Solving continuous variable optimization problems by factorization machine quantum annealing (FMQA) demonstrates the potential of Ising machines to be extended as a solver for integer and real optimization problems. However, the details of the Hamiltonian function surface obtained by the factorization machine (FM) have been overlooked. This study shows that in the widely used case where real numbers are represented by a combination of binary variables, the function surface of the Hamiltonian obtained by FM can be very noisy. This noise interferes with the inherent capabilities of quantum annealing and is likely to be a substantial cause of problems previously considered unsolvable because of the limitations of FMQA performance. The origin of the noise is identified and a simple, general method is proposed to prevent its occurrence. The generalization performance of the proposed method and its ability to solve practical problems is demonstrated.
Solving continuous variable optimization problems by factorization machine quantum annealing (FMQA) demonstrates the potential of Ising machines to be extended as a solver for integer and real optimization problems. However, the details of the Hamiltonian function surface obtained by the factorization machine (FM) have been overlooked. This study shows that in the widely used case where real numbers are represented by a combination of binary variables, the function surface of the Hamiltonian obtained by FM can be very noisy. This noise interferes with the inherent capabilities of quantum annealing and is likely to be a substantial cause of problems previously considered unsolvable because of the limitations of FMQA performance. The origin of the noise is identified and a simple, general method is proposed to prevent its occurrence. The generalization performance of the proposed method and its ability to solve practical problems is demonstrated.
Black-box optimization (BBO) is used in materials design, drug discovery, and hyperparameter tuning in machine learning. The world is experiencing several of these problems. In this review, a factorization machine with quantum annealing or with quadratic-optimization annealing (FMQA) algorithm to realize fast computations of BBO using Ising machines (IMs) is discussed. The FMQA algorithm uses a factorization machine (FM) as a surrogate model for BBO. The FM model can be directly transformed into a quadratic unconstrained binary optimization model that can be solved using IMs. This makes it possible to optimize the acquisition function in BBO, which is a difficult task using conventional methods without IMs. Consequently, it has the advantage of handling large BBO problems. To be able to perform BBO with the FMQA algorithm immediately, we introduce the FMQA algorithm along with Python packages to run it. In addition, we review examples of applications of the FMQA algorithm in various fields, including physics, chemistry, materials science, and social sciences. These successful examples include binary and integer optimization problems, as well as more general optimization problems involving graphs, networks, and strings, using a binary variational autoencoder. We believe that BBO using the FMQA algorithm will become a key technology in IMs, including quantum annealers.
This paper presents an initialization method that can approximate a given approximate Ising model with a high degree of accuracy using a factorization machine (FM), a machine learning model. The construction of an Ising models using an FM is applied to black-box combinatorial optimization problems using factorization machine with quantum annealing (FMQA). It is anticipated that the optimization performance of FMQA will be enhanced through an implementation of the warm-start method. Nevertheless, the optimal initialization method for leveraging the warm-start approach in FMQA remains undetermined. Consequently, the present study compares initialization methods based on random initialization and low-rank approximation, and then identifies a suitable one for use with warm-start in FMQA through numerical experiments. Furthermore, the properties of the initialization method by the low-rank approximation for the FM are analyzed using random matrix theory, demonstrating that the approximation accuracy of the proposed method is not significantly influenced by the specific Ising model under consideration. The findings of this study will facilitate advancements of research in the field of black-box combinatorial optimization through the use of Ising machines.
Ising machines including quantum annealers have attracted attention as efficient solvers for combinatorial optimization problems (COPs). To use Ising machines, the objective function and constraints of a COP must be expressed in a QUBO (quadratic unconstrained binary optimization) form. However, their application range is practically limited since some COPs are difficult to formulate in a QUBO form, or require a large number of binary variables. Quadratic three-dimensional assignment problem (Q3AP) is a challenging problem for Ising machines to solve since the number of required binary variables rapidly increases with the problem size N. In this paper, to overcome the difficulty, we propose a method to reduce the number of necessary variables fromN3to2N2by using a factorization machine with quantum annealing (FMQA) to estimate QUBO through machine learning. Experimental results with a real Ising machine demonstrate that the proposed method can obtain the best-known solution in a short time.
Factorization Machine (FM) is the most commonly used model to build a recommendation system since it can incorporate side information to improve performance. However, producing item suggestions for a given user with a trained FM is time-consuming. To address this problem, we propose a quadratic unconstrained binary optimization (QUBO) scheme to combine with FM and apply quantum annealing (QA) computation. Compared to classical methods, this hybrid algorithm provides a fast sub-optimal sampling of good user suggestions. We then demonstrate the aforementioned computational behavior on current noisy intermediate-scale quantum (NISQ) hardware by experimenting with a real example on a D-Wave annealer.
Quantum annealing (QA) is a heuristic optimization method for efficiently solving complex combinatorial problems with discrete variables. Thus, quantum annealing is proper for optical designs with multiple materials. The possible designs with multiple materials and various geometries form a huge solution space that is challenging for optimization. Due to their wave-like behavior, optical systems are typically coupled over long distances and can interfere with one another. Mathematically, this leads to high-order couplings in quantum annealing that are not supported by current quantum hardware. Factorization machines (FMs) can reduce high-order terms and provide an effective quadratic form suitable for quantum hardware. In this work, we adopt the previously proposed factorization machines with quantum annealing (FMQA) to design selective thermal absorbers. The FQMA, consisting of iterations between QA and FMs, has many hyperparameters, such as the initial inputs, latent factors, and selection of trained data. We study how these hyperparameters improve the convergence of the FMQA. We find that a balance between the initial random samplings and the sampling by QA is crucial for optimization. The principle underlying the entire optimization process of the FMQA requires a deliberate balance between depth and breadth, alongside an adiabatic approach. We design selective thermal absorbers to only absorb light with wavelengths ranging from 8 to 13 microns. We design the absorbers with silicon dioxide, PMMA, and silver. The FMQA with the optimized hyperparameters shows stably better figures of merit.
Quantum annealing (QA) uses the principles of quantum mechanics for solving unconstrained optimization problems. Since its initial proposal, there has been much interest in the search for practical problems for which QA has advantage over classical algorithms. Although it is believed that quantum computers cannot in general solve NP-complete problems efficiently, there has been evidence suggesting that effects such as quantum tunneling can bring quantum speedup over classical computation for some problems in NP. In this dissertation, we consider two problems that have both theoretical and practical importance. One is the Set Cover with Pairs problem. The other is the Factorization problem. The Set Cover with Pairs (SCP) problem is a generalization of the Set Cover problem. It requires each element in a set to be covered by at least two objects instead of one as in the Set Cover problem. The SCP problem is believed to be not only NP-hard, but also hard to approximate. In this dissertation, we explicitly construct the Ising Hamiltonians whose ground states encode the solutions of SCP instances. The resulting Ising Hamiltonian has the appealing property that the control precision required for encoding an SCP instance scales only linearly with the number of objects in the covering set. The Integer Factorization problem is of great interest in the quantum community because it is difficult to solve using classical methods and its importance in cryptography. In this dissertation, we propose a general framework for solving factorization problems using quantum annealing, by mapping the framework to an Ising Hamiltonian. We develop several methods to consider the specific requirements such as control precision, spin-spin connection in the current Ising machines. We test the methods on the latest D-Wave 2000Q machine.