Variational optimization of quantum annealing schedule for the Lechner-Hauke-Zoller scheme

The annealing schedule is optimized for a parameter in the Lechner-Hauke-Zoller (LHZ) scheme for quantum annealing designed for the all-to-all-interacting Ising model representing generic combinatorial optimization problems. We adapt the variational approach proposed by Matsuura et al. (arXiv:2003.09913) to the annealing schedule of a term representing a constraint for variables intrinsic to the LHZ scheme. Numerical results for a simple ferromagnetic model and the spin glass problem show that non-monotonic annealing schedules optimize the performance measured by the residual energy and the final ground-state fidelity. This improvement does not accompany a notable increase in the instantaneous energy gap, which suggests the importance of a dynamical viewpoint in addition to static analyses in the study of practically-relevant diabatic processes in quantum annealing.


I. INTRODUCTION
Quantum annealing (QA) is a quantum-mechanical metaheuristic for combinatorial optimization problems [1][2][3][4][5][6][7] and is related to adiabatic quantum computation [8,9]. An important goal of QA is to find the ground state of a classical Ising model, to which the cost function of a combinatorial optimization problem can be mapped [10]. One of the challenges in experimental implementation of QA is the realization of long-range interactions in the Ising model on locally-connected qubits on the device, especially in the superconducting technology [11]. The standard approach to this problem is embedding [12,13], where a logical qubit is represented by a set of strongly coupled physical qubits, but it leads to an overhead in the number of qubits as well as possible errors caused by misalignment among physical qubits. The latter problem is solved by the scheme proposed by Lechner, Hauke, and Zoller (LHZ) [14], in which a pair of interacting logical qubits are replaced by a single physical qubit with an appropriate constraint among the latter qubits being introduced to reproduce the original Hamiltonian. The classical part of the Hamiltonian in the LHZ scheme is composed of local longitudinal fields representing the original problem Hamiltonian and nearest-neighbor fourbody interactions for the constraint mentioned above. This scheme has been studied toward practical implementation [15][16][17][18][19][20][21][22]. It was also reported that the performance of QA under the LHZ scheme can be enhanced by counter-diabatic control of system parameters [23] and inhomogeneous driving of the transverse field [24]. In addition, a mean-field analysis assuming adiabatic parameter control shows that nonlinear driving of the coefficient of the constraint term is effective to avoid a problematic first-order phase transition [25].
Theoretical analyses of QA have traditionally been carried out in the framework of adiabatic computation, in which the * y-susa@nec.com system stays in the instantaneous static ground state during the annealing process. However, it has gradually been recognized that investigation of diabatic (i.e., non-static) processes may at least be equally important. The reasons are, first, a real device usually operates away from the adiabatic regime, second, a shorter annealing time would be beneficial to avoid decoherece, third, repeating short annealing processes may sometimes lead to better results than a single long adiabatic process, and last, diabatic annealing cannot be efficiently simulated classically generally, which may lead to a quantum speedup. See Ref. [26] for a review.
One of the promising approaches to diabatic annealing is a variational method recenlty proposed by Matsuura et al. [27,28], in which one iteratively improves the time dependence of coefficients in the Hamiltonian to minimize the energy expectation value. This is an idea inspired by variational quantum eigensolver [29,30], a quantum-classical hybrid algorithm, to find the eigenstate of a quantum system in the noisy near-term quantum device. They applied the variational method to a small-scale quantum chemistry problem and an Ising model with frustration and concluded that much shorter annealing times (strongly diabatic processes) are sufficient to achieve satisfactory results.
Given these developments, we follow Refs. [27,28] and adapt the method to the LHZ scheme toward performance enhancement under diabatic conditions. We numerically show that variational optimization of the annealing schedule of the constraint term in the LHZ scheme leads to a significant improvement of performance measured by the residual energy and ground-state fidelity using small-scale examples of a simple ferromagnet and the spin glass problem.
This paper is organized as follows. Section II introduces the LHZ scheme, and Sec. III explains the variational algorithm applied to the LHZ scheme. Numerical results are presented in Sec. IV and discussions are given in Sec. V.

II. LECHNER-HAUKE-ZOLLER SCHEME
We first recapitulate the LHZ scheme [14] to set the stage for developments in the following sections. The Hamiltonian describing QA has traditionally been chosen aŝ whereˆrepresents the component of the Pauli operator of qubit (spin) . The total annealing time is . The first term in the Hamiltonian is the transverse field, and its coefficient is decreased from 1 ( = 0) to 0 ( = ). The second term is the problem Hamiltonianˆfor the cost function of combinatorial optimization problem written in terms of the component of the Pauli operator. More explicitly, the problem Hamiltonian is often the fully connected Ising model, The LHZ scheme converts the all-to-all connectivity in the above Hamiltonian to local terms. The interactionˆˆ, where {ˆ} are logical qubits, is replaced by a longitudinalfield termˆ, whereˆis for a physical qubit, and four-body interactions among neighboring physical qubits are added as a constraint to recover the original Hamiltonian. The resulting Hamiltonian is written aŝ where = ( − 1)/2 is the total number of physical qubits and = ( −1) ( −2)/2 is the number of constraints. As long as the product of four qubits takes the value +1, the ground state of the first term is equivalent to that of the original Hamiltonian of Eq. (2). A transverse field is added to the above Hamiltonian in the usual way as in Eq. (1), withˆL HZ replacingˆ. Reference [31] reports that an appropriate value of can be determined according to the class of the problem and the system size. Reference [25] develops a static analysis and finds it better to control the value of differently from those of other coefficients to avoid a first-order phase transition. In the present paper, we introduce time dependence in the coefficient as ( / ) and iteratively update its time functional form to maximize the performance.

III. VARIATIONAL ALGORITHM
We adapt the variational method of Ref. [28] to the determination of optimal time dependence of the coefficient ( / ). The total Hamiltonian is noŵ For practical purposes of variational optimization, we discretize the time as / = /( + 1) ( = 0, 1, 2, · · · , + 1) and represent the function ( / ) at discrete points = ( /( + 1)) with the boundary condition 0 = (0) = 0 and +1 = (1) = 1. The principle of variational optimization is to iteratively change the value of toward minimization of the expectation value of the final Hamiltonianˆ(1) by repeating the process of short-time quantum annealing. The value of the coefficient at the th iteration will be denoted as ( ) . The update of the value of is performed on a classical computer. After each step of classical update, we fit a polynomial of degree to the discrete values of { ( +1) } and run the next annealing process using this ( +1) ( / ) following the continuous-time Schrödinger dynamics, a quantum process. This is thus a classical-quantum hybrid algorithm. Figure 1 summarizes the whole algorithm.

IV. NUMERICAL RESULT
We now show the results obtained by numerically solving the Schrödinger equation.

A. Ferromagnetic model
First we test the framework by the simplest ferromagnetic model with = 0.5 with the results in Fig. 2. The number of the variational parameters is = 3, and the order of fitting polynomial is  In an effort to clarify the significance of the obtained annealing schedule, we have calculated the energy gap between the instantaneous ground state and the first-excited state as a function of time. The result is displayed in Fig. 3. We also plot the corresponding energy gaps for the linear function ( / ) = / and a monotonically increasing nonlinear function ( / ) = ( / ) 5 , the latter of which avoids the firstorder transition in the static (adiabatic) phase diagram [25]. The minimum gap of the variationally optimal schedule (a) is almost the same as the simple linear case (b), and the nonlinear schedule (c) has a little larger gap, possibly reflecting the avoidance of first-order transition. This latter slightness of the increase of the gap would come from smallness of the system size. These results may suggest the difficulty to predict dynamical, diabatic, properties from the energy gap. We should nevertheless be cautious since the the data on the energy gap may reflect finite-size effects.
We have further calculated the probabilities of three lowest instantaneous eigenstates as functions of time for three schedules used in Fig. 3. As seen in Fig. 4(a), the ground state probability drops around the middle of annealing, and but finally increases toward 1 for the variationally obtained schedule. On the other hand, as observed in Figs succeeds to keep the system close to the instantaneous ground state at least for the present small-size problem with a short annealing time, where the traditional method fails. In all panels, we use the same instances of randomness.

B. Spin glass problem
Let us next consider the spin glass problem with uniformly chosen from the range −0.5 < < 0.5. This independent random choice of for local field under the constraint of four-body interaction realizes the spin glass problem in the LHZ scheme [14]. It is useful to focus on the final ground state fidelity because there exist many excited states with the energy very close to the ground state energy in the spin glass problem, which makes it difficult to measure the performance by the residual energy.   Fig. 6(a). The dotted curve displays (1− horizontal axis)/(horizontal axis) representing the upper bound of the fidelity improvement and the data points near the dotted curve show the cases that the variational algorithm improves the fidelity to 1. It is observed that an easy instance for the linear schedule is more likely to benefit from the variational method than a difficult instance. The ground state fidelity is improved mostly over 40 percent regardless of the difficulty. Figure 6(c) is the histogram of the degree of improvement shown in 6(b). Most of the instances have values between 0.4 and 0.8 with scattered outliers. The lower tail of the histogram in Fig. 6(c) may correspond to the final states with the energy close to the ground state but the fidelity or the hamming distance far away. Figure 6(d) is for optimized schedules, most of which look similar to the ferromagnetic case in Fig. 2(c).

V. DISCUSSION
We have adapted the variational method of Refs. [27,28] to the determination of optimal annealing schedule of the constraint term in the LHZ scheme for quantum annealing in the diabatic regime. Numerical studies of small-size systems with random (spin glass) and non-random (ferromagnetic) interactions reveal that non-monotonic schedules with a single local minimum lead to significant improvements in the ground-state fidelity in almost all cases. Although the present study was performed numerically on the classical computer due to the lack of quantum hardware for the LHZ scheme, the result may be suggestive for the design of future hardware under development. A similar approach should be applicable to other terms of the Hamiltonian, and it may serve as a generic framework to find the best possible diabatic annealing schedule not just for the LHZ scheme but also for the standard annealing model. In particular, the present method may greatly mitigate the difficulty of an appropriate time-dependent control of the coefficient of the constraint term, which almost always shows up when we embed a practical problem on the hardware with nearest-neighbor interactions only.
Another advantage is in the short annealing time for a single iteration, which allows the quantum device to operate with less effects of noise and thus with more reliable outputs, as is always the case for variational classical-quantum hybrid algorithms. Although quantum annealing is relatively stable against noise [37], it should nevertheless be better to run a quantum processor under an environment closer to isolation.

ACKNOWLEDGMENTS
This paper is based on results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO). In some calculation, we have used QuTip python library [38,39].