A ug 2 01 9 Localization in the constrained quantum annealing of graph coloring

Constrained quantum annealing (CQA) is a quantum annealing approach that is designed so that constraints are satisfied without penalty terms. There is an analogy between the model used for the CQA of graph coloring and a disordered spin chain. In a strongly disordered spin chain, localization phenomena such as Anderson localization and many-body localization occurs. In the model for the CQA of graph coloring, disorder corresponds to the fluctuation of local effective fields that increase in a CQA process. Several measures of entanglement show how localization appears in a CQA process, depending on the fluctuation of effective fields. Localization in CQA is also considered as Anderson localization in Fock space. The CQA approach of graph coloring causes extra localization due to degenerated ground states of the problem Hamiltonian and a specific choice of the driver Hamiltonian.


I. INTRODUCTION
Entanglement plays an essential role for the speedup of quantum computation [1][2][3]. However, the role of entanglement in adiabatic quantum computation and quantum annealing (QA) is currently unclear [4][5][6][7]. QA is well known as a quantum-mechanical approach for combinatorial optimization problems [8][9][10][11][12][13][14][15][16]. In a QA framework, the total Hamiltonian consists of a problem Hamiltonian H p and a driving Hamiltonian H d : where s is a time dependent parameter. The QA process starts with the ground state of H d at s = 0 and ends at s = 1. If the process is adiabatic, the solution of an optimization problem, in other words the ground state of H p , is obtained as the ground state of the total Hamiltonian at the end of the QA process. The computational efficiency of QA depends on the energy gap between the ground and first-excited states. However, the connection between the energy gap and entanglement is not well understood. Entanglement also characterizes many-body localization (MBL) in quantum systems where the interplay between disorder and the many-body interaction is important. In the MBL phase, where disorder is relatively strong, quantum dynamics is nonergodic and breaks the eigenstate thermalization hypothesis (ETH) [17][18][19]. While many-body eigenstates in the ETH phase are linked to a large amount (volume law) of entanglement, MBL eigenstates correspond to low (area-law) entanglement. Entanglement entropy (EE) is often employed to characterize the MBL transition [20][21][22][23][24][25][26][27][28][29][30][31]. In a disordered quantum spin chain, for instance, the standard deviation of the half-chain EE has a peak around the value of the MBL transition [27][28][29][30].
In this paper, we explore evidence of the MBL transition in QA by means of entanglement. While MBL occurs in systems with disordered fields, the system considered in QA does not usually have any external random fields. However, a spin can be sensible of inhomogeneous * kudo@is.ocha.ac.jp fields from neighboring spins, even if random fields are not applied externally. For example, the total Hamiltonian for the constrained quantum annealing (CQA) of graph coloring is analogous to an ensemble of spin chains with disordered fields [32]. In CQA approaches, the driving Hamiltonian is chosen so that it commutes with constraint operators but not with the problem Hamiltonian [32][33][34]. In the CQA of graph coloring, the driving Hamiltonian consists of independent spin chains. This implies that the system is made of almost independent spin chains interacting weakly with other chains in the early stages of CQA. In the late stage, as inter-chain interaction increases with time, spins are affected by effective fields arising from neighboring chains. These effective fields can cause the MBL transition. To examine this transition, we investigate quantum dynamics and entanglement evolution using real-time quantum simulations. Several types of measures of entanglement suggest that the MBL transition occurs in the CQA process of graph coloring.
The remainder of the paper is organized as follows. In Sec. II, the model for the CQA of graph coloring is introduced, and numerical methods are outlined. In Sec. III, the time dependence of effective fields is illustrated. The results on EE and concurrence, both measures of entanglement, are presented in Secs. IV and V, respectively. Discussion and conclusions are given in Sec. VI.

II. MODELS AND METHODS
We focus here on the CQA of graph coloring; this model was introduced in Ref. [32]. Graph coloring consists of coloring the nodes of a graph such that nodes directly connected through an edge do not share the same color. When coloring a graph G = (V, E) with q available colors, the classical Hamiltonian is given by where i and j represent nodes V = {1, . . . , N }, and (ij) ∈ E denotes the edge connecting the pair of nodes i, j ∈ V . If node i is colored a, S i,a = 1; otherwise, S i,a = −1. Thus, Eq. (2) counts the number of edges that connect nodes with the same color. The quantum version of Eq. (2) is the problem Hamiltonian, given by where σ z i,a denotes the Pauli matrix of the component z, and J has a unit of energy (J = 1 in the simulations below). As each node can only have one color, the following constraint is required. q a=1 σ z i,a = 2 − q. In standard QA approaches, the penalty term to satisfy this constraint is often incorporated into the problem Hamiltonian [35][36][37][38][39]. In the CQA approach, however, we choose the driver Hamiltonian so that the constraint is satisfied consistently, instead of adding a penalty term. Here, we give the driving Hamiltonian for this problem as where periodic boundary conditions are imposed for index a. As the total Hamiltonian, which consists of H p and H d , is a type of XXZ model, the magnetization q a=1 σ z i,a is conserved for each i. The model is a kind of an ensemble of tight-binding chains. In Fig. 1, which is a schematic of the model, each broken line corresponds to a tight-binding chain. Because each chain only contains one up-spin, there is no interaction between up-spins in a tight-binding chain. However, as represented in Fig. 1, each site of a chain (a broken line) interacts with the corresponding site of neighboring chains through the problem Hamiltonian (solid lines). This leads to the interpretation that a tight-binding chain is affected by the effective local fields that arise from the spins in its neighboring chains.
The benefits of the CQA approach are not only that constraints are consistently satisfied without penalty terms but also, that the dimension of the Hilbert space is reduced considerably. The dimension reduction is essential for simulations based on real-time Schrödinger evolution. In a standard approach to real-time quantum simulations, 2 qN dimensions are required for coloring a graph of N nodes with q colors. However, with the CQA approach described above, the number of required dimensions is reduced to just q N .
To perform the CQA based on real-time quantum simulations, we just deal with the subspace that satisfies the constraint q a=1 σ z i,a = 2 − q. The initial state is taken as the lowest-energy state of H d in the subspace. Although this initial state is not the global ground state, the state can be prepared in the whole Hilbert space of the system by adding additional Zeeman term with an appropriate magnetic field [34]. In the following sections, we refer to the lowest energy state in the subspace as the ground state. Time evolution is calculated by solving the time-dependent Schrödinger equation using the fourth Runge-Kutta method. The annealing schedule is taken as s(t) = t/τ , where t is time and τ is the final time, so that s(0) = 0 and s(τ ) = 1. Time is measured in units ofh/J.

III. EFFECTIVE FIELDS
When s > 0, spins are sensible of effective local fields caused by the interaction between neighboring nodes in a given graph. Note that the problem Hamiltonian (3) can be rewritten as where J ij = J/2 for (ij) ∈ E, otherwise 0. The effective field, which is time dependent, is defined as where |Ψ(s) is the wavefunction at s and Similarly, the fluctuation of the effective field is defined as Considering regular random graphs with connectivity c, the effective field at s = 1 is estimated as Similarly, we have (ĥ eff i,a ) 2 = (cJ 2 /4)[1 + (c − 1)(q − 2) 2 /q 2 ]. Thus, the fluctuation at s = 1 is estimated as Numerical simulations (Fig. 2) demonstrate that both the effective field h eff and its fluctuation ∆ eff change linearly with s. Although the data in Fig. 2 are only applies to i = a = 1, the behavior is independent of i and a. The simulations are conducted with N = 6 q = 4, c = 3, and τ = 20, data is averaged over 500 realizations of random regular graphs. The annealing process for this parameter setting is almost adiabatic [32]. Substituting J = 1, c = 3, and q = 4 into Eqs. (9) and (10), we have h eff = −3/4 and ∆ eff = 3/4. The numerical result for h eff is consistent with the estimated value, while the value of ∆ eff is a little larger than the estimated value.

IV. ENTANGLEMENT ENTROPY
Entanglement entropy (EE) is often used as an entanglement measure, and is defined by where ρ A is the reduced density matrix of the subsystem A. The behavior of EE depends on how the subsystem was chosen. Here, we examine two types of EE: half-chain EE and one-chain EE. The half-chain EE is often used in the case of a one-dimensional spin chain. In our system, the subsystem A is half of the system, i.e., spins with a = 1, . . . , N/2. With the one-chain EE, the subsystem A is taken as the chain (a broken line in Fig. 1) with i = 1.
In this section, we examine the time dependence of the two types of EE divided by the Page value [30,40,41], as defined by While m = n = q N/2 for the half-chain EE, m = q and n = q N −1 for the one-chain EE. In the simulations below, the number of colors and the final time are taken as q = 4 and τ = 20, respectively. The average and standard deviation of EE are calculated using the data from 500 realizations of regular random graphs with connectivity c and the number of nodes N . The time dependence of the half-chain EE is illustrated in Fig. 3. The average half-chain EE increases with time monotonically. Roughly speaking, the larger the connectivity c of random graphs, the larger the half-chain EE. The standard deviation of the half-chain EE tends to increase with time, although no systematic dependence on c or N can be discerned. By contrast, peaks are clearly visible in the standard deviation of the one-chain EE, as shown in Fig. 4, although the time dependence of the average one-chain EE looks similar to that of the half-chain EE. These peaks indicate the MBL transition. Actually, the peak of the standard deviation of EE corresponds to the MBL transition in a disordered quantum spin chain [27][28][29][30]. The standard deviation σ E also shows system-size dependence: The height of the peak is lower and the trough after the peak is shallower for N = 8 than it is for N = 6. In Fig. 4(b), σ E for c = 4 is almost 0 because c = 4 is too large to produce fluctuations for N = 6. We will discuss the second peaks for σ E later in Sec. VI.

V. CONCURRENCE
We here employ concurrence as another measure of entanglement. As concurrence is defined for two spins, it is useful for the comparison of entanglement in systems with different dimensions. The concurrence C i,j in spins i and j is defined from the eigenvalues of the matrix ρ ijρij , where ρ ij is the reduced density matrix, andρ ij = σ y ⊗ σ y ρ * ij σ y ⊗σ y . Suppose that the eigenvalues are . From the conservation of magnetization, the concurrence in the systems considered here can be expressed in a simple form [25,41]: where z = ↑↓ |ρ ij | ↓↑ , x = ↑↑ |ρ ij | ↑↑ , and y = ↓↓ |ρ ij | ↓↓ . The time dependence of the one-chain EE shown in the previous section implies that focusing on one chain is the key to find the relation between entanglement and MBL. Here, we introduce intra-chain concurrence as the concurrence for two spins in the same chain, and compare it with the concurrence in a disordered tight-binding chain.
A. intra-chain concurrence In our system, a spin has two indices, e.g., node i and color a. Let C(i, a; j, b) denote the concurrence for spins (i, a) and (j, b). We define intra-chain concurrence as which is scaled so that C ch = 1 at s = 0. Equation (14) expresses the average of the nearest-neighbor concurrence in each chain. We perform numerical simulations with the same parameter setting as in Sec. IV. The time dependence of the intra-chain concurrence is demonstrated in Fig. 5. The average intra-chain concurrence decreases monotonically. The larger the connectivity c of regular random graphs, the lower the average intra-chain concurrence. The decrease in the intra-chain concurrence implies a decay of the entanglement between neighboring spins. This is caused by effective local fields increasing with time. In contrast, the standard deviation σ ch of the intra-chain concurrence has peaks, resembling that of the one-chain EE. Peak positions in Figs. 5(b) and 5(d) are close to those in Figs. 4(b) and 4(d), which implies that these peaks also indicate an MBL transition.

B. disordered-chain concurrence
As mentioned earlier, there is an analogy between the system considered and disordered spin chains. Can similar peaks therefore appear in the standard deviation of concurrence in a disordered spin chain? To answer the question, let us consider a disordered tight-binding chain whose problem and driving Hamiltonians are respectively given by where h a is a local random field, and c † a and c a are the creation and annihilation operators of a spinless fermion, respectively. Suppose that there is only one fermion in a chain. Instead of real random numbers with a uniform distribution, we take h a = mh eff , where h eff is the effective field given in Sec. II. Integer m is given by m = i m i (i = 1, . . . , c), where m i = 1 with a probability 1/q and m i = −1 with a probability 1 − 1/q. Disordered-chain concurrence is simply defined as which is scaled so that C ds = 1 at s = 0. Figure 6 shows the time dependence of the disorderedchain concurrence. The behavior is different from that of Fig. 5. The average of disordered-chain concurrence decreases monotonically, and its standard deviation increases almost monotonically. We see no peak appearing in the middle of an annealing process, which implies that the MBL transition does not occur in this system. In fact, Anderson localization, rather than MBL, occurs in a disordered tight-binding chain. Because there is no many-body interaction, MBL does not occur in the tightbinding chain.

VI. DISCUSSION AND CONCLUSIONS
In Figs. 4 and 5, each curve for the standard deviation of the one-chain EE and the intra-chain concurrence exhibits a peak followed by a trough in the middle of a CQA process. Following the trough, the standard deviation increases again. The second peak is characteristic of the entanglement evolution in our system. In fact, in the case of a one-dimensional disordered spin chain, the standard deviation of the half-chain EE only has one peak [27][28][29][30]. It corresponds to the first peak in our case and indicates the fluctuation associated with the MBL transition. In the late stage of a CQA process, the wave-function is very close to the ground state of the problem Hamiltonian. As the problem Hamiltonian represents a classical system, the second peak reflects the variations in effective random fields rather than quantum effects.
We have found evidence of the MBL transition in a CQA process. The MBL transition corresponds to peaks in the standard deviation of the one-chain EE [Figs. 4(b) and 4(d)] and the intra-chain concurrence [Figs. 5(b) and 5(d)]. In a CQA process, intra-chain connection decreases, and the strength of the effective random fields increases. However, the increase in the strength of the fields, i.e., disorder, is not sufficient to cause an MBL transition. This is evidenced by the fact that the standard deviation of the disordered-chain concurrence only increases monotonically [Figs. 6(b) and 6(d)]. The characteristic behavior of entanglement in our system, compared with that of the disordered-chain concurrence, suggests that inter-chain interaction as well as effective fields are an essential factors for an MBL transition.