Search and state transfer between hubs by quantum walks
Stanislav Skoupy Martin Stefanak


]Search and state transfer between hubs by quantum walks


Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1 - Staré Město, Czech Republic




§ ABSTRACT 
Search and state transfer between hubs, i.e. fully connected vertices, on otherwise arbitrary connected graph is investigated. Motivated by a recent result of Razzoli et al. (J. Phys. A: Math. Theor. 55,
265303 (2022)) on universality of hubs in continuous-time quantum walks and spatial search, we extend the investigation to state transfer and also to the discrete-time case. We show that the continuous-time quantum walk allows for perfect state transfer between multiple hubs if the numbers of senders and receivers are close. Turning to the discrete-time case, we show that the search for hubs is successful provided that the initial state is locally modified to account for a degree of each individual vertex. Concerning state transfer using discrete-time quantum walk, it is shown that between a single sender and a single receiver one can transfer two orthogonal states in the same run-time. Hence, it is possible to transfer an arbitrary quantum  state of a qubit between two hubs. In addition, if the sender and the receiver know each other location, another linearly independent state can be transferred, allowing for exchange of a qutrit state. Finally, we consider the case of transfer between multiple senders and receivers. In this case we cannot transfer specific quantum states. Nevertheless, quantum walker can be transferred with high probability in two regimes - either when there is a similar number of senders and receivers, which is the same as for the continuous-time quantum walk, or when the number of receivers is considerably larger than the number of senders. Our investigation is based on dimensional reduction utilizing the invariant subspaces of the respective evolutions and the fact that for the appropriate choice of the loop weights the problem can be reduced to the complete graph with loops.  


[
    S. Skoupý and M. Štefaňák
    September 9, 2024
=============================





§ INTRODUCTION


One of the most promising applications of quantum walks <cit.> is the spatial search
<cit.> which can be seen as an extension of the Grover's algorithm <cit.> for search of an unsorted database on a quantum computer. Quantum walk search was shown to provide quadratic speed-up over classical search on various graphs and lattices <cit.>. In fact, search utilizing continuous-time quantum walk for a single marked vertex was shown to be optimal for almost all graphs <cit.>. More recently, quadratic speed-up over a classical random walk search for any number of marked vertices was achieved in both discrete-time <cit.> and continuous-time <cit.> quantum walk algorithms. 

Closely related problem to spatial search is state transfer <cit.> which is the basic tasks of any quantum communication network <cit.>. Apart from direct exchange of quantum information between nodes of a quantum network it can be utilized for distributed quantum computing <cit.> or more generally in quantum internet <cit.>. While in search we evolve the system from the equal weight superposition to a localized state on the marked vertex, in state transfer we aim to evolve from a localized state from the sender to the receiver vertex. One possibility to achieve state transfer is to design the dynamics on the whole graph accordingly. For the continuous-time evolution, where the dynamics is governed by the Schroedinger equation with a given Hamiltonian, the problem was over the years investigated on various types of graphs <cit.>. Discrete-time version was also studied in detail <cit.>. Second possibility to achieve state transfer is to employ spatial search, thus altering the dynamics only locally at the sender and the receiver vertex <cit.>. This approach was extensively investigated especially in the discrete-time quantum walk model <cit.>.

In a recent paper <cit.> the authors have investigated continuous-time quantum walk on an arbitrary graph with fully connected vertices, which we refer to as hubs for short. The Hamiltonian of the studied walk was given by the graph Laplacian, with additional potential on marked hubs. It was shown that such quantum walks have certain universal behaviour when the probability amplitudes at the hubs are concerned, guaranteed by the existence of invariant subspaces <cit.> which are independent of the rest of the graph. Applications to spatial search and quantum transport for single and multiple hubs were discussed, showing that the walk behaves the same as if the graph would be complete.

Our paper elaborates on the ideas of <cit.> and extends the investigation to state transfer between multiple hubs and also to the discrete-time scenario. Utilizing the effective Hamiltonian of the continuous-time quantum walk with marked hubs derived in <cit.>, we show that state transfer from S senders to R receivers can be achieved with high fidelity provided that S≈ R. Moving on to the discrete-time case, we first investigate a spatial search for M marked hubs. The dynamics is given by a coined walk with a flip-flop shift. As the quantum coin we consider a modified Grover coin with a weighted loop at each vertex <cit.>. The hubs which are a solution to the search problem are marked by an additional phase shift of π. We show that there is a five-dimensional invariant subspace of the walk provided that the initial state of the search is locally modified such that it has equal projection onto every vertex subspace. In this case the walk evolves as search on a complete graph with loops, which is known to be optimal. We provide a detailed investigation how do the terms contributing to the total success probability change with the number of solutions M. Next, we proceed to the state transfer between two marked hubs. Properly choosing the weights of the loops of the local coins turns the problem of state transfer between two hubs on an otherwise arbitrary graph to the state transfer on a complete graph with loops, which we have investigated earlier <cit.>. We expand the known results and show that it is possible to transfer multiple orthogonal states. We find that there is a 9-dimensional invariant subspace, which can be further split into symmetric and antisymmetric subspaces with respect to the exchange of the sender and the receiver vertices. Time evolution in the invariant subspace is investigated in detail for two orthogonal initial states, namely loop and the equal weight superposition of all outgoing arcs at the sender vertex. It is shown that in both cases the discrete-time quantum walk achieves transfer to the corresponding state at the receiver vertex in the same run-time. Hence, we can transfer an arbitrary quantum state of a qubit between the hubs. Since our analysis utilizes an approximation of effective evolution operator eigenstates for large graphs, we also numerically investigate the fidelity of qubit state transfer in dependence on the number of vertices N. It is shown that the fidelity behaves as 1-O(1/N). In addition, we show that if the sender and the receiver know each other's location, a third linearly independent state can be transferred in the same run-time, namely the arc on the edge connecting the sender and the receiver. Hence, in this case the sender and the receiver can exchange a qutrit state. Finally, we extend the study to state transfer from S sender hubs to R receiver hubs. For S≠ R the exchange symmetry between senders and receivers is broken and the splitting of the invariant subspace (which is now 11-dimensional) into symmetric and anti-symmetric subspaces is no longer possible. Nevertheless, the system can be investigated analytically. We show that while we can't send a qubit state, transfer of the walker from senders to receivers occurs with high probability in two regimes - either when S≈ R as for the continuous time case, or when the number of receivers is considerably larger than the number of senders.

The rest of the paper is organized as follows: in Section <ref> we review the results of <cit.> for the continuous-time quantum walk and extend the investigation to state transfer between multiple hubs. Section <ref> introduces the discrete-time quantum walk and investigates search for M marked hubs. Section <ref> is dedicated to the state transfer between two hubs by discrete-time quantum walk. In Section <ref> we investigate state transfer between multiple hubs. We conclude and present an outlook in Section <ref>. Technical details concerning the form of the individual evolution operators in the invariant subspaces and their eigenstates are left for Appendices.



§ STATE TRANSFER BETWEEN HUBS BY CONTINUOUS-TIME QUANTUM WALK




We begin with the continuous-time quantum walk. Let us have a graph G=(V,E), where V denotes the set of vertices and E denotes the set of edges. We consider simple graphs, i.e. the are no loops and no multiple edges in E. For a graphs with N=|V| vertices the Hilbert space of the continuous-time walk is an N-dimensional complex space spanned by vectors |v⟩, v∈ V, corresponding to a particle being localized at the vertex v. The authors of <cit.> consider M vertices w∈ W to be marked hubs, i.e. fully connected with degrees d_w = N-1. The Hamiltonian of the walk is taken as the weighted Laplacian with additional potential on the marked vertices

    Ĥ = γL̂ + ∑_w∈ Wλ_w |w⟩⟨w|.

Note that the Laplacian of a given graph G is determined by its adjacency matrix Â and the diagonal degree matrix D̂ as

    L̂ = D̂ - Â.

The results of <cit.> show that as far as the amplitudes at the marked vertices are concerned, the system evolves in a M+1 invariant subspace <cit.> spanned by |w⟩, w∈ W, and the equal weight superposition of all remaining vertices 

    |g⟩ = 1/√(N-M)∑_v∉ W|v⟩ .

The effective Hamiltonian in the invariant subspace acts according to (see Eq. (50) in <cit.> for μ=M)

    Ĥ|w⟩    =    γ(N-1+λ_w/γ)|w⟩  - γ∑_w ∈ W
    w'≠ w|w'⟩ - 
           - γ√(N-M)|g⟩,   ∀ w∈ W,
    Ĥ|g⟩    =     -γ√(N-M)∑_w∈ W|w⟩ + γ M |g⟩.

For spatial search, the weights of all marked hubs are chosen as λ_w=-1, and the optimal choice of the hopping amplitude γ is shown to be equal to 1/N. The dimensionality of the problem can be further reduced to 2, since the target state of the search algorithm is the equal weight superposition of the marked vertices

    |w̃⟩ = 1/√(M)∑_w∈ W|w⟩.

In the invariant subspace spanned by |w̃⟩ and |g⟩ the Hamiltonian is described by a matrix (see Eq. (72) in <cit.> for γ = 1/N)

    H = -1/N[         M √(M(N-M)); √(M(N-M))        -M ].

Search evolves from equal weight superposition of all vertices 

    |ψ⟩ = 1/√(N)∑_v|v⟩ = √(M/N)|w̃⟩ + √(N-M/N)|g⟩,

towards the target state |w̃⟩ with unit probability in time given by

    T = π/2√(N/M).


Let us now utilize the Hamiltonian (<ref>) for state transfer. Consider S sender and R receiver hubs from sets S and R, such that S+R=M. We keep the hopping rate γ = 1/N and the weights λ_s = λ_r = -1 for all s∈ S and r∈ R. In such a case, we can group together the states localized on the sender and the receiver vertices

    | S⟩    =    1/√(S)∑_s∈ S|s⟩, 
    | R⟩    =    1/√(R)∑_r∈ R|r⟩,

which we consider as the initial and the target state of state transfer. Together with (<ref>) they form a 3D invariant subspace. 
The Hamiltonian in the reduced subspace is given by the following 3x3 matrix

    H = -1/N[         S    √(R S) √(S(N-M));    √(R S)         R √(R(N-M)); √(S(N-M)) √(R(N-M))        -M ]

The energy spectrum of (<ref>) is found to be

    E_0 = 0,    E_± = ±√(M/N),

and the corresponding eigenvectors are

    |0⟩    =    1/√(S+R)(√(R)| S⟩ - √(S)| R⟩), 
    |±⟩    =    √(1- E_±/2M)(√(S)| S⟩ + √(R)| R⟩) ∓
          ∓√(1+ E_±/2)|g⟩.


Given the initial state |ψ(0)⟩ = | S⟩, the state of the walk at time t reads

    |ψ(t)⟩    =     e^-i H t| S⟩
        =    √(R/M)|0⟩ +   e^-i E_+ t√(S(1-E_+)/2 M)|+⟩ + 
           + e^i E_+ t√(S(1+E_+)/2M)|-⟩.

Overlap with the target state | R⟩ then equals

    ⟨ R|ψ(t)⟩ = √(RS)/M[1-cos(E_+ t) + i E_+ sin(E_+ t)].

Fidelity of state transfer is then given by

    F(t)     =     |⟨ R|ψ(t)⟩|^2 
        =    4 RS/M^2sin^4(E_+ t/2) + R S/N Msin^2(E_+ t).

Hence, the maximal fidelity 

    F_max = 4RS/(R+S)^2,

is reached at time T given by

    T = π/E_+ = √(N/R+S)π.

We find that state transfer with high fidelity is possible when R≈ S, see Figure <ref> for a visualization. Unit fidelity corresponding to perfect state transfer is achieved when the number of receivers and senders are equal. Run-time of the state transfer is twice of that for search for M=R+S vertices (<ref>). Indeed, in state transfer we evolve the system from the state localized on the sender vertices through the equal weight superposition of all vertices (<ref>) towards the state localized on the receiver vertices, effectively iterating the search dynamics twice. For S and R differing considerably the state transfer is suppressed. In particular, for S≫ R or R≫ S, maximal fidelity (<ref>) tends to zero. 




§ DISCRETE-TIME QUANTUM WALK SEARCH FOR HUBS



We turn to the discrete-time quantum walk search for M marked hubs labeled by indices from a set M. Let us start with the Hilbert space. Given a graph G=(V,E) the corresponding Hilbert space H_G can be decomposed as a direct sum of local Hilbert spaces H_v at each vertex v∈ V. Each H_v is spanned by vectors |v,w⟩ such that there is an edge between vertex v and w, and an additional state |v,v⟩ corresponding to the loop at the vertex v. The dimension of the local space H_v is thus equal to d_v + 1, where d_v is the degree of the vertex v in the simple graph G.

The unitary evolution operator of the walk, Û = ŜĈ, is a product of the shift operator Ŝ and the coin operator Ĉ. We consider the flip-flop shift, which is defined in the following way 

    Ŝ| v,w⟩ = | w,v⟩,   Ŝ|v,v⟩ = |v,v⟩ .

The coin operator is given by a direct sum

    Ĉ = ⊕_v Ĉ_v,

of local unitaries Ĉ_v acting on vertex spaces H_v. As the local coins at non-marked vertices we consider the Grover operator with a weighted loop <cit.>. The original Grover operator <cit.> is invariant under all permutations. Hence, when used as a coin in the discrete time quantum walk, it has the advantage that it does not matter how we order the basis states in the local Hilbert space H_v <cit.>. Adding the weighted loop adds a parameter, which can be tuned to improve the success probability of search <cit.>. We can write the Grover operator with a weighted loop as <cit.>

    Ĝ_v(l_v)= 2|Ω_v(l_v)⟩⟨Ω_v(l_v)| -Î_v ,

where Î_v is an identity operator at the subspace H_v and |Ω_v(l_v)⟩ is given by

    |Ω_v(l_v)⟩=1/√(d_v+l_v)(√(d_v)|Ω_v⟩ + √(l_v)|v,v⟩) .

By |Ω_v⟩ we have denoted the equal weight superposition of all outgoing arcs from the vertex v

    |Ω_v⟩=1/√(d_v)∑_w
    { v,w}∈ E| v,w⟩.

The free parameter l_v sets the weight of the loop state at the vertex v. In correspondence to the continuous-time case <cit.> we tune it according to the degree of the vertex and set l_v=N-d_v. On the marked hubs m∈ M this results in l_m = 1. In addition, we multiply the coin on the marked hubs by -1, which is analogous to the marking of the solutions in Grover algorithm <cit.>. Moreover, one can show that the phase shift of π is the optimal choice, as it results in large gap of the avoided crossing which is inversely proportional to the run-time of the search <cit.>. Hence, choosing different phase shift results in slower algorithm with lower success probability. The evolution operator of the search is then given by

    Û_ M = Ŝ (⊕_v∉ M(Ĝ_v(N-d_v))⊕_m∈ M  (-Ĝ_m(1))) .


The usual choice of the initial state for the search algorithm is the equal weight superposition of all arcs

    |Ω̃⟩ = 1/√(∑_v∈ V
    d_v)∑_v∈ V√(d_v)|Ω_v⟩ .

However, with this initial state the search does not perform well, as can be observed from numerical simulations. In addition, we do not find a simple closed invariant subspace of Û_ M which involves (<ref>). The reason for that is that |Ω̃⟩ has the same overlap with every arc, but not with every vertex of the graph, since the graph is not expected to be regular. Therefore, state (<ref>) is not an eigenstate of the unperturbed walk operator (i.e. without marking of hubs by a π phase shift), which is given by 

    Û = Ŝ(⊕_v∈ V(Ĝ_v(N-d_v)) ).

Such eigenstate reads

    |Ω⟩ = 1/√(N)∑_v∈ V|Ω_v(N-d_v)⟩,

which has the same overlap of 1/√(N) with all vertex subspaces ℋ_v. One can easily check that it satisfies Û|Ω⟩ = |Ω⟩. For this initial state we find that search for hubs behaves analogously as search on the complete graph with loops <cit.>. Let us construct the invariant subspace <cit.> of the evolution operator (<ref>). First, we include states where the walker is localized on the marked hubs 

    |ν_1⟩   =   1/√(M)∑_m∈ M|m,m⟩ , 
    |ν_2⟩   =   1/√(M(M-1))∑_m≠ m'∈ M|m,m'⟩,
    |ν_3⟩   =   1/√(M(N-M))∑_m∈ M∑_v∉ M|m,v⟩,

corresponding to superposition of loops (|ν_1⟩), arcs of the edges connecting different marked hubs (|ν_2⟩), and arcs leaving marked hubs to the rest of the graph (|ν_3⟩). Since the evolution operator (<ref>) treats all marked hubs equally we can group them together and form the superpositions of the corresponding internal states. Note that the flip-flop shift operator (<ref>) leaves the states |ν_1⟩ and |ν_2⟩ unchanged, however, it maps the vector |ν_3⟩ onto

    |ν_4⟩   =   Ŝ|ν_3⟩ =  1/√(M(N-M))∑_m∈ M∑_v∉ M|v,m⟩,

where the walker is localized on the non-marked vertices in the arcs heading towards the marked ones. We add this state as the fourth vector of the invariant subspace I. To complete the invariant subspace we consider the superposition of loops and arcs between non-marked vertices

    |ν_5⟩   =   1/N-M∑_v∉ M(√(N)|Ω_v(N-d_v)⟩-∑_m∈ M|v,m⟩).

These 5 vectors are required to decompose the initial state of the search (<ref>) according to

    |Ω⟩    =    1/N(√(M)|ν_1⟩+√(M(M-1))|ν_2⟩  + . 
           + . √(M(N-M))(|ν_3⟩ + |ν_4⟩) + (N-M)|ν_5⟩).

We note that in search for a single marked hub (M=1) the state |ν_2⟩ vanishes and the invariant subspace I is only 4-dimensional.

The action of the search evolution operator (<ref>) on the states |ν_j⟩ is described in the   Appendix <ref>, see Eq. (<ref>). We find that the spectrum of Û_ M reduced to the invariant subspace is composed of ± 1 and pair conjugate eigenvalues λ_±=e^± iω where the phase is given by

    ω = arccos(1-2M/N).

For M>1 eigenvalue 1 is doubly degenerate. We denote the corresponding eigenvectors as |(1)_1,2⟩, |-1⟩ and |±ω⟩, their explicit forms are given in the formulas (<ref>), (<ref>) and (<ref>.)
Decomposition of the initial state (<ref>) into the eigenbasis of Û_ M is given by

    |Ω⟩  =   - √(N-M/2N)|(1)_1⟩ + √(M/2N)|-1⟩ + 1/2(|+ω⟩ + |-ω⟩).

Evolution of the walk after t steps then can be written as 

    |ψ(t)⟩    =     - √(N-M/2N)|(1)_1⟩ + (-1)^t√(M/2N)|-1⟩ + 
           + 1/2(e^iω t|+ω⟩ + e^-iω t|-ω⟩).


The total probability to find one of the marked hubs after t steps can be written as a sum of transition probabilities from |ψ(t)⟩ into the basis states |ν_i⟩, i = 1,2,3

    P(t) = ∑_i=1^3 P_i(t) = ∑_i=1^3 |⟨ν_i|ψ(t)⟩|^2 .

We find that the corresponding probability amplitudes read

    ⟨ν_1|ψ(t)⟩    =    1/2√(M)(cos(ω t) - 1 + (1+(-1)^t)M/N), 
    ⟨ν_2|ψ(t)⟩    =    √(M-1)⟨ν_1|ψ(t)⟩ , 
    ⟨ν_3|ψ(t)⟩    =    1/2sin(ω t)  + (1+(-1)^t)√(M(N-M))/2N .

Considering first the case N≫ M, we can neglect the rapidly oscillating terms and obtain the transition probabilities

    P_1(t)    ≈   1/Msin^4(ω t/2), 
     P_2(t)    ≈   M-1/Msin^4(ω t/2) , 
    
    P_3(t)    ≈   1/4sin^2(ω t) .

The total success probability is then equal to

    P(t) ≈sin^4(ω t/2) + 1/4sin^2(ω t),

which is close to unity provided that we choose the number of steps t as the closest integer to

    T = π/ω≈π/2√(N/M)+O(√(M/N)) .

For N≫ M the maximum success probability comes from P_1(T) and P_2(T), while P_3(T) vanishes. When M=1 we find the particle in the loop at the unique marked vertex, since there is no |ν_2⟩ state and P_2(t) equals zero. As the number of marked hubs grows the contribution of P_2(t) increases, so we are more likely to find the walker on an edge connecting two marked hubs.

When the number of marked hubs is non-negligible relative to the total number of vertices the approximations (<ref>) are no longer valid. While P_1(t) vanishes the contribution of P_3(t) to the total success probability becomes more important. One can show that for all values of N and M<N the total success probability of search evolves with the number of steps according to

    P(2t) = P(2t+1) = sin^2(ω (2t+1)/2).

Indeed, as was shown in <cit.> two steps of the quantum walk search on a complete graph with loops are equivalent to a single iteration of the Grover algorithm. In conclusion, we find one of the marked hubs with probability close to one for the number of steps derived earlier (<ref>).

For comparison of analytical and numerical results see Fig. <ref>, where we display the course of the success probability on a graph with N = 100 vertices with different number of marked hubs. The upper plot shows the case of a single marked hub, i.e. M=1. In the middle plot we have considered M=3, and the lower plot shows the case M=15. The plot highlights how the contribution of different terms P_i(t) changes as the number of marked hubs increases. One can also notice that the total success probability changes only after each two steps, in accordance with (<ref>).






§ STATE TRANSFER BETWEEN TWO HUBS



Let us now turn to the state transfer between two hubs, sender and receiver vertices s and r. We consider the same local coins for the marked and non-marked vertices as for the search. The evolution operator for the state transfer then has the form

    Û_s,r    =    Ŝ (⊕_v≠ s,r(Ĝ_v(N-d_v)) ⊕  (-Ĝ_s(1)) . 
          . ⊕  (-Ĝ_r(1))) .

As for the search, the choice of the loop weights allows to map the problem of state transfer between hubs on otherwise arbitrary graph to state transfer on complete graph with loops. We have investigated this model earlier <cit.>, however, only a single initial state was considered. In the present paper we extend this investigation and show that it is possible to transfer two orthogonal states, namely the loop and the equal weight superposition of all outgoing arcs, in the same run-time. Hence, one can transfer a state of a qubit from one hub to another. In fact, we show that if the sender and the receiver knows their locations, then they can exchange another linearly independent state, allowing for exchange of a qutrit.

First, we perform the dimensional reduction, i.e. determine the invariant subspace I of the walk evolution operator Û_s,r containing both pairs |s,s⟩, |r,r⟩ and |Ω_s⟩, |Ω_r⟩. Construction of the invariant subspace follows the same logic as for the search problem in the previous Section. We begin with the loops on the marked vertices and the connecting arcs

    |ν_1⟩    = |s,s⟩ ,    |ν_2⟩  = |r,r⟩,
    |ν_3⟩    = |s,r⟩ ,    |ν_4⟩ = |r,s⟩.

Next, we add states connecting sender and receiver with the rest of the graph

    |ν_5⟩    = 1/√(N-2)∑_v≠ s,r|s,v⟩ ,  |ν_6⟩  = 1/√(N-2)∑_v≠ s,r|r,v⟩, 
    |ν_7⟩   = 1/√(N-2)∑_v≠ s,r|v,s⟩,   |ν_8⟩  = 1/√(N-2)∑_v≠ s,r|v,r⟩.

To complete the invariant subspace I we consider vectors corresponding to loops and arcs between non-marked vertices

    |ν_9⟩    =  1/N-2∑_v≠ s,r(√(N)|Ω_v(N-d_v)⟩  -  |v,r⟩-|v,s⟩) .

Note that the equal weight superpositions can be expressed in the form

    |Ω_s⟩    =    1/√(N-1)|ν_3⟩ + √(N-2/N-1)|ν_5⟩, 
    |Ω_r⟩    =    1/√(N-1)|ν_4⟩ + √(N-2/N-1)|ν_6⟩,

which tends to |ν_5⟩ and |ν_6⟩ for large graph size N. One can show by direct calculation that the 9-dimensional space is closed under action of Û_s,r. We note that in <cit.> we considered state transfer on the complete graph with loops from the state corresponding to the equal weight superposition of the loop and arcs leaving the sender vertex, which in the present notation reads

    |ψ⟩ = 1/√(N)(|ν_1⟩ + |ν_3⟩ + √(N-2)|ν_5⟩) .

For this particular initial state it was sufficient to consider a smaller 5 dimensional invariant subspace <cit.>.

We provide the detailed investigation of the evolution operator (<ref>) in the invariant subspace in Appendix <ref>. We find that the spectrum of the reduced operator consists of ± 1 and three conjugated pairs λ_j^(±) = e^± i ω_j where the phases ω_j are given by

    ω_1     =    arccos(1-4/N), 
    ω_2    =   arccos(√(1-2/N)) = ω_1/2, 
    ω_3    =   arccos(-√(1-2/N)) = π - ω_2 .

Eigenvalue 1 is doubly degenerate. The corresponding eigenstates are denoted by |(1)_1,2⟩, |-1⟩ and |±ω_j⟩, j=1,2,3. Their explicit forms are given by equations (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>). We proceed with the calculation of the time evolution of the walk for different choices of the initial and target states.




 §.§ Transfer of equal weight superposition states



Let us first consider the initial state as the equal superposition of all outgoing arcs at the sender vertex (<ref>). As follows from (<ref>), for large graph the initial and the target states, |Ω_s⟩ and |Ω_r⟩, can be approximated by

    |Ω_s⟩   ≈   |ν_5⟩ ,  |Ω_r⟩≈|ν_6⟩  .

In terms of the eigenbasis of Û_s,r the vectors |Ω_s⟩ and |Ω_r⟩ can be estimated by

    |Ω_s⟩   ≈   1/2√(2)(√(2)|-1⟩ + i(|+ω_1⟩-|-ω_1⟩) + .
          . / +i(|+ω_2⟩ - |-ω_2⟩)  - i(|+ω_3⟩ - |-ω_3⟩)),
    |Ω_r⟩   ≈   1/2√(2)(√(2)|-1⟩ + i(|+ω_1⟩-|-ω_1⟩) - . 
          . / - i (|+ω_2⟩ - |-ω_2⟩) + i (|+ω_3⟩ - |-ω_3⟩)).

where we have utilized the approximations of eigenstates (<ref>) and (<ref>) for large N. The state of the walk after t steps is then given by

    |ϕ_Ω(t)⟩   =   Û_s,r^t|Ω_s⟩ = (-1)^t/2|-1⟩ + 
           +  i/2√(2)(e^2iω_2 t|+ω_1⟩ - e^- 2iω_2 t|-ω_1⟩) + 
           + i/2√(2)(e^ iω_2 t|+ω_2⟩ - e^- iω_2 t|-ω_2⟩) - 
           - i(-1)^t/2√(2)(e^ -iω_2 t|+ω_3⟩ - e^ iω_2 t|-ω_3⟩).

For the amplitude of state transfer to |Ω_r⟩ we obtain

    ⟨Ω_r|ϕ_Ω (t)⟩   =   1/4[(-1)^t+cos(2ω_2 t)-. 
           -.cos(ω_2 t)(1+(-1)^t)].

We see that there are rapid changes between odd and even steps. For odd number of steps 2t+1 the amplitude turns into

    ⟨Ω_r|ϕ_Ω (2t+1)⟩ = -1/2sin^2(ω_2 (2t+1)),

so the fidelity can reach 1/4 at most. For even number of steps 2t the amplitude (<ref>) equals 

    ⟨Ω_r|ϕ_Ω (2t)⟩    =     
    -cos(2ω_2 t)sin^2(ω_2 t).

Hence, for the closest even integer to

    T  = 2t ≈π/ω_2≈π√(N/2)+O(1/√(N)) .

the amplitude (<ref>) is close to one, i. e. we obtain state transfer from |Ω_s⟩ to |Ω_r⟩ with high fidelity. We note that (<ref>) results in the same fidelity as for the initial state (<ref>) studied in <cit.> (see eq. (17) of the aforementioned paper).



 §.§ Transfer of loop states



In the case of transfer of the loop states the initial and target states are

    |s,s⟩  = |ν_1⟩ ,   |r,r⟩ = |ν_2⟩ .

In the eigenvector basis we find that the for large N the loop states can be approximated by

    |s,s⟩   ≈   1/2|(1)_1⟩ + 1/2√(2)|(1)_2⟩ + 1/4(|+ω_1⟩+|-ω_1⟩)+ 
            + 1/2(|+ω_2⟩ + |-ω_2⟩), 
    |r,r⟩   ≈   1/2|(1)_1⟩ + 1/2√(2)|(1)_2⟩ + 1/4(|+ω_1⟩+|-ω_1⟩) -
            -1/2(|+ω_2⟩ + |-ω_2⟩).

The state of the walk after 2t steps equals

    |ϕ_l(2t)⟩   =   Û_s,r^2t|s,s⟩
       =   1/2|(1)_1⟩+1/2√(2)|(1)_2⟩+ 
           + 1/4(e^4 i ω_2 t|+ω_1⟩+e^- 4 i ω_2 t|-ω_1⟩)+
           + 1/2(e^ 2i ω_2 t|+ω_2⟩ + e^- 2 iω_2 t|-ω_2⟩).

The scalar product of |ϕ_l(2t)⟩ and the target state is then given by

    ⟨r,r|ϕ_l(2t)⟩   =   sin^4(ω_2 t).

Hence, we achieve state transfer with high fidelity in the same number of steps as for the equal weight superposition states (<ref>).

For comparison of state transfer of loops and equal weight superposition states see Fig. <ref>.





 §.§ Transfer of qubit state and finite-size effects of the graph



Our results show that we can achieve state transfer of both |s,s⟩ to |r,r⟩ and |Ω_s⟩ to |Ω_r⟩ in the same run-time. Moreover, we can perform a state transfer of any superposition of these two orthonormal states. Hence, it is possible to transfer an arbitrary state of a qubit between two hubs in time given by the closest even integer to (<ref>). Consider a general qubit state at the sender encoded as

    |ψ_s⟩ = a |s,s⟩ + b|Ω_s⟩≡ a|0⟩ + b |1⟩.

The target qubit state at receiver vertex is

    |ψ_r⟩ = a |r,r⟩ + b|Ω_r⟩.

Time evolution of the initial state (<ref>) after 2t steps is given by

    |ψ(2t)⟩ = a|ϕ_l(2t)⟩ + b |ϕ_Ω(2t)⟩,

where |ϕ_l(2t)⟩ and |ϕ_Ω(2t)⟩ are given by (<ref>) and (<ref>). Amplitude of state transfer from (<ref>) to (<ref>) then reads

    ⟨ψ_r|ψ(2t)⟩    =     |a|^2⟨r,r|ϕ_l(2t)⟩ + |b|^2⟨Ω_r|ϕ_Ω(2t)⟩ + 
           + ab⟨Ω_r|ϕ_l(2t)⟩ + a b ⟨r,r|ϕ_Ω(2t)⟩,

where ⟨Ω_r|ϕ_Ω(2t)⟩ and ⟨r,r|ϕ_l(2t)⟩ were determined in (<ref>) and (<ref>). The remaining transition amplitudes are found to be

    ⟨Ω_r|ϕ_l(2t)⟩    =    1/√(2)sin^2(ω_2 t)sin(2ω_2 t) , 
    ⟨r,r|ϕ_Ω(2t)⟩    =     - ⟨Ω_r|ϕ_l(2t)⟩,

which both vanish when the number of steps T is taken as the closest even integer to (<ref>). Since ⟨Ω_r|ϕ_Ω(T)⟩ and ⟨r,r|ϕ_l(T)⟩ both tend to one for a large graph, we can transfer an arbitrary qubit state from the sender to the receiver vertex.

We note that our result were derived utilizing the approximation of eigenstates (<ref>), (<ref>) for large graph, which neglected the O(1/√(N)) terms in the eigenvectors. For smaller values of N these contributions might be non-negligible and affect the fidelity of state transfer. In addition, the influence can be state dependent. To elucidate the finite-size effects we numerically investigated the fidelity of state transfer of a qubit state

    |ψ⟩ = ρ|0⟩ + e^iφ√(1-ρ^2)|1⟩,

on a graph with 10 vertices. The results are displayed in the upper plot of Fig. <ref>. For both parameters ρ and φ we split the corresponding interval range into 100 elements and evaluate the fidelity at the number of steps given by the closest even integer to (<ref>). The plot indicates that the worst fidelity is obtained for the state |1⟩, i.e. |Ω_s⟩. On a graph of 10 vertices the fidelity ranges between 0.82 and 0.9.

In the lower plot of Fig. <ref> we show the fidelity of state transfer for the basis states as a function of the graph size N. The plot is on the log-log scale to unravel the power-law behaviour F(T) = 1- O(1/N). Note that the rapid drops and increases in the figure are due to the discreteness of the optimal time T, which has to be chosen as the closest even integer to (<ref>). The plot indicates that changing the qubit state to be transferred alters the pre-factor but not the power law dependence.







 §.§ Transfer of the connecting arcs



Let us now consider the situation when the sender and the receiver know their positions. In such a case, they can transfer the states corresponding to the arcs on the connecting edge, i.e. the state |ν_3⟩ will be mapped onto |ν_4⟩. We show that this is achieved in the same runtime as for the loop and the equal weight superposition. 

The initial and the target states decomposed into the eigenvectors of Û_s,r have the following form

    |ν_3⟩   ≈     -1/2|(1)_1⟩ + 1/2√(2)|(1)_2⟩ + 1/4(|+ω_1⟩ + |-ω_1⟩) + 
           +1/2 (|+ω_3⟩ + |-ω_3⟩), 
    |ν_4⟩   ≈    -1/2|(1)_1⟩ + 1/2√(2)|(1)_2⟩ + 1/4(|+ω_1⟩ + |-ω_1⟩) - 
           - 1/2 (|+ω_3⟩ + |-ω_3⟩).

In even steps 2t, the state of the walk is given by

    |ϕ_sr(2t)⟩    =     -1/2|(1)_1⟩ + 1/2√(2)|(1)_2⟩ + 
           + 1/4(e^4i ω_2 t|+ω_1⟩ + e^-4i ω_2 t|-ω_1⟩) + 
           + 1/2 (e^-2i ω_2 t|+ω_3⟩ + e^2i ω_2 t|-ω_3⟩).

For the amplitude of state transfer in even steps we find

    ⟨ν_4|ϕ_sr(2t)⟩ = sin^4(ω_2 t),

i.e., it behaves the same as for the transfer of loop states (<ref>). Hence, the arc from the sender to the receiver is transferred into its opposite in the runtime given by (<ref>)

We note that the vectors |ν_3,4⟩ and |Ω_s,r⟩ are linearly independent but not mutually orthogonal. Nevertheless, from (<ref>) we find

    ⟨ν_3|Ω_s⟩ = ⟨ν_4|Ω_r⟩ = 1/√(N-1)

so the overlap vanishes with increasing size of the graph N. Within the approximations we have used in our derivations, the states |s,s⟩ = |ν_1⟩, |Ω_s⟩≈|ν_5⟩ and |ν_3⟩ are orthogonal. Their superposition represent a qutrit state, which can be transferred from the sender to the receiver vertex, provided that the two communicating parties know which edge connects them. Note that |ν_1⟩, |ν_3⟩ and |ν_5⟩ are the only basis states which are localized on the sender vertex. Hence, in this model we cannot transfer a higher dimensional qudit state with d≥ 4.

We have also tested numerically the effect of finite graph size on the fidelity of state transfer of the connecting arcs. The simulation reveals that for every N the fidelity is exactly equal to the one for the state transfer of the loop state, which is depicted in the lower plot of Figure <ref> with the blue triangles.



§ STATE TRANSFER BETWEEN MULTIPLE SENDER AND RECEIVER HUBS



We now expand the investigation of the previous Section by considering S>1 senders and R>1 receivers from sets S and R, respectively, with S+R=M. We consider the same dynamics as for the search and state transfer between two marked vertices, i.e. the local loop weights are chosen as l_v = N-d_v and on the marked hubs the coin is multiplied by -1. Having multiple senders and receivers leads to the following modification of the invariant subspace we have constructed in the previous Section for S=R=1. First, states (<ref>) corresponding to the loops and connecting arcs between senders and receivers are adjusted according to  

    |ν_1⟩    =    1/√(S)∑_s∈ S|s,s⟩, 
    |ν_2⟩    =    1/√(R)∑_r∈ R|r,r⟩
    |ν_3⟩    =    1/√(RS)∑_s∈ S∑_r∈ R|s,r⟩ , 
    |ν_4⟩    =     1/√(RS)∑_s∈ S∑_r∈ R|r,s⟩ ,

where we have utilized the fact that evolution operator treats all senders and receivers equally and we can group them together. Next, states connecting sender and receiver vertices to the rest of the graph  (<ref>) are changed into 

    |ν_5⟩    =    1/√(S(N-M))∑_s∈ S∑_v ∉ S∪ R|s,v⟩ , 
    |ν_6⟩    =    1/√(R(N-M))∑_r∈ R∑_v ∉ S∪ R|r,v⟩, 
    |ν_7⟩    =    1/√(S(N-M))∑_s∈ S∑_v∉ S∪ R|v,s⟩ , 
    |ν_8⟩    =    1/√(R(N-M)))∑_r∈ R∑_v ∉ S∪ R|v,r⟩ .

The state (<ref>) corresponding to the loops and arc between non-marked vertices is modified as follows

    |ν_9⟩    =    1/(N-M)∑_v∉ S∪ R(/√(N)|Ω_v(N-d_v)⟩ - . 
          . - ∑_s∈ S|v,s⟩ - ∑_r∈ R|v,r⟩).

To complete the invariant subspace we have to add two states describing the superpositions of arcs connecting sender hubs together, and the same for receiver hubs, i.e. 

    |ν_10⟩    =    1/√(S(S-1))∑_s ≠ s'∈ S|s,s'⟩, 
    |ν_11⟩    =    1/√(R(R-1))∑_r ≠ r'∈ R|r,r'⟩,

 which are not present when we have only one sender and one receiver vertex.

We leave the detailed investigation of the evolution operator Û_ S, R

    Û_ S, R    =    Ŝ (⊕_v∉ S,  R(Ĝ_v(N-d_v)) . 
          .⊕_s∈ S   (-Ĝ_s(1)) ⊕_r∈ R  (-Ĝ_r(1))) .

for Appendix <ref>. The spectrum of Û_ S, R has a similar form as for S=R=1, i.e. it consists of eigenvalues ± 1 and three complex conjugated pairs λ_j^(±) = e^± i ω_j, where the phases are given by

    ω_1     =    arccos(1-2(R+S)/N), 
    ω_2     =    ω_1/2,   ω_3 = π - ω_2 .

Eigenvalue 1 now has a degeneracy of 4. The corresponding eigenvectors are denoted by |(1)_m⟩ , m = 1,…,4, (see formulas (<ref>)), |-1⟩ (equation (<ref>)) and |±ω_j⟩, j=1,2,3. Eigenvectors |±ω_1⟩ are given by (<ref>), and approximations of |±ω_2⟩ and |±ω_3⟩ for large N≫ R,S can be found in the formula (<ref>).

To investigate state transfer we consider evolution from two initial states supported on the sender vertices corresponding to the loops (|ν_1⟩) and the equal weight superposition of outgoing arcs (|ν_5⟩). We analyze the transition amplitudes to the states supported on the receiver vertices (|ν_k⟩, k=2,4,6,11). As we will see, for S,R≠ 1 it is not possible to transfer specific quantum states from the senders to the corresponding states on the receiver vertices. Nevertheless, if we relax the requirements and focus solely on transfer of the quantum walker from senders to receiver vertices irrespective of the internal state, we find that this task can be achieved in certain regimes of S and R.

Let us begin with the superposition of loop states on the sender vertices |ν_1⟩. In this case the state of the walk after t steps reads

    |ϕ_l(t)⟩   ≈   1/M(R/√(S)|(1)_1⟩ + √(S/2)|(1)_2⟩ + . 
          . + M√(S-1/S)|(1)_3⟩ + . 
           + .  √(2)/2[e^i 2ω_2 t|+ω_1⟩ + e^-i 2ω_2 t|-ω_1⟩] + . 
           + . √(R)[e^i ω_2 t|+ω_2⟩ + e^-i ω_2 t|-ω_2⟩] ) ,

within the approximations of (<ref>) and (<ref>) for N≫ R,S. Transition amplitudes to the states at the receiver vertices are given by

    ⟨ν_2|ϕ_l(t)⟩    =    4√(RS)/M^2sin^4(ω_2 t/2), 
    ⟨ν_4|ϕ_l(t)⟩    =     -2√(R)/M^2 (R + Scos(ω_2 t))sin^2(ω_2 t/2), 
    ⟨ν_6|ϕ_l(t)⟩    =      -2√(RS/M^3)sin(ω_2 t) sin^2(ω_2 t/2), 
    ⟨ν_11|ϕ_l(t)⟩    =    4√(RS(R-1))/M^2sin^4(ω_2 t/2).

We see that all amplitudes are vanishing with increasing number of sender and receiver vertices. The total probability of transfer to an arbitrary loop or arc originating from one of the receiver vertices is given by

    P_l(t) = ∑_k=2,4,6,11 |⟨ν_k|ϕ_l(t)⟩|^2 = 4R/(R+S)^2sin^4(ω_2 t/2) .

Hence, from the loop state |ν_1⟩ one can achieve unit transfer probability only in the case S=R=1 treated in the previous Section. Notice the asymmetry - for R≫ S the maximal transfer probability decreases as R^-1, while for S≫ R the decrease is faster and follows S^-2. 

Concerning the superposition of all outgoing arcs from all senders 
    |Ω_ S⟩ = √(RS/N-1)|ν_3⟩ + √(S(N-M)/N-1)|ν_5⟩,

we focus on a large graph with N≫ R,S and approximate this initial state as |ν_5⟩. After t steps the walk is described by the following vector 

    |ϕ_Ω(t)⟩   ≈   1/2√(R+S)( (-1)^t √(2S)|-1⟩ + .
          . + i √(S)[e^2iω_2 t|+ω_1⟩ - e^-2iω_2 t|-ω_1⟩] + . 
          . + i√(R)[e^iω_2 t|+ω_2⟩ - e^-iω_2 t|-ω_2⟩] -   . 
          . - i(-1)^t√(R)[e^-iω_2 t|+ω_3⟩ - e^iω_2 t|-ω_3⟩] ) .

Turning to the transition amplitudes, we investigate odd and even steps separately. For even number of steps 2t we obtain

    ⟨ν_2|ϕ_Ω(2t)⟩    =       -⟨ν_6|ϕ_l(2t)⟩ , 
    ⟨ν_4|ϕ_Ω(2t)⟩    =    √(S)⟨ν_2|ϕ_Ω(2t)⟩ , 
    ⟨ν_6|ϕ_Ω(2t)⟩    =     -2√(RS)/Mcos(2ω_2 t)sin^2(ω_2 t), 
    ⟨ν_11|ϕ_Ω(2t)⟩    =    √(R-1)⟨ν_2|ϕ_Ω(2t)⟩ .

The total transfer probability in even steps is given by 

    P_Ω(2t) = 4RS/(R+S)^2sin^4(ω_2 t),

which reaches the maximum

    P_max^(even) =  4RS/(R+S)^2,

for number of steps given by the closest even integer to

    T = 2t ≈π/ω_2≈π√(N/R+S).

We point out that the result (<ref>) coincide with the one for the continuous time quantum walk (<ref>).

Considering odd steps, the amplitudes at time 2t+1 are equal to

    ⟨ν_2|ϕ_Ω(2t+1)⟩    =     2√(RS/M^3)sin(ω_2 (2t+1)) ×
          ×sin^2(ω_2 (2t+1)/2) , 
    ⟨ν_4|ϕ_Ω(2t+1)⟩    =     -√(R/M^3)[R + S cos(ω_2(2t+1))] ×
          ×sin(ω_2 (2t+1)) , 
    ⟨ν_6|ϕ_Ω(2t+1)⟩    =     -√(RS)/Msin^2(ω_2 (2t+1)) 
    ⟨ν_11|ϕ_Ω(2t+1)⟩    =    √(R-1)⟨ν_2|ϕ_Ω(2t+1)⟩ .

The total transfer probability in odd steps is then given by

    P_Ω(2t+1)     =    R/R+Ssin^2(ω_2(2t+1)).

The maximum of value

    P_max^(odd) = R/R+S,

is reached in a number of steps given by the closest odd integer to T/2.

The derived results show that the discrete time quantum walk allows to transfer the walker initialized at the sender vertices in the state |Ω_ S⟩ to the receiver vertices with high probability in two regimes. The first regime occurs for R≈ S, which is the same as for the continuous time quantum walk. Here the probability in even steps reaches close to one (<ref>) for T given by (<ref>). The second regime is given by R≫ S, which is absent for the continuous time case. In this case the transfer probability approaches one in odd steps (<ref>) in half the runtime. We illustrate our results in Figures <ref>, <ref> and <ref>.



In Figure <ref> we show the maximal transfer probabilities in odd steps (<ref>) as a function of the number of senders S and the number of receivers R. For even steps, we refer to the continuous time case and Figure <ref>, since P_max^(even) (<ref>) and F_max (<ref>) are equal.



Figure <ref> displays the course of the transfer probability for the case of equal number of senders and receivers. We see that the transfer probability is close to 1 for even number of steps T given by (<ref>). In odd number of steps, the maximal value reached is 1/2 in approximately T/2 steps.



In Figure <ref> we consider transfer from a single sender vertex to multiple receivers. The two plots display the course of the transfer probability on a graph of N=1000 vertices with 3 receivers (upper plot) and 20 receivers (lower plot). We observe that for R=3 the peaks for even (<ref>) and odd (<ref>) steps reach almost the same maximal value close to 0.75. For R=20 the peak for even steps is suppressed considerably, while in odd number of steps the transfer probability reaches close to 1.




§ CONCLUSIONS



Search and state transfer between multiple hubs on otherwise arbitrary graphs were investigated in detail. We have shown that both continuous-time and discrete-time quantum walks can be utilized for this purpose, since the models can be mapped to the complete graph with loops for a proper choice of the loop weights. In the continuous-time case the state transfer is achieved with high fidelity when the number of senders S is close to the number of receivers R. The discrete-time walk has proven to be more versatile. Indeed, state transfer between multiple hubs is achieved also in the regime R≫ S, which is not possible in the continuous-time case. Moreover, when only a single sender and a single receiver is considered, the coin degree of freedom allows us to transfer multiple orthogonal states in the same run-time, expanding our earlier results <cit.>. We have shown that sender and receiver can always exchange an arbitrary qubit state, and in the case they know each others position this can be extended to a qutrit. This is not possible in the continuous time model, since there the walker does not have an internal degree of freedom, only the position.

There are several directions for generalizing our results. Our aim is to investigate if one can achieve state transfer from an ordinary vertex (i.e. not fully connected) to a hub. If the answer is positive we can perform state transfer between arbitrary vertices through the hub by switching the marked coin from the sender to the receiver after the transfer to the hub was completed, while keeping the hub vertex marked throughout the whole duration. Another approach to state transfer on arbitrary graphs is to utilize the framework based on the ergodic reversible Markov chains developed for both discrete-time <cit.> and continuous-time models <cit.>. This method was used to prove that quantum spatial search is quadratically faster over the classical search for an arbitrary number of marked vertices. We aim to modify this approach to investigate state transfer between multiple vertices which are not fully connected, where the reduction to a complete graph as in this paper is not possible.  



MŠ acknowledges the financial support from Czech Grant Agency project number GAČR 23-07169S. SS is grateful for financial support from SGS22/181/OHK4/3T/14. 










§ EVOLUTION OPERATOR OF THE SEARCH




In this Appendix we treat the search evolution operator (<ref>) in the invariant subspace I spanned by vectors (<ref>), (<ref>) and (<ref>). The action of Û_ M on the basis states is given by

    Û_ M|ν_1⟩    =    (1-2/N) |ν_1⟩  - 2√(M-1)/N|ν_2⟩  -2√(N-M)/N|ν_4⟩ , 
    Û_ M|ν_2⟩    =     - 2√(M-1)/N|ν_1⟩  + (1 - 2M-1/N)|ν_2⟩ - 2√((M-1)(N-M))/N|ν_4⟩ , 
    Û_ M|ν_3⟩    =     -2√(N-M)/N|ν_1⟩  -2√((M-1)(N-M))/N|ν_2⟩ - (1-2M/N) |ν_4⟩ ,
    Û_ M|ν_4⟩    =     - (1-2M/N) |ν_3⟩ + 2√(M(N-M))/N|ν_5⟩ , 
    Û_ M|ν_5⟩    =    2√(M(N-M))/N|ν_3⟩ + (1-2M/N) |ν_5⟩,

which shows that I is indeed invariant. 

Spectrum of the reduced evolution operator in the invariant subspace consists of eigenvalues ± 1 and e^± i ω with the phase ω given in (<ref>). We find that eigenvalue 1 is doubly degenerate and the corresponding eigenvectors read

    |(1)_1⟩    =    1/√(2 N M)(√(N-M)(|ν_1⟩ + √(M-1)|ν_2⟩) - M (|ν_3⟩ + |ν_4⟩) - √(M(N-M))|ν_5⟩), 
    |(1)_2⟩    =    1/√(M)(√(M-1)|ν_1⟩ - |ν_2⟩).

Eigenvector corresponding to -1 is given by

    |-1⟩    =    1/√(2N)(|ν_1⟩ + √(M-1)|ν_2⟩  - √(M)|ν_5⟩ + √(N-M)(|ν_3⟩+|ν_4⟩) ).

For the conjugate pair λ_± we find the eigenvectors

    |±ω⟩    =    1/2(1/√(M)|ν_1⟩ + √(M-1/M)|ν_2⟩ + |ν_5⟩∓ i(|ν_3⟩-|ν_4⟩)).

Note that for a single marked hub the state |ν_2⟩ vanishes. In that case we have only 4-dimensional invariant subspace. Nevertheless, all results remain valid for M=1, except that eigenvalue 1 is simple as the state |(1)_2⟩ equals zero.





§ EVOLUTION OPERATOR OF STATE TRANSFER BETWEEN TWO HUBS



In this Appendix we investigate the evolution operator of the state transfer between two hubs (<ref>) in the invariant subspace spanned by vectors |ν_j⟩ given in equations (<ref>), (<ref>) and (<ref>). To simplify further calculations we utilize the invariance of Û_s,r under exchange of sender and receiver vertices <cit.>. This symmetry reduces the dimensionality of the problem, since we can decompose Û_s,r into a direct sum of lower-dimensional operators. Consider the swap operator P̂ which acts on the basis states |ν_j⟩ as

    P̂|ν_1⟩   =   |ν_2⟩,  P̂|ν_3⟩ = |ν_4⟩,  P̂|ν_5⟩ = |ν_6⟩,
    P̂|ν_7⟩   =   |ν_8⟩,  P̂|ν_9⟩ = |ν_9⟩.

Clearly, P̂ commutes with Û_s,r and therefore they have common eigenvectors. Since the square of P̂ is the identity it has eigenvalues ± 1. Hence, it divides the invariant subspace I =  I_+⊕ I_- into a direct sum of symmetric states I_+ and antisymmetric states I_-. We find that I_+ has dimension 5 and I_- has dimension 4. Subspace I_+ is spanned by vectors 

    |σ_1⟩   =   1/√(2)(|ν_1⟩+|ν_2⟩),
    |σ_2⟩   =   1/√(2)(|ν_3⟩+|ν_4⟩),
    |σ_3⟩   =   1/√(2)(|ν_5⟩+|ν_6⟩),
    |σ_4⟩   =   1/√(2)(|ν_7⟩+|ν_8⟩),
    |σ_5⟩   =   |ν_9⟩


Let us denote the restriction of the evolution operator Û_s,r onto the symmetric subspace as Û_+. We find that it acts on the basis states |σ_j⟩ as follows 

    Û_+|σ_1⟩   =   (1-2/N)|σ_1⟩-2/N|σ_2⟩-2√(N-2)/N|σ_4⟩,
    Û_+|σ_2⟩   =   -2/N|σ_1⟩+(1-2/N)|σ_2⟩-2√(N-2)/N|σ_4⟩,
    Û_+|σ_3⟩   =   -2√(N-2)/N|σ_1⟩-2√(N-2)/N|σ_2⟩ - 
           -(1-4/N)|σ_4⟩,
    Û_+|σ_4⟩   =    -(1-4/N) |σ_3⟩+2√(2)√(N-2)/N|σ_5,⟩
    Û_+|σ_5⟩   =   2√(2)√(N-2)/N|σ_3⟩ + (1-4/N)|σ_5⟩ .

Spectrum of Û_+ is composed of eigenvalues ± 1 and a conjugate pair λ_1^(±)=e^± iω_1 with eigenphase given by (<ref>). Eigenvalue 1 has degeneracy 2, the corresponding eigenvectors are given by 

    |(1)_1⟩   =   1/√(2)(|σ_1⟩-|σ_2⟩),
    |(1)_2⟩   =   1/2√(N)(√(N-2)(|σ_1⟩+|σ_2⟩) - . 
                   - . 
     2(|σ_3⟩+|σ_4⟩) - √(2)√(N-2)|σ_5⟩).

Eigenvector corresponding to -1 reads

    |-1⟩   =   1/√(2N)(|σ_1⟩+|σ_2⟩+√(N-2)(|σ_3⟩+|σ_4⟩) - .
                  . -√(2)|σ_5⟩) .

For large N the eigenvectors |(1)_2⟩ and |-1⟩ can be approximated by neglecting the O(1/√(N)) terms with

    |(1)_2⟩   ≈   1/2(|σ_1⟩+|σ_2⟩) - 1/√(2)|σ_5⟩ , 
    |-1⟩   ≈   1/√(2)(|σ_3⟩+|σ_4⟩).

Concerning the conjugated pair we find that the corresponding  eigenvectors read

    |±ω_1⟩   =   1/2(1/√(2)(|σ_1⟩ + |σ_2⟩) + |σ_5⟩± i(|σ_4⟩ - |σ_3⟩)).

   
Lets now move to the antisymmetric subspace I_- which is spanned by vectors 

    |τ_1⟩   =   1/√(2)(|ν_1⟩-|ν_2⟩),
    |τ_2⟩   =   1/√(2)(|ν_3⟩-|ν_4⟩),
    |τ_3⟩   =   1/√(2)(|ν_5⟩-|ν_6⟩),
    |τ_4⟩   =   1/√(2)(|ν_7⟩-|ν_8⟩).

We denote by Û_- the restriction of Û_s,r to I_-, which acts as follows

    Û_-|τ_1⟩   =   (1-2/N) |τ_1⟩ + 2/N|τ_2 ⟩ -2√(N-2)/N|τ_4⟩,
    Û_-|τ_2⟩   =   -2/N|τ_1⟩ -(1-2/N)|τ_2⟩-2√(N-2)/N|τ_4⟩,
    Û_-|τ_3⟩   =   -2√(N-2)/N|τ_1⟩+2√(N-2)/N|τ_2⟩ - 
           - (1-4/N)|τ_4⟩,
    Û_-|τ_4⟩   =   -|τ_3⟩.

We find that Û_- has two pairs of conjugated eigenvalues λ_2^(±)=e^± iω_2 and λ_3^(±)=e^± iω_3 with eigenphases given by equations (<ref>). The corresponding eigenvectors read

    |±ω_2⟩    =    cos(ω_2/2)/√(2)|τ_1⟩∓ i sin(ω_2/2)/√(2)|τ_2⟩±
          ±i/2(e^± i ω_2/2|τ_4⟩ - e^∓ i ω_2/2|τ_3⟩),
    |±ω_3⟩   =   ± i sin(ω_2/2)/√(2)|τ_1⟩ + cos(ω_2/2)/√(2)|τ_2⟩±
          ±i/2( e^± i ω_2/2|τ_3⟩ + e^∓ i ω_2/2|τ_4⟩).

For large N the phase ω_2 tends to zero as √(2/N). Omitting the O(1/√(N)) terms the eigenvectors can be approximated by

    |±ω_2⟩   ≈   1/√(2)|τ_1⟩±i/2(|τ_4⟩ - |τ_3⟩) ,
    |±ω_3⟩   ≈   1/√(2)|τ_2⟩±i/2(|τ_3⟩ + |τ_4⟩) .




§ EVOLUTION OPERATOR OF STATE TRANSFER BETWEEN MULTIPLE HUBS



In this Appendix we present the analysis of the evolution operator Û_ S, R for state transfer between multiple senders and receivers (<ref>). We find that Û_ S, R acts on the basis vectors of the invariant subspace (<ref>), (<ref>), (<ref>) and (<ref>) according to


    Û_ S, R|ν_1⟩    =    (1-2/N)|ν_1⟩ - 2√(R)/N|ν_4⟩ -    2 √(N-M)/N|ν_7⟩ - 2√(S-1)/N|ν_10⟩, 
    Û_ S, R|ν_2⟩    =    (1-2/N) |ν_2⟩ -2√(S)/N|ν_3⟩  - 2 √(N-M)/N|ν_8⟩ -    2 √(R-1)/N|ν_11⟩, 
    Û_ S, R|ν_3⟩    =     - 2√(R)/N|ν_1⟩ + (1-2R/N)|ν_4⟩  -2 √(R(N-M))/N|ν_7⟩ - 2 √(R(S-1))/N|ν_10⟩, 
    Û_ S, R|ν_4⟩    =      -2√(S)/N|ν_2⟩ + (1-2/N)|ν_3⟩  - 2 √(S(N-M))/N|ν_8⟩ -    2 √(S(R-1))/N|ν_11⟩ , 
    Û_ S, R|ν_5⟩    =     -2 √(N-M)/N(|ν_1⟩ + √(R)|ν_4⟩ +  √(S-1)|ν_10⟩)   -    (1-2M/N) |ν_7⟩ , 
    Û_ S, R|ν_6⟩    =     - 2 √(N-M)/N(|ν_2⟩ + √(S)|ν_3⟩  +    √(R-1)|ν_11⟩) -(1-2M/N) |ν_8⟩, 
    Û_ S, R|ν_7⟩    =     -(1-2S/N)|ν_5⟩ - 2√(RS)/N|ν_6⟩ -    2 √(S(N-M))/N|ν_9⟩, 
    Û_ S, R|ν_8⟩    =    2√(RS)/N|ν_5⟩ - (1-2R/N)|ν_6⟩ -    2 √(R(N-M))/N|ν_9⟩, 
    Û_ S, R|ν_9⟩    =    2 √(N-M)/N(√(S)|ν_5⟩ + √(R)|ν_6⟩) +    (1-2M/N)|ν_9⟩, 
    Û_ S, R|ν_10⟩    =     - 2 √(S-1)/N(|ν_1⟩ + √(R)|ν_4⟩ + √(N-M)|ν_7⟩)  +     (1-2(S-1)/N)|ν_10⟩, 
    Û_ S, R|ν_11⟩    =     - 2 √(R-1)/N(|ν_2⟩ + √(S)|ν_3⟩ + √(N-M)|ν_8⟩)  +     (1-2(R-1)/N)|ν_11⟩ .

For S≠ R the splitting of the invariant subspace into the symmetric and antisymmetric subspaces is no longer possible. Nevertheless, the evolution operator Û_ S, R can be diagonalized analytically. The eigenvalues are again ± 1 and three conjugated pairs λ_j = e^± i ω_j, j=1,2,3,  where the phases ω_j are given in (<ref>). Eigenvalue 1 has a degeneracy of 4, one can choose the basis in the degenerate subspace e.g. according to 

    |(1)_1⟩    =    1/M(R/√(S)|ν_1⟩ + S/√(R)|ν_2⟩ - √(RS)(|ν_3⟩ 
    + |ν_4⟩) + R √(S-1/S)|ν_10⟩ + S √(R-1/R)|ν_11⟩) ,
    |(1)_2⟩    =    1/√(2N)M[√(N-M)(√(S)|ν_1⟩ + √(R)|ν_2⟩ + √(RS)(|ν_3⟩ + |ν_4⟩) - M|ν_9⟩ + . .  
          . . + √(S(S-1))|ν_10⟩ + √(R(R-1))|ν_11⟩) -  M(√(S)(|ν_5⟩ + |ν_7⟩) + √(R) (|ν_6⟩ + |ν_8⟩) )], 
    |(1)_3⟩    =    1/√(S) (√(S-1)|ν_1⟩ - |ν_10⟩), 
    |(1)_4⟩    =    1/√(R) (√(R-1)|ν_2⟩ - |ν_11⟩).

The eigenvector corresponding to -1 reads

    |-1⟩    =    1/√(2NM)(√(S)|ν_1⟩  + √(R)|ν_2⟩ + √(RS)(|ν_3⟩ + |ν_4⟩) + √(N-M)[√(S)(|ν_5⟩ + |ν_7⟩) + √(R)(|ν_6⟩ + |ν_8⟩)] - . 
          .  - M|ν_9⟩ + √(S(S-1))|ν_10⟩ + √(R(R-1))|ν_11⟩).

In the limit N≫ R,S we can approximate the eigenstates |(1)_2⟩ and |-1⟩ with

    |(1)_2⟩   ≈   1/√(2)M(√(S)|ν_1⟩ + √(R)|ν_2⟩ + √(RS)(|ν_3⟩ + |ν_4⟩) - M|ν_9⟩ + √(S(S-1))|ν_10⟩ +  √(R(R-1))|ν_11⟩), 
    |-1⟩   ≈   1/√(2M)[√(S)(|ν_5⟩ + |ν_7⟩) + √(R)(|ν_6⟩ + |ν_8⟩) ].

For the complex conjugated pair λ_1^(±) we find the following

    |±ω_1⟩    =    1/2M(√(S)|ν_1⟩  + √(R)|ν_2⟩ + √(RS)(|ν_3⟩ + |ν_4⟩)  + M|ν_9⟩ + √(S(S-1))|ν_10⟩ + √(R(R-1))|ν_11⟩±. 
          . ± i√(M)[√(S)(|ν_7⟩ - |ν_5⟩) + √(R)(|ν_8⟩ - |ν_6⟩)]) .

The exact form of the eigenvectors corresponding to λ_2,3^(±) is rather complicated, however, in the limit of N ≫ R,S they can be approximated by

    |±ω_2⟩   ≈   1/2M(2√(R)|ν_1⟩ - 2√(S)|ν_2⟩ + (R-S)(|ν_3⟩ + |ν_4⟩) + 2√(R(S-1))|ν_10⟩  -2√(S(R-1))|ν_11⟩±. 
          . ± i√(M)[√(R)(|ν_7⟩ -|ν_5⟩ + √(S)(|ν_6⟩ - |ν_8⟩))]  ), 
    |±ω_3⟩   ≈   1/2M(M(|ν_3⟩ - |ν_4⟩) ± i √(M)[√(R)(|ν_5⟩ + |ν_7⟩) - √(S)(|ν_6⟩ + |ν_8⟩)] )

We point out that for S=R=1 the spectrum and the eigenvectors reduce to those derived in the Appendix <ref>. Note that in this case the states |ν_10⟩ and |ν_11⟩ need to be eliminated, since there no connecting edges when we have only a single sender and a single receiver.









