this ancilla-free frontier zone <cit.> and is also difficult to scale to any finite-dimensional quantum systems due to its complex circuit structure. In <cit.>, the authors have shown an efficient implementation for d-dimensional quantum walks in qudit systems where d is odd, but the circuit for even dimensions remains inefficient because it is still based on the increment-decrement approach due to engineering challenges. 
    
    Contribution: With the background of the implementation of DTQW, in this article, we have successfully cut down the circuit cost concerning gate count and circuit depth by half through our proposed methodology in qubit systems as compared to the naive increment-decrement approach. Certainly, this is the first-of-its-kind approach apart from naive increment-decrement, which is generalized for any n-qubit systems. Because of its engineering excellence, we also show that with slight modification, the proposed approach is generalized for any finite-dimensional quantum system. 
    
    
    
    Our key contributions are as follows:
    
        
  * We optimize the circuit for DTQW by reducing the circuit depth and gate count. We have developed a generalized algorithm that is theoretically twice as efficient as the naive increment-decrement approach. From now on we will call it the Enhanced Increment-Decrement approach.
         
  * Our proposed algorithm is generalized in the binary space, meaning it works for any number of qubits. However, our further aim is to extend this generalization to higher dimensions as well. Through our proposed approach, by setting the Least Significant Bit (LSB) as any even dimension when d ≥ 4  and adjusting the dimensionality of the other Most Significant Bits (MSB) for any qudits, our proposed algorithm can be generalized for any d-dimensional space.

        
  * We have further compared the naive approach and our proposed approach in terms of gate count, circuit depth, and effect of noise after Toffoli decomposition and we have observed that in every comparison, our proposed approach has outperformed the naive approach significantly.

       
    

 In this paper, first, we discuss some basic concepts and gates in quantum computing in section <ref>. Then we discuss the mathematical foundation and implementation of the naive quantum walk approach in section <ref>. Next, we explain our proposed methodology in binary systems in detail in section <ref> and compare our approach with the naive approach in section <ref>. We then discuss the implementation of our approach in higher dimensions in section <ref>. Finally, we conclude our remarks in <ref>.




§ BACKGROUND

In this section, our discussion will focus on some basics of quantum computing and quantum walks.

    

 §.§ Quantum computing

        The bit is the fundamental concept of classical computation and classical information. Quantum computation and quantum information are built upon an analogous concept, the quantum bit or qubit for short <cit.>. As classical bits can be either 0 or 1, a qubit can be in a continuum of states between 0 and 1 until it is observed or measured which is called superposition. For example, a qubit can be in the probabilistic state α|0⟩+β|1⟩, where α and β can be complex numbers. Changes occurring in a quantum state can be described using the language of quantum computation. Analogous to the way a classical computer is built from an electrical circuit containing wires and logic gates, a quantum computer is built from a quantum circuit containing wires and elementary quantum gates to carry around and manipulate the quantum information <cit.>. We have used the following gates for our proposed approach: Pauli X (NOT) Gate, Hadamard Gate, Controlled NOT (CX) Gate, Toffoli (CCX) Gate, and Multi-controlled Toffoli (C^nX) Gate and its generalized versions. A decomposition using the Clifford+T gate set of the Toffoli gate in binary systems is shown in Figure <ref>, which is used to estimate the gate resources in this paper in binary quantum systems.

    
 
  
   
        
       
    

 §.§ Discrete-time quantum walks (DTQW)
 
    

  §.§.§ Mathematical background of DTQW

        As a mathematical illustration, DTQW is applied over 1D space which is an Integer line. A coin is tossed, and according to that, the walker shifts its position to the right or left. Let H_c be the Hilbert space of the coin spanned by the basis set {|↑⟩, |↓⟩} and H_x be the Hilbert space of the position states spanned by the basis set {|x⟩}_x∈ℤ. The unitary operator for a single step of the quantum walk is given by
        
    U=S(C⊗ I)

        where C is the coin toss operator and S is the shift operator that is applied to the walker after the toss. ⊗ is tensor product. In the case of a general coin,
        
    .75C=cosθ|↑⟩⟨↑|+e^iϕ_1sinθ|↑⟩⟨↓|+e^iϕ_2sinθ|↓⟩⟨↑|-e^i(ϕ_1+ϕ_2)cosθ|↓⟩⟨↓|

        where θ∈[0, 2π) and ϕ_1, ϕ_2∈[0, π). We refer to θ as rotation and ϕ_1, ϕ_2 as phase parameters of the general coin <cit.>. But here we have considered only the Haramard coin for simplicity. So having θ=π/4 and ϕ_1, ϕ_2=0 we get,
        
    C=1/√(2)[  1  1;  1 -1 ]

        Finally, the unitary operator for shifting the walker is defined as <cit.>,
        
    S=∑_x∈ℤ(|↑⟩⟨↑|⊗|x+1⟩⟨x|+|↓⟩⟨↓|⊗|x-1⟩⟨x|)


    

  §.§.§ Implementation of quantum walks in naive approach

        The components for increment and decrement in a 3-qubit system in the naive implementation of DTQW are shown in Figure <ref>. After a coin toss, which is a simple Hadamard transform in this case, we need to do both increment and decrement operations based on the state of the coin. The circuit for a single step of DTQW in this naive setting is shown in Figure <ref>. For n qubits, we can have 2^n-1-1 steps of DTQW. The position state mapping for the circuit shown in Figure <ref>, after it is applied 3 times is as Table <ref>.
        
        
        



§ OUR PROPOSED APPROACH: ENHANCED INCREMENT-DECREMENT APPROACH FOR DTQW IN BINARY QUANTUM SYSTEMS
 
    

 §.§ Mathematical formulation of our proposed approach

     As an instance, DTQW is applied over a 1D integer line. A coin is tossed and according to that, the walker shifts its position. Let H_c be the Hilbert space of the coin spanned by the basis set {|↑⟩,|↓⟩} and H_x be the Hilbert space of the position state by the basis set {|x⟩}_x∈ℤ. |x⟩. |x⟩ can be represented by multiple qubits like |q_n q_n-1...q_0 ⟩ where q_n is the MSB and q_0 is the LSB.
        In this approach two unitary operators are needed for the walk, unlike the naive approach, one operator is used for the even position state and another one is used for the odd position state. The operators are
        
    U_even=S_even(C⊗ I)

        
    U_odd=S_odd(C⊗ I)

        where C is the coin toss operator and S_even and S_odd are the shift operators that are applied to the walker after the toss. A tensor product is denoted by ⊗.
        The unitary operator for shifting the walker to an even position state is defined as
         
    S_even=∑_x∈ℤ(|↑⟩⟨↑|⊗|x_e1⟩⟨x|+|↓⟩⟨↓|⊗|x_e2⟩⟨x|)

        where |x_e1⟩ consists of q_nq_n-1...q_0. It can be written as |x_n-1+1⟩|x_0+1⟩ where x_n-1 consists of q_nq_n-1...q_1 and x_0 consists of q_0.
        |x_e2⟩ consists of q_nq_n-1...q_0. It can be written as |x_n-1⟩|x_0+1⟩ where x_n-1 consists of q_nq_n-1...q_1 and x_0 consists of q_0.
        The unitary operator for shifting the walker to an odd position state is defined as
         
    S_odd=∑_x∈ℤ(|↑⟩⟨↑|⊗|x_o1⟩⟨x|+|↓⟩⟨↓|⊗|x_o2⟩⟨x|)

        where |x_o1⟩ consists of q_nq_n-1...q_0. It can be written as |x_n-1⟩|x_0-1⟩ where x_n-1 consists of q_nq_n-1...q_1 and x_0 consists of q_0.
        |x_o2⟩ consists of q_nq_n-1...q_0. It can be written as |x_n-1-1⟩|x_0-1⟩ where x_n-1 consists of q_nq_n-1...q_1 and x_0 consists of q_0.
    
    

 §.§ Implementation of the proposed approach

       
        In the naive implementation of the quantum walk, two components were required for increment and decrement in each coin toss. Inspired by the work in <cit.>, in our proposed algorithm, only one component is required in each coin toss. Figure <ref> shows both components for a 3-qubit system. Extending the circuit for more than 3 qubits requires more multi-controlled Toffoli gates similar to the naive approach. Figure <ref> shows the generalized circuit diagram. It can be easily observed that, in our proposed algorithm, Algorithm <ref>, we can cut down the depth of the circuit for a single step of DTQW by half.

        
            


             


         
  
        

  §.§.§ Explanation

         In our approach, the basic concept of increment and decrement (shown in Figure <ref>) is used with some modification. Indeed, it's widely understood that the increment and decrement circuit can be extended to n qubits. In our proposed approach, each step incorporates either the enhanced increment or the enhanced decrement circuit (Shown in Figure <ref>).
         
             
  * The enhanced increment operation: The walk starts at the initial state of |0⟩^⊗ n. Following the coin toss, executed through a simple Hadamard transform, we proceed to select a single shift operator. Referring back to our discussion in Figure <ref>, since the walker currently resides in an even position state, our selection will be the enhanced increment circuit. This circuit performs an increment operation on every qubit except for the least significant bit (LSB) when the coin state happens to be |1⟩. As an example, if we consider a 3 qubit system, then from |000⟩ we get two states, |001⟩ when the coin is |0⟩, and |011⟩ when the coin is |1⟩. We can consider |011⟩ as +1 state and |001⟩ as -1 state. From this example, it's evident that for even position states, the least significant bit (LSB) will always be |0⟩, while for odd position states, it will be |1⟩.
             
  * The enhanced decrement operation:
             After the first toss, the walker is on |001⟩ and |011⟩. Now we choose the enhanced decrement circuit as the shift operator. Akin to the enhanced increment circuit, it decrements all the qubits except for the LSB. So, from |001⟩, we get |110⟩ (a new state) when the coin is |0⟩ and |000⟩ (the previous state) when the coin is |1⟩. We can consider the new state |110⟩ as the -2 state. Similarly, from |011⟩, we get |000⟩ (the previous state) when the coin is |0⟩ and |010⟩ (a new state) when the coin is |1⟩. We can consider the new state |010⟩ as the +2 state.
         

         The position state mapping employing the naive approach, shown in Table <ref>, uses binary 2’s complement counting scheme. Similarly, in our proposed approach, for all even position states, the LSB is always |0⟩, and |1⟩ for all odd position states. This implies that the value of LSB always flips at every step of the DTQW. We have exploited this fact in our proposed approach and removed the flipping of the LSB in control with the coin state. In our approach, as it is also portrayed by the diagram (Fig. <ref>), the LSB is flipped irrespective of the coin state. For a 3-qubit system, the position state mapping employing our approach is shown in Table <ref>.

         

          



      
        
        

  §.§.§ Mathematical steps for a 3-qubit DTQW using our proposed approach 
We have taken |000⟩ as the initial position state of the walker. A total of three steps are as follows:
        a_0|0⟩|000⟩.
        ↓ U_even
        a_1|0⟩|001⟩ + b_1|1⟩|011⟩.
        ↓ U_odd
        a_2|0⟩|110⟩ + c_0|1⟩|000⟩ 
         + c_1|0⟩|000⟩ + b_2|1⟩|010⟩.
         ↓ U_even
        a_3|0⟩|111⟩ + c_30|1⟩|001⟩ 
         + c_10|0⟩|001⟩ + c_11|1⟩|011⟩ 
         + c_20|0⟩|001⟩ + c_21|1⟩|011⟩ 
         + c_31|0⟩|011⟩ + b_3|1⟩|101⟩.
where a, b, and c are the amplitudes.

 

 §.§ Experimental results in binary systems
    
    

  §.§.§ Comparative study between naive approach and our proposed approach
 
 
   We consider three metrics for the comparison, which are as follows:     
        
            
  * Circuit depth: It is an important property of a quantum circuit. The circuit depth of a quantum circuit is the measure of how many layers of quantum gates, executed in parallel, it takes to complete the total computation. The depth of a circuit roughly corresponds to the amount of time it takes the quantum computer to execute the circuit. Therefore, we always want to reduce the depth of our quantum circuits as much as possible.
            
  * Gate count: In Figure <ref> and Figure <ref>, we observe that multi-controlled Toffoli gates are used in abundance and for circuits with increasing number of qubits, this number of multi-controlled Toffoli's will increase substantially. However, these gates are not directly implementable on any quantum hardware. We need to decompose these gates into single and two-qubit gates before implementing them on real hardware. Gate count is the total number of gates after decomposing Toffoli gates. We also want to minimize the gate count of our circuits as much as possible. It is also possible to use intermediate qudits to decompose the Toffoli gates as suggested by the authors in <cit.> for ancilla-free decomposition, which is thoroughly discussed in the next section.
            
  * Probability of success: After the multi-controlled Toffoli gates are decomposed, certainly the depth and gate count of the circuit increase because we are replacing one gate with more than one other gate. We want a higher probability of success for our circuits. Small errors in the gates employed in quantum circuits are frequently present; these errors can be represented as an ideal gate followed by an unwanted Pauli operator. Instead of comparing the probability of small circuit errors, we compare the probability that the circuit will stay error-free (probability of success) in this comparison, maintaining the same level of generality as in <cit.>. The product that each component fails independently is the generalized formula for the probability of success (P_success) for any decomposition. The general equation for calculating success probability is,
            
    P_success=∏_gates((P_success of gate)^#gates× e^-depth/T_1)

            where P_success is the success probability of the decomposed circuit, P_success of gate is the success probability of each type of gate (single qubit/qudit or multi-qubit/qudit) as per the given hardware, depth is the depth of the circuit after decomposition and T_1 is the relaxation time of quantum hardware. For further experiments, we take the value of T_1 as per <cit.>.
        

The comparative resource estimation for the naive approach and our proposed approach is exhibited in Table <ref>.

            
 
    If we roughly estimate, the naive approach requires around 2n multi-controlled Toffoli gates for n qubits, whereas our proposed approach requires roughly half that amount. Not only is the gate count approximately halved compared to the naive approach, but the circuit depth is also roughly halved.
        
    
Compared to the naive approach, our proposed approach provides a better probability of success. The result is as Figure <ref>.

            

   
            
    

  §.§.§ Runs and Simulations

        We use IBM's Qiskit package and the BasicAer backend to simulate and validate our approach. We decompose the circuit into 1-qubit and 2-qubit gates. We use Qiskit's 'transpile' function for decomposition. For decomposition, the basic gate set is ['cx', 'u1', 'u2', 'h', 'x', 'p']. In this set, only 'cx'(CNOT) is the 2-qubit gate, others are 1-qubit gates. The optimization level is set to 1. We also use ancilla qubits to decompose multi-qubit gates for better circuit depth. The total number of shots considered is 1024.
        
        The results of DTQW using our proposed approach are as follows:
        
        

  
3-qubit quantum walks using our proposed approach
            We have implemented a 3-qubit quantum walk using our proposed approach. The circuits are shown in Figure <ref> and the corresponding position state mapping is as Table <ref>. The probability distribution after three steps of quantum walks is portrayed in Figure <ref>.

            
            
        

  
4-qubit Quantum Walk Using Our Proposed approach
            For a 4-qubit system, the circuits of the DTQW are shown in Figure <ref> and the corresponding position state mapping is as Table <ref>. The probability distribution after seven steps of quantum walks is portrayed in Figure <ref>.

            
            
            
    
            
            
            
            
            
            

            
        
   These results validate our proposed approach and exhibit efficiency as compared to the naive approach.    
   
   
       
    

 §.§ Runs and Simulations

        We have used IBM's Qiskit package to implement our approach. We have used the BasicAer backend to simulate our approach. We have decomposed the circuit into 1 and 2 qubit gates. We have used Qiskit's 'transpile' function for decomposition. For decomposition the basic gate set is ['cx', 'u1', 'u2', 'h', 'x', 'p']. In this set, only 'cx'(CNOT) is the 2 qubit gate others are 1 qubit gate. The optimization level is set to 1.
        
        The results of DTQW using our proposed methodology are as below:
        
        

  §.§.§ 3-qubit Quantum Walk Using Our Proposed Methodology

            We have implemented a 3-qubit quantum walk using our proposed methodology. The circuits are shown in Figure <ref> and the corresponding position state mapping is as Table <ref>.

            

            
            
            
        

  §.§.§ 4-qubit Quantum Walk Using Our Proposed Methodology

            For a 4-qubit system, the circuits of the DTQW are shown in Figure <ref> and the corresponding position state mapping is as Table <ref>.

            
            
            
    
            
        
           
    

 §.§ Comparative study between naive approach and our proposed methodology
 
        

  §.§.§ Metrics of comparison

        
            
  * Circuit depth: It is an important property of a quantum circuit. The circuit depth of a quantum circuit is the measure of how many layers of quantum gates, executed in parallel, it takes to complete the total computation. The depth of a circuit roughly corresponds to the amount of time it takes the quantum computer to execute the circuit. Therefore, we always want to reduce the depth of our quantum circuits as much as possible.
            
  * Gate count: In Figure <ref> and Figure <ref>, we see that multi-controlled Toffoli gates are used in abundance and for circuits with increasing number of qubits, this number of multi-controlled Toffoli's will increase substantially. However, these gates are not directly implementable on any quantum hardware. We need to decompose these gates into single and double qubit gates before implementing them on real hardware. Gate count is the total number of gates after decomposing Toffoli gates. We also want to minimize the gate count of our circuits as much as possible. It is also possible to use intermediate qudits to decompose the Toffoli gates as suggested by the authors in <cit.>.
            
  * Probability of success: After the multi-controlled Toffoli gates are decomposed, certainly the depth and gate count of the circuit increase because we are replacing one gate with more than one other gates. The probability of success is the probability value that denotes how much the working of the multi-controlled Toffoli is intact after the decomposition. We want a higher probability of success for our circuits. The general equation for calculating success probability is as, <cit.>
            
    P_success=∏_gates((P_success of gate)^#gates× e^-depth/T_1)

            where P_success is the success probability of the decomposed multi-controlled Toffoli, P_success of gate is the success probability of each type of gate (single qubit/qudit or multi-qubit/qudit), depth is the depth of the circuit after decomposition and T_1 is a constant.
        
        
        

  §.§.§ Gate count and circuit depth analysis

            If we consider roughly the naive approach takes 2n multi-controlled toffoli gate for n qubits, but our proposed methodology takes roughly half of the amount. The comparison is shown in Figure <ref>.
            
            Just as the gate count is half compared to the naive approach, circuit depth is also roughly half when compared to the naive approach. The comparison is shown in Figure <ref>
            

            

        

  §.§.§ Noise analysis

            Compared to the naive approach our proposed methodology gives a better probability of success. The result is as Figure <ref>.

            

        

  §.§.§ Resource Estimation

            Below is the comparative resource estimation table for the naive approach and our proposed approach in Table <ref>:

            

     
   
    

§ ENHANCED INCREMENT-DECREMENT APPROACH FOR DTQW IN HIGHER DIMENSIONAL QUANTUM SYSTEMS
 
        In this section, at first, we try to map our proposed binary method to higher dimensional quantum systems by considering LSB is in the binary system. Therefore, for n qudits with dimension d, we can have a total of 2d^(n-1) number of state spaces and we can implement a total of d^n-1 number of steps. Thus implementing our proposed methodology in higher dimensions is limiting us from leveraging the whole state space of higher dimensions because of the LSB which is set to d=2. 
        
        
            
        
        As an instance, the circuit for 3-qudit DTQW is shown in Figure <ref>. For the sake of simplicity, here we also take a Hadamard coin for each toss.
        
        
   

    
     

         
            As an instance, in a ququad system (Four-dimensional quantum systems), the circuit for the 3-ququad DTQW is as Figure <ref>, and the corresponding position state mapping is shown in Table <ref>. In the last step of Table <ref>, we get the states |200⟩ as the mid-state of the total state set in both directions,  which restricts the walker to use the rest of the 32 number of state spaces further.  We also check upto 8-dimensional quantum systems and observe that this direct mapping technique in higher dimensions can not be generalized for all n-qudit systems since it can not use all possible state spaces. Therefore, we need some changes in our proposed technique of binary systems while applying in higher dimensions so that it also utilizes all the state spaces in the higher dimensional quantum systems.
         
    
        


       
            To solve the state-space issue in higher dimensions, with a slight modification of the proposed binary-only approach, we propose a modified technique where we set the Least Significant Bit (LSB) as any even dimension when d ≥ 4. We propose modified circuits for both the even and odd dimensions when d > 2, which are portrayed in Figures <ref> and <ref> respectively. 
            
            
           
           
           
           In our modified circuit, as shown in Figures <ref> and <ref>, instead of applying only a NOT/Pauli-X gate on the LSB, a conditional increment or decrement gate is applied on the LSB for both the increment and decrement circuits so that the same mid-state of the total state set can be avoided in both directions of DTQW. With the modified logic as per Algorithm <ref>, when the walker reaches the middle point, it increments the value of LSB by 2 because of the increment circuit and decrements the value of LSB by 2 because of the decrement circuit. Hence, with this modified circuit, all possible states can be traversed using the even-dimensional circuit, which is d^n and the total number of steps that can be implemented is (d^n/2)-1. In our modified circuit, the LSB is not only dependent on control operations from other qudits for a conditional increment or decrement operation, but also it must increment the value of LSB at one step, and in the very next step, it decrements the value of LSB. For that reason, we need to restrict the LSB as an even dimension, but the other qudits can be in any dimension. Henceforth, the total number of states that can be traversed using the odd-dimensional circuit is d_even*d^n-1 and the total number of steps that can be implemented is ((d_even*d^n-1)/2)-1, where d_even represents the LSB.


            

        
           
           
           
           For better understanding, we take two separate example circuits, (i) an even dimension (3-ququad DTQW) circuit and (ii) an odd dimension (3-qutrit DTQW) circuit.

    
        

   

  
Implementation of 3-ququad (even dimension) DTQW with a Hadamard coin:
   As an example, we again take ququad systems, but this time with 3 ququads, including the LSB as ququad as per the modified circuit. Earlier, we observed in Table <ref>, at step 16, we get |200⟩ in both the directions of DTQW. As per the circuit in Figure <ref>, we get these states because we have used only binary (|0⟩ and |1⟩) for the LSB. Since the modified enhanced increment-decrement approach for DTQW can utilize |2⟩ and |3⟩ states in the LSB as shown in Figure <ref>, we get the following states in Table <ref>. In Table <ref>, We observe that at step 16, |x=+16⟩=|200⟩ becomes |x=+16⟩=|202⟩ by incrementing the value of LSB by 2 with the help of controlled increment operator as per modified circuit's logic. After this step, the previous process of DTQW again starts working and we get all possible state spaces as required.

        
            
        
       
    
       
    

  
Implementation of 3-qutrit (odd dimension) DTQW with a Hadamard coin:

    
        We have also shown an example of our modified enhanced increment decrement approach for DTQW in a 3-qutrit system (Figure <ref>) and we get the results shown in Table <ref> by considering LSB as in ququad system. If we had used only our enhanced increment decrement circuit, we could get up to step 9, where we have to stop the DTQW with |x=+9⟩=|121⟩ and |x=-9⟩=|121⟩. Because of our modified circuit, it can be well-avoided. The |x=-9⟩=|121⟩ now becomes |123⟩ since the conditional increment activates at the midpoint as per the logic to get access to the full state spaces. 
    
        
            
        
           
           
        
        
       
       
        
        
       
       
       
        
        
     


     
        
 
 
      
  
      
  
    
        Compared to the circuit before applying the improvement (Figure <ref>), after the improvement is applied (Figure <ref>), we can get twice as more states. The comparison is shown in Figure <ref>.

        
        
         




 §.§ Decomposition of Multi-controlled Toffoli Gates with Intermediate Qudit

    To achieve more improvement, we use multi-controlled Toffoli decomposition using intermediate qudits <cit.> in our proposed quantum walk circuit, where momentarily the binary system goes into higher dimensions, and using this concept, after the decomposition, gate count and circuit depth are reduced. No ancilla bits are required for this decomposition. In <cit.>, the authors have shown how we can efficiently decompose a multi-controlled Toffoli gate in the binary system by employing qutrits (3-ary quantum systems). In <cit.>, the authors have extended and generalized the previous work to decompose n-qubit Toffoli gates by considering both the intermediate-qutrit and the intermediate-ququad (4-ary quantum systems). For instance, in Figure <ref>, a 2-controlled Toffoli gate is decomposed into 2-qutrit gates. We have also portrayed 7-controlled Toffloi gate decomposition into 2-qutrit gates and 2-ququad gates in Figure <ref>. 
    
    
    
 
  
   


    
    
    For the sake of better understandability, we have divided the steps of decomposition into three steps - 
        
            
  * Preparation: In this step, the controls are prepared for the operation on the target. We must use only 2-qudit gates, and in the original circuit, we had multiple controls (more than 1). Therefore, based on the state of all those controls, we need to manipulate a single control in such a way that if all the other controls are set to |1⟩, then only the last single control will be set to |2⟩. However, in the process of setting a single control to |2⟩, the state of the other controls may get disturbed. Thus, in the final step, we need to correct those disturbances. We have highlighted this step with purple color.
            
  * Target Operation: Following the initial step, where a single control is set to |2⟩ if all other controls were |1⟩, this step involves simply applying a CNOT gate. The control for this gate is the single control that was set to |2⟩ in the previous step, while the target remains the original target. We have highlighted this step with green color.
            
  * Correction: As previously discussed, this final step aims to correct the states of the controls. We need to reverse the operation performed in the first step, ensuring that only the target changes if all the controls were set to |1⟩. We have highlighted this step with red color.
        
        Using this logic, we can decompose binary multi-controlled Toffoli gates of any size using intermediate qudits. This can be generalized for any finite d-dimensional quantum system as per <cit.> using two higher dimensions i.e., |d+1⟩ and |d+2⟩, which is thoroughly discussed in Appendix <ref>.

        


    

    
        
    We further observe that by using intermediate qudit decomposition, we have acutely reduced the gate count and circuit depth. For the N-controlled Toffoli gate, we get a circuit depth of logN <cit.>. By considering that we can formulate a generalized formulation for our proposed quantum walk circuit. Thus, the total circuit depth for a N qubit circuit is 2^N-1(∑_x=2^N-1 (logx) + 1) + 3*2^N-2 + 2^N-2.
        Similarly the total gate count for a N qubit circuit is 2^N-1(∑_x=2^N-1 (2x-3) + 1) + 2*2^N-2 + 4*2^N-2.   We have compared circuit depth among naive approach <cit.> with intermediate qudits, enhanced increment decrement with intermediate qudits, and increment decrement without using intermediate qudits. The comparative analysis is shown in Figure <ref>.  

        
    
       With these circuit costs, we have also compared the probability of success by considering Equation <ref> among the naive approach, enhanced increment decrement approach using intermediate qudits, and increment decrement approach without using intermediate qudits. The comparative analysis is shown in Figure <ref>. Using intermediate qudit decomposition for our approach increases the probability of success. We have also simulated and validated our proposed quantum walk circuits with intermediate qudits on the QuDiet platform <cit.>. 
        
     




§ CONCLUSION

This paper achieved a 50% reduction in circuit cost, including gate count and circuit depth, compared to the current increment-decrement method, through our proposed technique in qubit systems. The experimental results extended the significance beyond a few time steps, establishing a reliable foundation for quantum computer utilization. Furthermore, the engineering excellence of our proposed approach achieves the generalization for any finite-dimensional quantum system. Additionally, to ensure efficient implementation of quantum walks without ancillary requirements, we integrated an intermediate qudit technique for multi-qubit gate decomposition. This proposed approach can be easily employed for higher dimensional lattices as well. How this proposed circuit can enhance the applications of quantum walks <cit.> remains a prospect outlined in this paper. The code for our proposed implementation of DTQW is available in https://github.com/Biswayan375/Discrete-Time-Quantum-Walk/tree/enh-enc-dechttps://github.com/Biswayan375/Discrete-Time-Quantum-Walk/tree/enh-enc-dec. 













§ GENERALIZATION OF INTERMEDIATE QUDIT APPROACH


    The intermediate qudit approach is generalized and further optimized in <cit.> using two higher dimensions i.e., |d+1⟩ and |d+2⟩. First, we discuss how it is generalized and then how it is optimized.
        
            
  * Generalization: The decomposition circuit for a d-ary 3-controlled Toffoli gate using the approach is shown in Figure <ref>. In this case, we are required to use only one higher-level state, |d+1⟩. But to generalize the work of <cit.> for an n-controlled d-ary Toffoli gate, we need to use two higher-level states, |d+1⟩ and |d+2⟩, as in Figure <ref>.
                


    
                    
                    (a) 3-controlled d-ary Toffoli decomposition using intermediate qudits of dimension d and d+1. Here U_d is used instead of ⊕ or X_d to represent any unitary operation in general and not just Addition modulo d. (b) Decomposition of 7-controlled d-ary Toffoli gate using the proposed approach of <cit.>.
                    
                

            
  * Optimization: In Figure <ref>, we observe that  there are some redundant gates. Hence, an optimization has been used, which is proposed in <cit.>. We now need to remove these redundant gates for a more optimized decomposition of the 7-controlled d-ary Toffoli gate, which is shown in Figure <ref>. 
            



               
            
         
          
           
          
          
           
        







*





 Paper Title: Perfect Wave Transfer in Continuous Quantum Systems.







per.moosavi@fysik.su.se

Department of Physics, Stockholm University, 10691 Stockholm, Sweden



christandl@math.ku.dk





gmgraf@phys.ethz.ch

Institute for Theoretical Physics, ETH Zurich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland



spyros.sotiriadis@physics.uoc.gr









§ ABSTRACT 
We study the perfect transfer of information in 1+1D continuous quantum systems. This includes effective descriptions of inhomogeneous spin chains, for which the notion of perfect state transfer in quantum information was introduced, and here phrased in terms of waves. We show that reflection symmetry is necessary for perfect wave transfer (PWT) in any inhomogeneous conformal field theory, and even sufficient when restricted to one-particle excitations. To determine if or when it is sufficient more generally, we first break conformal invariance and study a broad class of 1+1D bosonic theories. We show that the question can then be posed as an inverse Sturm-Liouville problem that determines when the bosonic theory exhibits PWT. We demonstrate how to uniquely solve this problem, which also shows that reflection symmetry is sufficient for the special case with conformal invariance. Using bosonization, our continuum results extend these notions to interacting quantum field theories.




Perfect Wave Transfer in Continuous Quantum Systems

    Spyros Sotiriadis
    
August 1, 2024

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







Introduction.—
phantomsection
tocsectionIntroduction


Non-photonic quantum communication enables the efficient transmission of information from one quantum register to another <cit.>.
This is desirable for quantum networks and computation <cit.>, not least since it
allows for the perfect transfer of states even without dynamical control <cit.>.
First shown theoretically for communication channels in the form of a spin chain with spatially varying hopping, such perfect state transfer (PST) was extended to spins on graphs and other networks <cit.> and demonstrated experimentally <cit.>.
However, the notion has remained squarely in the discrete realm.

In this paper, we extend PST to continuous systems in one space and one time dimension (1+1D).
In so doing, a rephrasing in terms of waves becomes natural, motivating the term perfect wave transfer (PWT).
As for states in PST, a system on the interval [-L/2, L/2] of length L with suitable boundary conditions (BCs) exhibits PWT if the propagation of any initial wave, after some time T, amounts to its reflection; see Fig. <ref>.
Concretely: The system enjoys PWT if

    ⟨(x, 0) ⟩ = ⟨(-x, T) ⟩,

for all time-evolved observables (x, t) and any expectation value ⟨·⟩. 
We focus on realizations that do not need dynamical control, but the same problem is also interesting for driven systems <cit.>.

Related recent advances in the continuum include non-equilibrium studies of inhomogeneous versions of conformal field theory (CFT).
These are conformally-invariant quantum field theories deformed so that the usual constant propagation velocity becomes a profile v(x) <cit.>.
Among their applications, such inhomogeneous CFTs effectively describe gapless quantum many-body systems with mesoscopic inhomogeneities, such as spin chains with spatially varying hopping, trapped ultracold atoms, and arrays of quantum circuits; cf. <cit.>.
Special cases include the square-root profile v(x) = v √(1 - (2x/L)^2) for some constant v > 0 <cit.>—the continuum counterpart of the inhomogeneous spin chain in <cit.> that first exhibited PST—and profiles produced by Möbius transformations <cit.>.
Note that for PWT, as for other transport properties of finite systems, the BCs <cit.> are crucial.
Furthermore, inhomogeneous CFT can be viewed as a theory in curved spacetime.
As such, besides applications to quantum communication in solid-state devices <cit.>, our work also relates to relativistic quantum information <cit.>.
Since the same theories are also natural to study on the half or full infinite line, our results may likewise be relevant for long-distance quantum communication <cit.>.






Engineering PST/PWT is at  heart a spectral problem, amounting to whether the propagator coincides with the parity operator at certain times.
We address this engineering question for quantum field theories describing a continuous quantum communication channel.
Remarkably, for bosonic theories <cit.>, we arrive at an inverse problem for Sturm-Liouville operators, which we solve and whose solution determines if the channel enjoys PWT.
Using bosonization <cit.>, our results extend the notion of PST/PWT to a large class of systems, including interacting fermionic quantum field theories.


Inhomogeneous CFT.—
phantomsection
tocsectionInhomogeneous CFT


To begin with, we study the continuum limit of the inhomogeneous spin chains in <cit.>.
In the gapless regime, the continuous counterpart is an inhomogeneous CFT with a velocity profile v(x) > 0, x ∈ [-L/2, L/2] <cit.>.
The simplest example is that of inhomogeneous free fermions, given by the Hamiltonian (up to an overall constant and setting ħ = 1)

    H_iFF≡∫_-L/2^L/2 x v(x)/2[ ψ^†_+ (-) ∂_x ψ_+
      + ψ^†_-∂_x ψ_-
      + ]

with fields ψ_±(x) of right(+)/left(-) fermions satisfying {ψ_±(x), ψ^†_±(x') } = δ(x-x') and {ψ_±(x), ψ^(†)_∓(x') } = 0.
Here, ⋯ denotes Wick (normal) ordering with respect to the ground state |Ω⟩ (which for bilinears in the fields amounts to an additive renormalization).
For simplicity, we focus on the theory (<ref>), while general CFTs are discussed in the Supplementary Material (SM) <cit.>.

We assume open (Neumann) BCs at x = ± L/2.
These can be incorporated by standard unfolding methods; cf. <cit.>:
Couple right and left movers by identifying ψ_-(x) = ψ_+(L-x), giving a chiral theory of only right-moving fermions on [-L/2, 3L/2] with periodic BCs and velocity v(x) for x ∈ [-L/2, L/2] and v(L-x) for x ∈ [L/2, 3L/2].
This implies zero particle flow at x = ± L/2 since the two fermion densities ρ_±(x) = ψ^†_±(x)ψ_±(x) are equal at each boundary and the current is ∝ρ_+ - ρ_-.

PST was probed in <cit.> using a time-dependent overlap between one-particle states that are reflections of each other.
Its continuum counterpart is to study states created by fields

    ψ(x)
    ≡^ xψ_+(x) + ^- xψ_-(x),

where  denotes the Fermi momentum.
Eq. (<ref>) corresponds to the low-energy linearization of the spin-chain dispersion relation around its Fermi points ±, followed by recentering the resulting fields so they describe right/left-moving excitations around zero momentum.
As a preparatory subset of possible states, consider

    |Ψ_j⟩≡∫_-L/2^L/2 x ξ_j(x) ψ^†(x) |Ω⟩     (j = 1,2)

for smooth enough functions ξ_j(x), interpreted as distributions or “wave packets” of fermionic excitations in each state, and study the time-dependent overlap

    F(t)
    ≡⟨Ψ_2| ^- H_iFF t |Ψ_1⟩.

The states (<ref>) form a restricted class as only one-particle excitations are possible and as right/left movers have the same particle distributions; cf. the more general treatment in the SM <cit.> where we lift the latter restriction.

To probe PWT for one-particle excitations as done in <cit.>, we restrict to specular waves, ξ_2(x) = ξ_1(-x), and compare F(t) in Eq. (<ref>) with ⟨Ψ_1|Ψ_1⟩.
We say that:
There is one-particle PWT if there exists T > 0 so that

    |F(T)|
    = |⟨Ψ_1|Ψ_1⟩|

for all initial waves ξ_1(x) in Eq. (<ref>) [If Eq. (<ref>) holds for any state |Ψ_1⟩ in the Fock space, then general PST/PWT (i.e., for any number of excitations) would be enjoyed.].

Eq. (<ref>) amounts to a comparison between two-point correlation functions at equal and non-equal times.
Such can be computed in any inhomogeneous CFT by mapping them to the corresponding homogeneous correlations <cit.>.
The details are given in the SM <cit.> and boil down to comparing spacetime points after conformal transformations to coordinates given by

    y = y(x) ≡∫_0^x s v_0/v(s),
        1/v_0≡1/L∫_-L/2^L/2 s/v(s).

The outcome of the comparison shows that reflection symmetry is essential:


An inhomogeneous CFT exhibits one-particle PWT 

if and only if v(x) is even.


The corresponding times are odd multiples, (2m+1)T for any integer m, of the time T = L/v_0 that excitations take to cross the entire system's length.
For later usage, we note that y ∈ [-L/2, L/2] if v(x) is even.

As an example, consider the square-root profile v(x) = v √(1 - (2x/L)^2), which gave the effective low-energy theory of the spin chain in <cit.>.
This continuum theory exhibits PWT, just like for any even profile v(x), in contrast to <cit.> where that profile was exceptional in this respect.
Notably, the square-root profile is positive everywhere except at x = ± L/2.
Thus, strictly speaking, v(x) should be regularized before using the tools in <cit.>, but since the integral defining 1/v_0 in Eq. (<ref>) converges, it has no consequence for PWT. Oppositely, the limiting case of inhomogeneous CFT given by Möbius transformations so that v(x) decays to zero at the boundaries cannot exhibit PWT since 1/v_0 diverges.


Bosonic theories.—
phantomsection
tocsectionBosonic theories


The above result and observations should not come as a surprise.
After all, if one tries to compare discrete and continuous theories, then an inhomogeneous CFT can only describe low-energy excitations of a given lattice model.
Moreover, one can show <cit.> that any Hamiltonian (with pure point spectrum) must commute with the parity operator for PWT to be possible, i.e., v(x) must be even.
The general structure of the spectrum due to conformal invariance in 1+1D <cit.> opens up the possibility that it might also be sufficient; e.g., the example (<ref>) has a linear spectrum and is one-particle reducible, suggesting that if PWT holds for one-particle excitations then this should extend to many particles.
The remainder of this paper concerns to what extent this is true more generally.
Namely:
What are necessary and sufficient conditions for PWT (<ref>) for any number of excitations?

To answer this question, we first break conformal invariance. 
Examples include 1+1D fermions with non-local interactions <cit.>, while here we focus on bosons, specifically inhomogeneous Tomonaga-Luttinger liquids (TLLs) <cit.>.
This theory is formulated using a compactified bosonic field φ(x) (modulo 2π) with conjugate Π(x) satisfying
[∂_x φ(x), Π(x')] = ∂_x δ(x-x').
The Hamiltonian is (up to an overall constant)

    H_iTLL≡∫_-L/2^L/2 x/2π v(x)
      (
          [πΠ(x)]^2/K(x)
          + K(x)[∂_x φ(x)]^2
        )
       ,

where, besides v(x) > 0, the so-called Luttinger parameter K(x) > 0 also depends on x.
For simplicity, we assume that both functions are smooth.
When K(x) = K is constant, Eq. (<ref>) is an inhomogeneous CFT, and when K = 1, it is the bosonized version of the free fermions (<ref>) <cit.>.
However, K'(x) ≠ 0 locally couples right and left movers non-trivially, thus they do not commute and the theory is not conformal <cit.>.
Again, we impose open BCs, namely zero current at the boundaries:

    (x) |_x = ± L/2 = 0,
      where  (x)
    ≡
    - v(x)K(x)/π∂_x φ(x)

is the total particle current.
Unless stated otherwise, we also assume that K'(x)|_x = ± L/2 = 0 so that right and left movers are only coupled via the BCs (<ref>) at those points.


Sturm-Liouville (SL) theory.—
phantomsection
tocsectionSturm-Liouville (SL) theory


Following <cit.>, we use SL theory to find eigenfunctions, using which the fields can be expanded and the Hamiltonian (<ref>) diagonalized.
(This generalizes <cit.> as v(x), K(x) are unrelated and differs from the approach in <cit.>.)
Indeed, using integration by parts and Eq. (<ref>),

    H_iTLL
    =
    ∫_-L/2^L/2 x/2π w(x)
      (
          [ πΠ(x)/w(x)]^2
          + φ(x) φ(x)
        )

with the SL operator

    ≡
    - 1/w(x)[ ∂_x p(x) ∂_x + q(x) ],
 


    w(x)
    ≡K(x)/v(x),
      
    p(x)
    ≡
    v(x)K(x),
      
    q(x)
    ≡
    0,

where  acts on the space  of smooth [More precisely, twice continuously differentiable.] functions u(x) on [-L/2, L/2] that satisfy v(x)K(x) ∂_x u(x)|_x = ± L/2 = 0, i.e., Neumann BCs when v(x), K(x) > 0.

We want to solve the SL problem

    u = λ u,
      
    u ∈.

A distinction is warranted: Eq. (<ref>) is regular if certain technical conditions are satisfied [Here, these amount to v(x), K(x) being strictly positive and v(x), K(x), ∂_x p(x) [and q(x) if non-zero] continuous on the whole interval [-L/2, L/2].]; otherwise it is irregular.
Its importance stems from properties that hold for regular problems <cit.>:



  * 
The spectrum consists of eigenvalues λ_n that are real and ordered: 0 ≤λ_0 < λ_1 < …. 


  * 
Each λ_n has a unique square-integrable eigenfunction u_n(x) with n isolated zeros in [-L/2, L/2].


  * 
The u_n(x) form a complete orthonormal basis for square-integrable functions on [-L/2, L/2] with weight w(x):
∫_-L/2^L/2 x  w(x) u_n(x) u_n'(x)
= δ_n,n'.


Furthermore, assuming that the integral defining 1/v_0 in Eq. (<ref>) converges:




  * 
The λ_n can at most grow as n^2 for large n, i.e., λ_n = O(n^2).


The last follows by Weyl's law; see the SM <cit.> and, e.g., <cit.>.
Simply put, it gives a semi-classical estimate of the number of eigenvalues below a given large λ > 0 for Laplace-type operators.

Unless otherwise stated, our SL problem will be regular.
If irregular, the above properties are not guaranteed, while in certain cases they survive.
Note also that, since q(x) = 0 here, u_0(x) = const solves Eq. (<ref>) with λ_0 = 0.

The stated properties allow us to expand

    φ(x)
    = ∑_n = 0^∞φ_n u_n(x),
        Π(x)
    = ∑_n = 0^∞Π_n w(x) u_n(x),

whereby we obtain [ φ_n, Π_n'] = δ_n,n' and can diagonalize H_iTLL in Eq. (<ref>).
Indeed, the latter becomes

    H_iTLL
    = 1/2π[ πΠ_0]^2
      + ∑_n = 1^∞
        E_n a^†_n a_n

with eigenenergies

    E_n = √(λ_n),

where we expressed φ_n = √(π/2E_n)( a_n + a^†_n) and Π_n = -√(E_n/2π)( a_n - a^†_n) for n ≥ 1 in bosonic operators a^†_n, a_n with [ a_n, a^†_n'] = δ_n,n'.
Note that φ_0 (and zero modes in general) should be treated separately, but this will have no consequence here.
In the SM <cit.>, we also discuss SL operators (<ref>) with q(x) ≠ 0, corresponding to an inhomogeneous (possibly imaginary) mass M(x) by adding v_0^2 M(x)^2 φ(x)^2 inside the big parentheses in Eq. (<ref>).

The time evolution of any observable, such as φ(x), will contain terms like a^†_n^ E_n t u_n(x) and a_n^- E_n t u_n(x).
Thus, as shown in the SM <cit.>, to exhibit PWT as defined by Eq. (<ref>), all u_n(x) must have definite parity, i.e., be invariant under reflections x ↦ -x up to a sign, which must equal the value of each phase factor
^± E_n T for a common T > 0.
Moreover, the u_n(x) have definite parity if and only if v(x), K(x) [and q(x) if non-zero] are even, i.e., H_iTLL is reflection symmetric.
In conclusion, PWT boils down to requirements on the spectrum:


An inhomogeneous TLL exhibits PWT if and only if v(x) and K(x) are even and all eigenvalues are of the form E_n = π m_n/T for certain integers m_n.


From Properties <ref> and <ref> and since λ_0 = 0 here, these integers m_n satisfy 0 = m_0 < m_1 < … with m_n even (odd) if n is even (odd).
Moreover, from Property <ref>, they scale as m_n ∼ n for large n, i.e., m_n = n + o(n).

Lastly, to probe PWT based on Eq. (<ref>), we note that the results above allow one to compute correlation functions for inhomogeneous TLLs; see the SM <cit.>.
In particular, the ground-state φφ correlation function is

    ⟨Ω| φ(x,t)φ(x',t') |Ω⟩
    = ∑_n = 1^∞π/2 E_n
      u_n(x) u_n(x') ^- E_n(t-t').

Thus, for a spectrum satisfying the PWT requirements,
⟨Ω| φ(x,T)φ(x',0) |Ω⟩ = ⟨Ω| φ(-x,0)φ(x',0) |Ω⟩,
meaning that, for fixed x', the correlations at t = T are reflections of those at t = 0.
In Fig. <ref>, we plot Eq. (<ref>) for K(x) = 1, which, by bosonization, is related to log F(t) in Eq. (<ref>), in that it encodes the same information; see the SM <cit.>.


Inverse Sturm-Liouville problem.—
phantomsection
tocsectionInverse Sturm-Liouville problem


Reversing the logic, the PWT requirements on the spectrum yield an inverse spectral problem for SL operators with Neumann BCs
[and q(x) = 0]:


Given a spectrum E_n = π m_n/T for integers m_n as above, determine the even functions v(x) and K(x).


A related prominent problem is the inverse scattering occurring in the theory of integrable PDEs, whose solution is given by the Gelfand-Levitan-Marchenko equation.
However, that case is formulated on an infinite line, while here we are interested in a finite interval.

Before a general discussion, we consider two examples:

The inverse SL problem has a known solution when the spectrum is exactly linear:
For K(x) = K > 0 constant but v(x) > 0 general, the SL operator is = - v(x) ∂_x v(x) ∂_x = -v_0^2 ∂_y^2 using Eq. (<ref>) with eigenenergies E_n = π n v_0/L and eigenfunctions u_n > 0(x) = √(2 v_0/L)cos( nπ [L + 2y]/2L ) and u_0(x) = √(v_0/L).
This is precisely the conformal case of non-interacting right/left movers, enjoying PWT for any even v(x) > 0.

In the previous example, the SL operator was regular.
Relaxing this condition by taking v(x) = v √(1 - (2x/L)^2) and K(x) = K [ 1 - (2x/L)^2 ]^α for α > -1/2 and constants v, K > 0, we get an irregular problem for which Properties <ref>–<ref> remain valid; the BCs being that a solution is sufficiently well-behaved for Eq. (<ref>) to hold.
The eigenfunctions are given by Gegenbauer polynomials C^(α)_n(2x/L) <cit.> up to normalizations, which have definite parity and satisfy Eq. (<ref>) with λ_n = (π v_0/L)^2 n(n + 2α) for v_0 = 2v/π.
Clearly, only the case α = 0 (Chebyshev polynomials of the first kind) exhibits PWT, while all α≠ 0 do not, including Legendre polynomials (α = 1/2) and Chebyshev polynomials of the second kind (α = 1) <cit.>.
The respective φφ correlations are plotted in Fig. <ref>.

To solve the inverse SL problem in general, we write the operator (<ref>) in Liouville normal form, as is always possible for regular problems <cit.>: Let (y) ≡ u(x) √(K(x)) for y = y(x) given by Eq. (<ref>) and define 











    q̂(y)
    ≡
    - q(x)/w(x) + 1/√(K(x)) v_0^2 
     ∂_y^2 √(K(x)).

The eigenvalue problem (<ref>) then becomes

    v_0^2 ∂_y^2 (y) + [λ - q̂(y)] (y) = 0

with unchanged BCs
[For even v(x) > 0, they are v_0∂_y(y) - v(x) Λ(x) (y) = 0 at y = ± L/2 with Λ(x) ≡∂_xln√(K(x)) <cit.>, which imply Neumann BCs if K'(x)|_x = ± L/2 = 0.] and if reflection symmetric in x, then so in y.
Clearly, _n(y) have the same eigenvalues as the respective u_n(x) and similarly form a complete orthonormal basis but with identity weight.

Results in the literature often require two sets of eigenvalues for different BCs to get a unique solution to the inverse problem.
However, one set suffices for reflection-symmetric SL operators and certain BCs, including ours.
Indeed, <cit.> and <cit.> say that an even q̂(y) is uniquely determined by the spectrum for Neumann BCs.
Thus, viewing q̂(y) as given and since q(x) = 0, Eq. (<ref>) becomes a second-order differential equation for √(K(x(y))).
By general uniqueness results [assuming q̂(y) is continuous] and since K(x) must be even, this determines K(x) up to changing coordinates (<ref>) and an overall rescaling (discarding solutions that are not positive).

The results <cit.> do not reconstruct q̂(y).
However, since we know that K(x) = K gives an SL problem with m_n = n and q̂(y) = 0, a constant Luttinger parameter is the unique solution for the inverse problem given an exactly linear spectrum.
Beyond this special case, nothing a priori prevents the spectrum from deviating from the leading n behavior as long as the requirements on the m_n remain satisfied.
This would then give rise to a class of possible bosonic theories that exhibit PWT, where, by uniqueness, q̂(y) is non-zero, thus K(x) non-constant.
While not usually analytically tractable, there are constructive approaches to find q̂(y) <cit.>.


Discussion.—
phantomsection
tocsectionDiscussion


We introduced the notion of PWT and showed that it is exhibited by one-particle excitations in inhomogeneous CFT whenever the velocity profile v(x) > 0 is even.
Breaking conformal invariance, by considering inhomogeneous TLLs with a Luttinger-parameter profile K(x) > 0, we showed that PWT for general excitations amounts to an inverse spectral problem, which we solved.
This continuum result for bosons is remarkably similar to earlier results: 
For spin chains with nearest neighbor hopping, the corresponding class of operators can be represented as Jacobi matrices, and the resulting inverse eigenvalue problem was studied <cit.>.
As in the continuum, two sets of eigenvalues are generally needed for uniqueness <cit.>, while one set suffices for matrices symmetric with respect to the anti-diagonal (corresponding to reflection symmetry) <cit.>.

By bosonization, our results for inhomogeneous TLLs cover fermionic theories with position-dependent interactions <cit.>.
Indeed, even when K(x) = K is constant, as long as K ≠ 1, the theory describes an interacting quantum many-body system, albeit at low-energies.

Lastly, we showed that an inhomogeneous TLL with constant K(x) = K enjoys PWT for general excitations whenever v(x) > 0 is even.
Such a theory is an example of an inhomogeneous CFT.
It would be interesting to study PWT for any number of excitations in general CFTs, and also extend to systems on the half or full infinite line.




  
*Acknowledgments.



We are thankful to Pawel Caputa and Shinsei Ryu for motivating discussions on related topics.
P.M. gratefully acknowledges financial support from the Wenner-Gren Foundations (Grant No. FT2022-0002).
M.C. acknowledges financial support from the European Research Council (ERC Grant Agreement No. 818761), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the Novo Nordisk Foundation (Grant No. NNF20OC0059939 ‘Quantum for Life’), and thanks the NCCR SwissMAP of the Swiss National Science Foundation and the Section of Mathematics at the University of Geneva for their hospitality.












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Supplemental Material


phantomsection
tocpartSupplemental Material



This supplemental material has three parts.
Part A gives details for two-point correlations in inhomogeneous 1+1D CFT with open (Neumann) boundary conditions (BCs).
Part B describes how to diagonalize general inhomogeneous Tomonaga-Luttinger liquids (TLLs) using Sturm-Liouville (SL) theory, leading to a formulation of perfect wave transfer (PWT) as an inverse SL problem, including extension to massive TLLs.
Part C contains details for correlation functions in inhomogeneous TLLs.






§ PART A: CORRELATIONS IN INHOMOGENEOUS 1+1D CFT WITH OPEN BOUNDARIES





The general inhomogeneous CFT Hamiltonian is (setting ħ = 1)

    H_iCFT≡∫_-L/2^L/2 x  v(x) [ T_+(x) + T_-(x) ],

where T_±(x) are the right/left-moving components of the stress-energy tensor in light-cone coordinates, whose Fourier transforms yield two commuting copies of the Virasoro algebra.
In position space, they satisfy the equal-time commutation relations

    [ T_±(x), T_±(x') ]
    = ∓ 2∂_xδ(x-x') T_±(x')
      ±δ(x-x') ∂_x'T_±(x')
      ±c/24π∂_x^3δ(x-x')

and [ T_±(x), T_∓(x') ] = 0.
Note that, at non-equal time, T_± should be understood to only depend on the “right/left-moving coordinate” x^∓; for now, we only view these as labels, distinguishing dependencies on right/left components.

Besides the T_±(x) fields, an important class of operators are Virasoro primary fields Φ(x^-, x^+), defined by how they behave under conformal transformations.
The latter are given by orientation-preserving circle diffeomorphisms f_+, f_- and primary fields transform as

    Φ(x^-, x^+)
    →
    f'_+(x^-)^Δ_Φ^+ f'_-(x^+)^Δ_Φ^-Φ(f_+(x^-), f_-(x^+)),

where (Δ_Φ^+, Δ_Φ^-) denote the field's conformal weights.
To impose open BCs at x = ± L/2, as in the main text, we will assume that primary fields come in pairs Φ_+, Φ_- with conformal weights satisfying Δ^+_Φ_+ = Δ^-_Φ_-, Δ^-_Φ_+ = Δ^+_Φ_-, since this allows one to couple excitations so that right movers are reflected to left movers when reaching x = L/2, and vice-versa at x = -L/2.
To ease notation, we will use Δ^+_Φ≡Δ^+_Φ_+, Δ^-_Φ≡Δ^-_Φ_+ for each such pair.

Before continuing, we mention two examples:



  * 
For the inhomogeneous free fermions in the main text, the components of the stress-energy tensor are
T_±(x)
= (1/2) [
    ψ^†_±(x)
    (∓∂_x ) ψ_±(x)  + ]
  - π/ 12L^2,
and the primary fields of the theory are the fermionic fields ψ_±(x) themselves, whose conformal weights are
Δ^+_ψ_+ = Δ^-_ψ_- = 1/2, Δ^-_ψ_+ = Δ^+_ψ_- = 0.


  * 
For an inhomogeneous TLL with constant Luttinger parameter K(x) = K, one has
T_±(x)
= (1/4π K)
  [ πΠ(x) ∓ K ∂_x φ(x) ]^2
  - π/ 12L^2,
and the primary fields include vertex operators constructed as exponentials of the bosonic fields.
Applied to the Luttinger model <cit.>, these vertex operators are (renormalized) fermionic fields ψ_± with conformal weights Δ^+_ψ_+ = Δ^-_ψ_- = (1+K)^2/8K, Δ^-_ψ_+ = Δ^+_ψ_- = (1-K)^2/8K.
(Clearly, free fermions are recovered for K = 1.)



As in the main text, open BCs can be incorporated by standard unfolding methods; cf. <cit.>:
Couple right- and left-moving excitations by identifying

    T_-(x) = T_+(L-x),
        Φ_-(x^-, x^+) = Φ_+(L-x^+, L-x^-)

for all pairs Φ_+, Φ_- of Virasoro primary fields.
It follows that we can equivalently consider a chiral inhomogeneous CFT of only right movers on the interval [-L/2, 3L/2] of length 2L with velocity

    (x)
    ≡
      v(x)    for  x ∈ [-L/2, L/2], 
    
      v(L-x)    for  x ∈ [L/2, 3L/2]

and with every quantity satisfying periodic BCs.
Note that the Hamiltonian (<ref>) can then be written as

    H_iCFT≡∫_-L/2^3L/2 x (x) T_+(x).


For later reference, we define

    (x) 
    ≡∫_0^x s _0/(s),
        1/_0≡1/2L∫_-L/2^3L/2 s/(s),
        _t^±(x)
    ≡^-1( (x) ±_0t ).

Naturally,  lies in the universal cover _+(S^1_2L) of the group of orientation-preserving diffeomorphisms of the circle S^1_2L with circumference 2L. 
Given any v(x), one can show the following properties:

    (L-x) = (x),
        (L-x) = (L) - (x),
        ^-1( (L)-y ) = L - ^-1(y),
        _t^±(L - x) = L - _t^∓(x).

Lastly, we introduce the undeformed Hamiltonian

    H_0 = ∫_-L/2^3L/2 x _0 T_+(x),

whose ground state we denote by |Ω_0⟩.
Similarly, we let |Ω⟩ denote the ground state of H_iCFT.
Following <cit.>, the latter are related to their undeformed counterparts by

    U_+() H_iCFT U_+()^-1
    = H_0 + const,
        
    |Ω⟩
    = U_+()^-1 |Ω_0⟩,

where U_+() is a projective unitary representation of  on the Hilbert space of the CFT.

Similar to the main text, we define

    |Φ_j⟩≡∫_-L/2^L/2 x ( ^- k xξ^+_j(x) Φ_+^†(x,x) + ^ k xξ^-_j(x) 
     Φ_-^†(x,x) ) |Ω⟩     (j = 1,2)

for smooth enough functions ξ^±_j(x), interpreted as distributions or “wave packets” of right/left-moving excitations in each state, and some k ∈ℝ in order to accommodate for  in Eq. (<ref>).
(The latter will have no effect on PWT, and one might as well set k = 0.)
As in the main text, this is a restricted class of states which only allow for one-particle excitations, while we eased other restrictions in the sense that the excitations are not necessarily fermionic and the distributions of right/left movers can be different.

To probe PWT for one-particle excitations, we seek to compute

    ⟨Φ_j| Φ_j⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_j(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_j(x_2)Φ_-(x_2,x_2) ) 
    ×( ^- k x_1ξ^+_j(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_j(x_1) Φ_-^†(x_1,x_1) )
    |Ω⟩

and

    F(t)
    ≡⟨Φ_2| ^- H_iCFTt |Φ_1⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_2(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_2(x_2)Φ_-(x_2,x_2) )
      ^- H_iCFTt
    ×( ^- k x_1ξ^+_1(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_1(x_1) Φ_-^†(x_1,x_1) )
    |Ω⟩,

in order to eventually compare |F(t)| with |⟨Φ_1| Φ_1⟩| for specular waves, ξ^±_2(x) = ξ^∓_1(-x).
Using the unfolding relations (<ref>), we obtain

    ⟨Φ_j| Φ_j⟩
    = ∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ( ^ k x_2ξ^+_j(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_j(x_2)Φ_+(L-x_2,L-x_2) ) 
    ×( ^- k x_1ξ^+_j(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_j(x_1) Φ_+^†(L-x_1,L-x_1) )
    |Ω⟩

and

    F(t)
    = ^- Ct∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 ⟨Ω|
      ^ H_iCFTt( ^ k x_2ξ^+_2(x_2)Φ_+(x_2,x_2) + ^- k x_2ξ^-_2(x_2)Φ_+(L-x_2,L-x_2) )
      ^- H_iCFTt
    ×( ^- k x_1ξ^+_1(x_1) Φ_+^†(x_1,x_1) + ^ k x_1ξ^-_1(x_1) Φ_+^†(L-x_1,L-x_1) )
    |Ω⟩,

where we used that H_iCFT |Ω⟩ = C |Ω⟩ for some real constant C that encodes the Casimir energy.
(This only gives rise to an overall phase that one could safely ignore.)
Clearly, as stated in the main text, the above amounts to evaluating two-point correlations for primary fields, specifically those of right movers in a chiral CFT on [-L/2, 3L/2].

First, as a preliminary step, for primary fields with periodic boundary conditions on the system with length 2L, conformal invariance implies that the undeformed two-point correlations are

    ⟨Ω_0| Φ_+(x_2^-, x_2^+) Φ^†_+(x_1^-, x_1^+) |Ω_0⟩
    = 1/2π
      G^+_ΦΦ(x_2^- - x_1^-)^2Δ^+_Φ
      G^-_ΦΦ(x_2^+ - x_1^+)^2Δ^-_Φ,

where G^±_ΦΦ(x) are 2L-periodic functions.
For example, as can be seen from, e.g., <cit.>, the correlations for right-moving free fermions (cf. Example <ref> above) on the system with length 2L and periodic BCs are

    ⟨Ω_0| ψ_+(x_2) ψ^†_+(x_1) |Ω_0⟩
    = 1/2π G^+_FF(x_2 - x_1)

with

    G^+_FF(x)
    ≡πexp( -π/2L x )
    /
      2L sin( π/2L [x + 0^+] )
    ,

where we used that Δ^+_ψ = 1/2, Δ^-_ψ = 0.
Similarly, for the Luttinger model (cf. Example <ref> above), the correlations for the (renormalized) right-moving fermionic fields are

    ⟨Ω_0| ψ_+(x_2^-, x_2^+) ψ^†_+(x_1^-, x_1^+) |Ω_0⟩
    = 1/2π
      G^+_FF(x_2^- - x_1^-)^(1+K)^2/4K
      G^+_FF(x_1^+ - x_2^+)^(1-K)^2/4K,

where we used that

    G^-_FF(x)
    ≡
      -πexp( +π/2L x )
    /
      2L sin( π/2L [x - 0^+] )
    
    = πexp( - π/2L [-x] )
      /
        2L
        sin( π/2L [- x + 0^+] )
      
    = G^+_FF(-x).


Second, by results spelled out in Proposition 3.1 in <cit.> for inhomogeneous CFT, the time dependence in F(t) means that those correlations are evaluated at spacetime points given by ^±_t(x) in Eq. (<ref>).
In addition, the action of U_+() from the expression (<ref>) for |Ω⟩ also leads to a conformal transformation of the fields.
It follows from this and the properties (<ref>) that

    ⟨Φ_j| Φ_j⟩
    = 1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_j(x_2)ξ^+_j(x_1)
    G^+_ΦΦ((x_2) - (x_1))^2Δ^+_Φ
    G^-_ΦΦ((x_2) - (x_1))^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_j(x_2)ξ^-_j(x_1)
    G^+_ΦΦ((x_2) + (x_1) - (L))^2Δ^+_Φ
    G^-_ΦΦ((x_2) + (x_1) - (L))^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_j(x_2)ξ^+_j(x_1)
    G^+_ΦΦ( - (x_2) - (x_1) + (L))^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) - (x_1) + (L))^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_j(x_2)ξ^-_j(x_1)
    G^+_ΦΦ( - (x_2) + (x_1))^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) + (x_1))^2Δ^-_Φ)

and

    F(t)
    = ^- Ct1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_2(x_2)ξ^+_1(x_1) 
    G^+_ΦΦ((x_2) - (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((x_2) - (x_1) + _0t)^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_2(x_2)ξ^-_1(x_1) 
    G^+_ΦΦ((x_2) + (x_1) - (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((x_2) + (x_1) - (L) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_2(x_2)ξ^+_1(x_1) 
    G^+_ΦΦ( - (x_2) - (x_1) + (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) - (x_1) + (L) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_2(x_2)ξ^-_1(x_1) 
    G^+_ΦΦ( - (x_2) + (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (x_2) + (x_1) + _0t)^2Δ^-_Φ),

where we used that
(_t^±(x) )
= (x) ±_0t
and
(_t^±(L - x) )
= (L) - (_t^∓(x))
= (L) - (x) ±_0t.

Finally, for specular waves, ξ^±_2(x) = ξ^∓_1(-x), we obtain

    F(t)
    = ^- Ct1/2π∫_-L/2^L/2 x_1 ∫_-L/2^L/2 x_2 '(-x_2)^Δ^+_Φ'(x_1)^Δ^+_Φ'(-x_2)^Δ^-_Φ'(x_1)^Δ^-_Φ
    ×(
       ^ k (x_2-x_1)ξ^+_1(x_2)ξ^+_1(x_1)
    G^+_ΦΦ( - (-x_2) - (x_1) + (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (-x_2) - (x_1) + (L) + _0t)^2Δ^-_Φ
        +
    ^ k (x_2+x_1)ξ^+_1(x_2)ξ^-_1(x_1)
    G^+_ΦΦ( - (-x_2) + (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ( - (-x_2) + (x_1) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2+x_1)ξ^-_1(x_2)ξ^+_1(x_1)
    G^+_ΦΦ((-x_2) - (x_1) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((-x_2) - (x_1) + _0t)^2Δ^-_Φ
        +
    ^- k (x_2-x_1)ξ^-_1(x_2)ξ^-_1(x_1)
    G^+_ΦΦ((-x_2) + (x_1) - (L) - _0t)^2Δ^+_Φ
    G^-_ΦΦ((-x_2) + (x_1) - (L) + _0t)^2Δ^-_Φ),

after a change of variable and rearranging.

By comparing the terms appearing in Eqs. (<ref>) and (<ref>), it follows that |F(t)| for certain t equals |⟨Φ_1| Φ_1⟩| for all ξ^±_1(x) if and only if (x) is even, meaning (x) must be odd.
This in turn implies (L) = L and _0 = v_0 given by Eq. (<ref>), which means that equality holds for all t = (2m+1)L/v_0 due the 2L periodicity of G^±_ΦΦ(x).
[Moreover, for ξ^±_2(x) = ξ^∓_1(-x), one has |⟨Φ_1| Φ_1⟩| = |⟨Φ_2| Φ_2⟩| when v(x) is even.]
Since (x) = v(x) for all x ∈ [-L/2, L/2], this proves the claim that one-particle PWT is exhibited for all even inhomogeneous CFTs.

As a last remark, one can unfold in a different way to take the spin/statistics of the primary fields into account, so that, e.g., one gets anti-periodic BCs for fermions
on the system of size 2L, and twisted BCs for general anyonic fields.
This can be used to obtain simple explicit formulas for the two-point correlations of any primary fields.
However, since we are interested in recombinations of waves traveling in different directions, it is necessary to identify so as to always have periodic BCs.
In any case, for our purposes, all we needed was the spacetime dependence of the correlations, which are fixed on general grounds in CFT, circumventing any need for explicit formulas.






§ PART B: SPECTRAL PROPERTIES OF INHOMOGENEOUS TLLS





In this part, we first show that reflection symmetry is a necessary condition for PWT for any theory; here under the assumption that the Hamiltonian has pure point spectrum.
Second, we give the details for how to use SL theory to diagonalize the inhomogeneous TLL Hamiltonian (<ref>) and to formulate the PWT requirements on the spectrum.
Third, we review some details on Weyl's law applied to SL operators and, fourth, extend the previous discussion to TLL Hamiltonians with a mass term.
Fifth, we comment on the Liouville normal form used to state the results for the inverse SL problem in the main text.






 §.§ Necessity of reflection symmetry





Before considering TLL theory, we note that all models studied in the main text have pure point spectrum [as long as the integral defining 1/v_0 in Eq. (<ref>) converges].
Thus, we first consider a general Hamiltonian H with pure point spectrum and show that for such a system to exhibit PST/PWT, it is necessary that H commutes with the parity operator , which maps x ↦ -x.
In other words, we will show that reflection symmetry is a necessary condition.

To prove the statement, let E_n denote the eigenenergies of H with the corresponding energy eigenstates |n⟩.
On general grounds, the latter form a complete orthonormal basis for the Hilbert space  of the theory.
Let |Φ_1⟩ be an arbitrary state in  and let |Φ_2⟩≡ |Φ_1⟩.
(We recall that  is a unitary operator satisfying ^2 = 1, meaning that ^-1 = ^† =.)
As usual, define F(t) ≡⟨Φ_2| ^- H t |Φ_1⟩ =  ⟨Φ_1| ^- H t |Φ_1⟩.
For general PST/PWT, we require that there exists T > 0 such that |F(T)| = |⟨Φ_1|Φ_1⟩| for any state |Φ_1⟩.
In particular, this must hold for any energy eigenstate |n⟩, implying |⟨ n||n⟩| = 1.
If |n⟩| is not an eigenstate of , it is straightforward to show that this equality cannot hold.
Consequently, all energy eigenstates must also be eigenstates of , which implies that [, H] = 0.






 §.§ Inhomogeneous TLLs and SL theory





We now turn to inhomogeneous TLLs studied in the main text.
Inserting the expansions (<ref>) into Eq. (<ref>) yields

    H_iTLL
    = 1/2π [πΠ_0]^2
      + 1/2π∑_n = 1^∞( [πΠ_n]^2 + λ_n φ_n^2 )^2 ,

where φ_0 is absent (when E_0 = 0), no different from the standard case with constant K(x) [and v(x)].
For n ≥ 1, let

    φ_n
    = √(π/2)1/λ_n^1/4( a_n + a^†_n),
        Π_n
    = -λ_n^1/4/√(2π)( a_n - a^†_n)

in terms of bosonic creation and annihilation operators a^†_n and a_n, satisfying [ a_n, a^†_n'] = δ_n,n', as in the main text.
Inserting Eq. (<ref>) into Eq. (<ref>) yields Eqs. (<ref>)–(<ref>).

Using the above, one can compute the time evolution of any observable in the algebra of operators generated by the fields φ(x) and Π(x).
For instance, for φ(x, t) ≡^ H_iTLL tφ(x) ^- H_iTLL t and Π(x, t) defined similarly, we obtain

    φ(x, t)
    = ∑_n = 1^∞√(π/2E_n)( a_n^- E_nt + a^†_n^ E_nt)
      u_n(x),
        Π(x, t)
    = ∑_n = 1^∞ (-) √(E_n/2π)( a_n^- E_nt - a^†_n^ E_nt)
      w(x) u_n(x),

where we omitted the zero modes (as they have to be treated separately when E_0 = 0) and used ^ H_iTLL t a_n^- H_iTLL t = a_n^- E_nt and ^ H_iTLL t a^†_n^- H_iTLL t = a^†_n^ E_nt.
Generally, any observable (x, t) in this algebra will contain contributions of the form a^†_n^ E_n t u_n(x) and a_n^- E_n t u_n(x).
Now, in order for PWT to be exhibited, Eq. (<ref>) must hold for any state, which means that it holds true without expectation values, i.e., (x, 0) must equal (-x, T) at some later time T > 0 as an operator identity.
Taking = φ and using the linear independence of the generators a_n and a^†_n, Eq. (<ref>) yields

    u_n(x) = u_n(-x) ^± E_n T  ∀ n = 0, 1, …

for a common T > 0, where we included the trivial case n = 0 [recall that u_0(x) = const and E_0 = 0 when q(x) = 0 in Eq. (<ref>)].
Since u_n(x) = u_n(-x), Eq. (<ref>) implies that each u_n(x) must be an eigenfunction of the parity operator , whose eigenvalues are +1 or -1 (since ^2 = 1).
It follows that all eigenfunctions must have definite parity, and the only possibility for PWT to hold is that the corresponding eigenenergies satisfy E_n T = π m_n for integers m_n for each n, with m_n even (odd) if u_n(x) has even (odd) parity.

Finally, it is straightforward to show the following statement:
The eigenfunctions u_n(x) of a regular SL operator  have definite parity if and only if [, ] = 0.
For our operator in Eqs. (<ref>)–(<ref>), this means that both v(x) and K(x) must be even functions (using that they must be positive to exclude the possibility that both are instead odd).

Given the above, together with Properties <ref>–<ref> for regular SL operators stated in the main text, we have derived our main result on the spectral requirements for an inhomogeneous TLL to exhibit PWT.

Moreover, as stated in the main text, reversing the logic, namely, given a spectrum that satisfy the spectral properties for PWT to be exhibited, our formulation of PWT for inhomogeneous TLLs as an inverse SL problem follows.






 §.§ Weyl's law for SL operators





Weyl's law applied to SL operators, see, e.g., <cit.>, states that the number N(λ) of eigenvalues below a given λ > 0 asymptotically behaves as

    N(λ)/√(λ)∼1/π∫_-L/2^L/2 x √(w(x)/p(x))

for large λ.
This holds under certain technical conditions on the coefficients w(x), p(x) in Eq. (<ref>) [which are satisfied assuming our v(x), K(x) are positive and smooth functions].
Using Eqs. (<ref>) and (<ref>) as well as Property <ref> in the main text, it follows that

    n/E_n∼1/π∫_-L/2^L/2 x/v(x)
    = L/π v_0

for large n, where in the last step we used Eq. (<ref>).
Thus, assuming that the integral defining 1/v_0 converges, then this gives the stated asymptotic behavior, E_n = π v_0 n/L + o(n), in the main text.
(Another possibility would be if the integral diverges, in which case E_n∼const× n^γ for γ < 1, but this is inconsistent with that the eigenenergies must be proportional to integers m_n with a common proportionality constant in order for the theory to exhibit PWT.)






 §.§ “Massive” inhomogeneous TLLs





For massless TLL bosons, as discussed above and in the main text, given a spectrum satisfying the PWT requirements, we saw that K(x) can be found uniquely up to rescaling and changing coordinates.

We now extend to SL operators (<ref>) with q(x) ≠ 0, corresponding to an inhomogeneous (possibly imaginary) mass M(x) = √(-q(x)/v_0^2 v(x)) by adding v_0^2 M(x)^2 φ(x)^2 inside the big parentheses in Eq. (<ref>):

    H_iTLL≡∫_-L/2^L/2 x/2π v(x)
      (
          [πΠ(x)]^2/K(x)
          + K(x)[∂_x φ(x)]^2
          + v_0^2 M(x)^2 φ(x)^2
        )
       ,

Indeed, it is straightforward to extend the previous discussion to such models.
For instance, the PWT requirements on the spectrum are the same, except the spectrum does not start at zero:
For q(x) ≠ 0, a constant u_0(x) no longer solves Eq. (<ref>), thus E_0 > 0.
Instead, we have a “massive” inhomogeneous TLL, for which one can show:


A “massive” inhomogeneous TLL exhibits PWT if and only if v(x), K(x), q(x) are even
 and all eigenvalues are of the form E_n = πm̃_n/T for certain integers m̃_n.


Here, these integers m̃_n = m_n + in terms of the integers m_n for the massless case in the main text, where we allow for an overall shift by a finite even integer .
[The latter must be even so that m̃_n is even (odd) if n is even (odd), as required due to Property <ref> in the main text.]
Moreover, given such a spectrum, the results in <cit.> still imply that q̂(y) is uniquely determined by the spectrum, but this can no longer uniquely determine K(x) since the first term in Eq. (<ref>) is now non-zero in general.

As an example, we revisit the case with v(x) = v √(1 - (2x/L)^2) and K(x) = K [ 1 - (2x/L)^2 ]^α for α > -1/2 and constants v, K > 0 encountered in the main text, whose eigenfunctions were given by Gegenbauer polynomials C^(α)_n(2x/L), but now we include a fine-tuned potential q(x) = - (2vα/L)^2 w(x), where we recall that w(x) = K(x)/v(x).
This fine-tuned potential allows us to complete the square so that the eigenvalues in Eq. (<ref>) are λ_n = (π v_0/L) (n + α)^2  with v_0 = 2v/π, meaning that E_n = π (n + α) v_0/L.
Consequently, any c = α = 0, 2, 4, … works, yielding a discrete family of models with fine-tuned inhomogeneous mass that exhibit PWT.
We stress that the mass in this example satisfies M(x) ∝ 1/L and therefore vanishes in the thermodynamic limit.
Thus, one should not expect hallmarks of massive theories, such as exponential decay of two-point correlations with respect to the spatial separation.






 §.§ Liouville normal form and compactified free bosons





It is curious to note the following “shortcut” to writing our massless bosons (<ref>) in Liouville normal form:
Recall that an inhomogeneous TLL is a theory of compactified free bosons with a position-dependent compactification radius R(x) in curved spacetime given by the metric (h_μν) = (v(x)^2/v_0^2, -1) for coordinates (x^0, x^1) ≡ (v_0t, x); see, e.g., <cit.>.
Given a Luttinger parameter profile K(x), the compactification radius is R(x) = √(2α') K(x) (α' has dimension length squared and is commonly set equal to 2).
In terms of the bosonic field φ(x) (with values taken modulo 2π), the action corresponding to the Hamiltonian (<ref>) is

    S
    = 1/4πα'∫^2 x √(-h)
      R(x)^2 (∂_μφ) (∂^μφ),

where we recall that h ≡(h_μν).
This can be viewed as a non-exactly marginal deformation of a theory with constant compactification radius; cf. <cit.>.
The corresponding action for X(x) ≡ R(x) φ(x) becomes

    S
    = 1/4πα'∫^2 x √(-h)[ (∂_μ X) (∂^μ X) - q̂ X^2/v(x)^2 ]

with q̂ = q̂(y) in Eq. (<ref>) for y = y(x) given by Eq. (<ref>), which agrees with Eq. (<ref>) for (y) = X(x)/√(2α').
I.e., as a theory for X(x), even when q(x) = 0, the position-dependent K(x) corresponds a (possibly imaginary) inhomogeneous mass √(q̂(y))/v(x).
This also extends to the case

    S 
    = 1/2π∫^2 x √(-h)[ K(x) (∂_μφ) (∂^μφ) - v_0^2 M(x)^2 φ^2 ]

with a non-zero mass M(x) = √(-q(x)/v_0^2 v(x)) for the bosonic field φ(x), corresponding to the Hamiltonian (<ref>), the only difference being that the first term in Eq. (<ref>) is then no longer zero.






§ PART C: CORRELATION FUNCTIONS IN INHOMOGENEOUS TLLS





Consider the inhomogeneous TLL theory with Hamiltonian H_iTLL in Eq. (<ref>).
Using the eigenfunctions obtained from SL theory, we can compute any ground-state two-point correlation functions for the fields φ(x) and ϑ(x), where the latter is defined by

    ∂_x ϑ(x)
    ≡πΠ(x).

Indeed, using the expansions (<ref>), that a_n annihilates the ground state |Ω⟩ of H_iTLL, as well as the commutation relations [ a_n, a^†_n'] = δ_n,n', it is straightforward to show that

    ⟨Ω| φ(x,t) φ(x',t') |Ω⟩    = ∑_n = 1^∞π/2E_n u_n(x)u_n(x') ^- E_n (t-t'), 
    ⟨Ω| φ(x,t) ϑ(x',t') |Ω⟩    = ∑_n = 1^∞π/2 u_n(x)U_n(x') ^- E_n (t-t'), 
    ⟨Ω| ϑ(x,t) ϑ(x',t') |Ω⟩    = ∑_n = 1^∞π E_n/2 U_n(x)U_n(x') ^- E_n (t-t'),

where we introduced U_n(x) defined by

    ∂_x U_n(x)
    ≡
    w(x) u_n(x).

Similar correlation functions can be computed for other objects, including vertex operators (i.e., normal-ordered exponentials of the bosonic fields) <cit.>.
It is also straightforward to compute Green's functions; see, e.g., <cit.>, the only difference with the present case being that one must account for the non-trivial weight function w(x) that was simply a constant in <cit.>.

As an important example, consider a theory with K(x) = K > 0 constant but v(x) > 0 general [and q(x) = 0].
We recall that the SL operator in this case is = - v(x) ∂_x v(x) ∂_x = -v_0^2 ∂_y^2 using Eq. (<ref>) with eigenenergies E_n = π n v_0/L and eigenfunctions u_n > 0(x) = √(2 v_0/L)cos( nπ [L + 2y]/2L ) and u_0(x) = √(v_0/L).
Then, one can show that

    ⟨Ω| φ(x,t) φ(x',t') |Ω⟩
    
    = 1/4   (
      logexp( -π/2L [f(x)-f(x')-v_0(t-t')] )
      /
        2 sin( π/2L [f(x)-f(x')-v_0(t-t')+0^+] )
      
    + logexp( -π/2L [f(x)+f(x')+L-v_0(t-t')] )
      /
        2 sin( π/2L [f(x)+f(x')+L-v_0(t-t')+0^+] )
      . 
          + .
      log
        -exp( π/2L [f(x)+f(x')+L+v_0(t-t')] )
      /
        2 sin( π/2L [f(x)+f(x')+L+v_0(t-t')-0^+] )
      
    + log
        -exp( π/2L [f(x)-f(x')+v_0(t-t')] )
      /
        2 sin( π/2L [f(x)-f(x')+v_0(t-t')-0^+] )
      ),

where we included the necessary regularizations in order to interpret coincident spacetime points and used the identity ∑_n = 1^∞1/n^-ξ n = - log( 1 - ^-ξ) for any ξ such that ξ > 0.
Eq. (<ref>) is the analytical expression for what was plotted in Fig. <ref> (assuming K = 1).
Moreover, for delta-like waves, the terms appearing in log F(t) given by Eq. (<ref>) for the free-fermion case (<ref>) closely resemble those in Eq. (<ref>).
Indeed, the former agrees with 4 ⟨Ω| φ(x,t) φ(x',t') |Ω⟩ to leading order, and one can furthermore show that the agreement between the two ways of computing (CFT methods versus bosonization) is exact (as it should be) by evaluating and summing all combinations of two-point correlation functions of the vertex operators describing the right/left-moving free fermionic fields.





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