Quantum annealing is a heuristic algorithm for searching the ground state of an Ising model. Heuristic algorithms aim to obtain near-optimal solutions with a reasonable computation time. Accordingly, many algorithms have so far been proposed. In general, the performance of heuristic algorithms strongly depends on the instance of the combinatorial optimization problem to be solved because they escape the local minima in different ways. Therefore, combining several algorithms to exploit their complementary strength is effective for obtaining highly accurate solutions for a wide range of combinatorial optimization problems. However, quantum annealing cannot be used to improve a candidate solution obtained by other algorithms because it starts from an initial state where all spin configurations are found with a uniform probability. In this study, we propose an Ising formulation of integer optimization problems to utilize quantum annealing in the iterative improvement strategy. Our formulation exploits the biased sampling of degenerated ground states in transverse magnetic field quantum annealing. We also analytically show that a first-order phase transition is successfully avoided for a fully connected ferromagnetic Potts model if the overlap between a ground state and a candidate solution exceeds a threshold. The proposed formulation is applicable to a wide range of integer optimization problems and enables us to hybridize quantum annealing with other optimization algorithms.
Quantum annealing is a heuristic algorithm for searching the ground state of an Ising model. Heuristic algorithms aim to obtain near-optimal solutions with a reasonable computation time. Accordingly, many algorithms have so far been proposed. In general, the performance of heuristic algorithms strongly depends on the instance of the combinatorial optimization problem to be solved because they escape the local minima in different ways. Therefore, combining several algorithms to exploit their complementary strength is effective for obtaining highly accurate solutions for a wide range of combinatorial optimization problems. However, quantum annealing cannot be used to improve a candidate solution obtained by other algorithms because it starts from an initial state where all spin configurations are found with a uniform probability. In this study, we propose an Ising formulation of integer optimization problems to utilize quantum annealing in the iterative improvement strategy. Our formulation exploits the biased sampling of degenerated ground states in transverse magnetic field quantum annealing. We also analytically show that a first-order phase transition is successfully avoided for a fully connected ferromagnetic Potts model if the overlap between a ground state and a candidate solution exceeds a threshold. The proposed formulation is applicable to a wide range of integer optimization problems and enables us to hybridize quantum annealing with other optimization algorithms.
The Potts model is a generalization of the Ising model with Q>2 components. In the fully connected ferromagnetic Potts model, a first-order phase transition is induced by varying thermal fluctuations. Therefore, the computational time required to obtain the ground states by simulated annealing exponentially increases with the system size. This study analytically confirms that the transverse magnetic-field quantum annealing induces a first-order phase transition. This result implies that quantum annealing does not exponentially accelerate the ground-state search of the ferromagnetic Potts model. To avoid the first-order phase transition, we propose an iterative optimization method using a half-hot constraint that is applicable to both quantum and simulated annealing. In the limit of Q→∞, a saddle point equation under the half-hot constraint is identical to the equation describing the behavior of the fully connected ferromagnetic Ising model, thus confirming a second-order phase transition. Furthermore, we verify the same relation between the fully connected Potts glass model and the Sherrington--Kirkpatrick model under assumptions of static approximation and replica symmetric solution. The proposed method is expected to obtain low-energy states of the Potts models with high efficiency using Ising-type computers such as the D-Wave quantum annealer and the Fujitsu Digital Annealer.
Reverse annealing is a variant of quantum annealing that starts from a given classical configuration of spins (qubits). In contrast to the conventional formulation, where one starts from a uniform superposition of all possible states (classical configurations), quantum fluctuations are first increased and only then decreased. One then reads out the state as a proposed solution to the given combinatorial optimization problem. We formulate a mean-field theory of reverse annealing using the fully connected ferromagnetic p-spin model, with and without random longitudinal fields, and analyze it in order to understand how and when reverse annealing is effective at solving this problem. We find that the difficulty arising from the existence of a first-order quantum phase transition, which leads to an exponentially long computation time in conventional quantum annealing, is circumvented in the context of this particular problem by reverse annealing if the proximity of the initial state to the (known) solution exceeds a threshold. Even when a first-order transition is unavoidable, the difficulty is mitigated due to a smaller jump in the order parameter at a first-order transition, which implies a larger rate of quantum tunneling. Reverse annealing has not been studied analytically before, and this study paves the way toward a systematic understanding of this relatively unexplored protocol in a broader context.
We show that the quantum approximate optimization algorithm (QAOA) can construct, with polynomially scaling resources, the ground state of the fully connected p-spin Ising ferromagnet, a problem that notoriously poses severe difficulties to a vanilla quantum annealing (QA) approach due to the exponentially small gaps encountered at first-order phase transition for $p\ensuremath{\ge}3$. For a target ground state at arbitrary transverse field, we find that an appropriate QAOA parameter initialization is necessary to achieve good performance of the algorithm when the number of variational parameters 2P is much smaller than the system size N because of the large number of suboptimal local minima. Instead, when P exceeds a critical value ${P}_{N}^{*}\ensuremath{\propto}N$, the structure of the parameter space simplifies, as all minima become degenerate. This allows achieving the ground state with perfect fidelity with a number of parameters scaling extensively with N and with resources scaling polynomially with N.
A combinatorial optimization problem is the problem of minimizing the energy function among many combinations of variables, which is often difficult to solve with conventional classical computers. Recently, Ising machines, including quantum annealers, have gained attention as a promising architecture for efficiently solving combinatorial optimization problems. Among various methods for solving such problems using Ising machines, one prominent approach is the three-stage annealing method. The approach effectively solves a combinatorial optimization problem, utilizing an initial solution, but it performs the annealing process only once. Repeating the annealing process several times may enhance the solution more efficiently. In this paper, we propose a novel hybrid iterative annealing method that consists of an initial process using a classical computer, an annealing process using a quantum annealer, and a correction process/selection process using a classical computer. The proposed method repeats the annealing process and the correction process/selection process until the solution is sufficiently converged. In the experimental evaluations through the two types of typical combinatorial optimization problems, the proposed method shows improvements by up to 50.5% compared to the three-stage annealing method.
A combinatorial optimization problem is the problem of minimizing the energy function among many combinations of variables, which is often difficult to solve with conventional classical computers. Recently, Ising machines, including quantum annealers, have gained attention as a promising architecture for efficiently solving combinatorial optimization problems. Among various methods for solving such problems using Ising machines, one prominent approach is the three-stage annealing method. The approach effectively solves a combinatorial optimization problem, utilizing an initial solution, but it performs the annealing process only once. Repeating the annealing process several times may enhance the solution more efficiently. In this paper, we propose a novel hybrid iterative annealing method that consists of an initial process using a classical computer, an annealing process using a quantum annealer, and a correction process/selection process using a classical computer. The proposed method repeats the annealing process and the correction process/selection process until the solution is sufficiently converged. In the experimental evaluations through the three types of typical combinatorial optimization problems, the proposed method shows improvements by up to 54.0% compared to the three-stage annealing method.
Combinatorial optimization has wide and high-value applications in many fields of science and industry, but solving general combinatorial optimization problems is non-deterministic polynomial time (NP) hard. Many such problems can be mapped onto the ground-state-search problems of the Ising model. Here, an iterative quantum algorithm based on quantum gradient descent to solve combinatorial optimization problems is introduced, where the initial state of a quantum register evolves over several iterations to a good approximation of the Ising-Hamiltonian ground state. We verified the effectiveness of the proposed algorithm in solving the MaxCut problem for different types of undirected graphs by numerical simulations, and analyzed the robustness of the algorithm to errors by simulating random error and Gaussian error. We compared the performance of the algorithm with the quantum approximate optimization algorithm, and the results indicate that the proposed algorithm has comparable convergence performance. We also verified the feasibility of the algorithm by conducting experiments on a real quantum computer through the quantum cloud platform. Our work provides a potential method for solving combinatorial optimization problems on future quantum devices without the use of complex classical optimization loops.