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arXiv:2501.03518

Transfer Learning for Deep-Unfolded Combinatorial Optimization Solver with Quantum Annealer

Ryo Hagiwara, Shunta Arai, Satoshi Takabe

Published Jan 7, 2025• 0 citations
arXivHTMLPDF

Abstract

Quantum annealing (QA) has attracted research interest as a sampler and combinatorial optimization problem (COP) solver. A recently proposed sampling-based solver for QA significantly reduces the required number of qubits, being capable of large COPs. In relation to this, a trainable sampling-based COP solver has been proposed that optimizes its internal parameters from a dataset by using a deep learning technique called deep unfolding. Although learning the internal parameters accelerates the convergence speed, the sampler in the trainable solver is restricted to using a classical sampler owing to the training cost. In this study, to utilize QA in the trainable solver, we propose classical-quantum transfer learning, where parameters are trained classically, and the trained parameters are used in the solver with QA. The results of numerical experiments demonstrate that the trainable quantum COP solver using classical-quantum transfer learning improves convergence speed and execution time over the original solver.

DOI
10.48550/arxiv.2501.03518

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Quantum annealing (QA) has attracted research interest as a sampler and combinatorial optimization problem (COP) solver. A recently proposed sampling-based solver for QA significantly reduces the required number of qubits, being capable of large COPs. In relation to this, a trainable sampling-based COP solver has been proposed that optimizes its internal parameters from a dataset by using a deep learning technique called deep unfolding. Although learning the internal parameters accelerates the convergence speed, the sampler in the trainable solver is restricted to using a classical sampler owing to the training cost. In this study, to utilize QA in the trainable solver, we propose classical-quantum transfer learning, where parameters are trained classically, and the trained parameters are used in the solver with QA. The results of numerical experiments demonstrate that the trainable quantum COP solver using classical-quantum transfer learning improves convergence speed and execution time over the original solver.
Solving combinatorial optimization problems (COPs) is a promising application of quantum computation, with the Quantum Approximate Optimization Algorithm (QAOA) being one of the most studied quantum algorithms for solving them. However, multiple factors make the parameter search of the QAOA a hard optimization problem. In this work, we study transfer learning (TL), a methodology to reuse pre-trained QAOA parameters of one problem instance into different COP instances. This methodology can be used to alleviate the necessity of classical optimization to find good parameters for individual problems. To this end, we select small cases of the traveling salesman problem (TSP), the bin packing problem (BPP), the knapsack problem (KP), the weighted maximum cut (MaxCut) problem, the maximal independent set (MIS) problem, and portfolio optimization (PO), and find optimal βββ and γγγ parameters for p layers. We compare how well the parameters found for one problem adapt to the others. Among the different problems, BPP is the one that produces the best transferable parameters, maintaining the probability of finding the optimal solution above a quadratic speedup over random guessing for problem sizes up to 42 qubits and p = 10 layers. Using the BPP parameters, we perform experiments on IonQ Harmony and Aria, Rigetti Aspen-M-3, and IBM Brisbane of MIS instances for up to 18 qubits. The results indicate that IonQ Aria yields the best overlap with the ideal probability distribution. Additionally, we show that cross-platform TL is possible using the D-Wave Advantage quantum annealer with the parameters found for BPP. We show an improvement in performance compared to the default protocols for MIS with up to 170 qubits. Our results suggest that there are QAOA parameters that generalize well for different COPs and annealing protocols.
A combinatorial optimization problem (COP) is the problem of finding the optimal solution in a finite set. When the size of the feasible solution set is large, the complexity of the problem increases, and it is not easy to solve in a reasonable time with the current classical computer technology. Quantum annealing (QA) is a method that replaces classical simulated annealing (SA) methods that do not solve these cases. Therefore, several attempts have been made to solve this problem using a special-purpose quantum annealer to which the QA method is applied. In this survey, we analyze recent studies that solve real-scale COPs using quantum annealers. Through this, we discuss how to reduce the size of the COP to be input to overcome the hardware limitations of the existing quantum annealer. Additionally, we demonstrated the applicability of quantum annealer to COP on a practical scale by comparing and analyzing the results of the classical simulated annealing (SA) and QA method from each study.
Quantum annealing (QA) has the potential to significantly improve solution quality and reduce time complexity in solving combinatorial optimization problems compared to classical optimization methods. However, due to the limited number of qubits and their connectivity, the QA hardware did not show such an advantage over classical methods in past benchmarking studies. Recent advancements in QA with more than 5000 qubits, enhanced qubit connectivity, and the hybrid architecture promise to realize the quantum advantage. Here, we use a quantum annealer with state-of-the-art techniques and benchmark its performance against classical solvers. To compare their performance, we solve over 50 optimization problem instances represented by large and dense Hamiltonian matrices using quantum and classical solvers. The results demonstrate that a state-of-the-art quantum solver has higher accuracy (~0.013%) and a significantly faster problem-solving time (~6561×) than the best classical solver. Our results highlight the advantages of leveraging QA over classical counterparts, particularly in hybrid configurations, for achieving high accuracy and substantially reduced problem solving time in large-scale real-world optimization problems.
Quantum annealing realizes quantum computers specialized for combinatorial optimization problems (COPs). A COP is formulated as a Hamiltonian, and quantum annealing obtains a solution by finding the ground state of the Hamiltonian. The ease of finding a solution depends on the weights assigned to the cost and constraint functions when formulating the problem. In other words, parameter tuning is essential in solving problems with quantum annealing. In the present paper, the problem of searching an ordering that reduces the fill-in for a sparse direct solver is formulated as a Hamiltonian, and quantum annealing finds the solution to this problem. We discuss the necessity and effectiveness of parameter tuning for solving COPs with quantum annealing. The results after weight tuning show that we can improve the rate of an optimal solution obtained by a maximum of 94% for5 × 5{5\,\times \,5}5×5matrices, 68% for6 × 6{6\,\times \,6}6×6matrices, and 27% for7 × 7{7\,\times \,7}7×7matrices. Moreover, it is shown that giving high weights to the constraints we want to satisfy will not provide an optimal solution.
Local quantum annealing (LQA), an iterative algorithm, is designed to solve combinatorial optimization problems. It draws inspiration from QA, which utilizes adiabatic time evolution to determine the global minimum of a given objective function. In the original LQA, the classical Hamiltonian is minimized via gradient descent. The performance of LQA depends on the choice of the parameters. Owing to the nonconvex nature of the original cost function, LQA often becomes trapped in local minima, limiting its effectiveness. To address this challenge, we combine LQA with a deep unfolding scheme, which enables us to tune the parameters from the training data via back-propagation. As a demonstration, we apply the deep unfolded LQA to the Sherrington-Kirkpatrick model, which is a fundamental model in statistical physics. Our findings exhibit that deep unfolded LQA outperforms the original LQA, exhibiting remarkable convergence speed and performance improvement. As the trained parameters can be generalized to unknown instances and different system sizes, our results have significant practical implications and provide valuable insights for real-world applications. Published by the American Physical Society 2024
Recently, quantum computers have drawn attention to their potential to solve problems faster than classical computers. However, quantum hardware’s limited practicality and scalability have led to an interest in alternative approaches to solving optimization problems. One such approach is classical quantum-inspired annealers, which provide efficient and scalable solutions for combinatorial optimization problems (COPs) using classical hardware. To use annealers, COPs must be formulated as quadratic unconstrained binary optimization (QUBO) forms. Current tools require coding expertise and manual parameter tuning, posing barriers to entry. To address these challenges, we developed a user-friendly software toolchain that offers several advantages. Our toolchain features a friendly input and output interface, an automated parameter tuner, and a library of commonly encountered COPs. Our software toolchain’s accessibility promotes the use of quantum-inspired annealers and accelerates the development of practical solutions for real-world problems.