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arXiv:2403.02045 · Quantum

Recursive Quantum Relaxation for Combinatorial Optimization Problems

Ruho Kondo, Yuki Sato, Rudy Raymond, Naoki Yamamoto

Published Jan 15, 2025• 10 citations
arXivHTMLPDF

Abstract

Quantum optimization methods use a continuous degree-of-freedom of quantum states to heuristically solve combinatorial problems, such as the MAX-CUT problem, which can be attributed to various NP-hard combinatorial problems. This paper shows that some existing quantum optimization methods can be unified into a solver to find the binary solution which is most likely measured from the optimal quantum state. Combining this finding with the concept of quantum random access codes (QRACs) for encoding bits into quantum states on fewer qubits, we propose an efficient recursive quantum relaxation method called recursive quantum random access optimization (RQRAO) for MAX-CUT. Experiments on standard benchmark graphs with several hundred nodes in the MAX-CUT problem, conducted in a fully classical manner using a tensor network technique, show that RQRAO not only outperforms the Goemans-Williamson and recursive QAOA methods, but also is comparable to state-of-the-art classical solvers. The code is available at https://github.com/ToyotaCRDL/rqrao.

DOI
10.22331/q-2025-01-15-1594

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Quantum optimization methods use a continuous degree-of-freedom of quantum states to heuristically solve combinatorial problems, such as the MAX-CUT problem, which can be attributed to various NP-hard combinatorial problems. This paper shows that some existing quantum optimization methods can be unified into a solver to find the binary solution which is most likely measured from the optimal quantum state. Combining this finding with the concept of quantum random access codes (QRACs) for encoding bits into quantum states on fewer qubits, we propose an efficient recursive quantum relaxation method called recursive quantum random access optimization (RQRAO) for MAX-CUT. Experiments on standard benchmark graphs with several hundred nodes in the MAX-CUT problem, conducted in a fully classical manner using a tensor network technique, show that RQRAO not only outperforms the Goemans-Williamson and recursive QAOA methods, but also is comparable to state-of-the-art classical solvers. The code is available at https://github.com/ToyotaCRDL/rqrao.
There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.
Among quantum algorithms, the quantum approximation optimization algorithm (QAOA) is an algorithm that finds approximate solutions. Recursive QAOA is proposed to overcome the limitation that QAOA has lower performance than Goemans-Williamson algorithm in the MAX-CUT problem [Physical Review Letter, 125, 260504, 2020]. In this paper, we introduce a method with lower complexity than RQAOA and compare its numerical results in the MAX-CUT problem. At the same level, the performance of the proposed method is lower than that of RQAOA, but the number of computations of quantum and classical parts is very small.
Quantum Random Access Optimizer (QRAO) is a quantum-relaxation based optimization algorithm proposed by Fuller et al. that utilizes Quantum Random Access Code (QRAC) to encode multiple variables of binary optimization in a single qubit. The approximation ratio bound of QRAO for the maximum cut problem is 0.5550.5550.555 if the bit-to-qubit compression ratio is 333x, while it is 0.6250.6250.625 if the compression ratio is 222x, thus demonstrating a trade-off between space efficiency and approximability. In this research, we extend the quantum-relaxation by using another QRAC which encodes three classical bits into two qubits (the bit-to-qubit compression ratio is 1.51.51.5x) and obtain its approximation ratio for the maximum cut problem as 0.7220.7220.722. Also, we design a novel quantum relaxation that always guarantees a 222x bit-to-qubit compression ratio which is unlike the original quantum relaxation of Fuller~et~al. We analyze the condition when it has a non-trivial approximation ratio bound (>12)\left(>\frac{1}{2}\right)(>21​). We hope that our results lead to the analysis of the quantum approximability and practical efficiency of the quantum-relaxation based approaches.
Relaxation is a common way for dealing with combinatorial optimization problems. QRAO (Quantum Random Access Optimization) is a quantum-relaxation based optimizer that uses fewer qubits than the number of bits in the original problem, by encoding multiple variables per qubit using QRAC (Quantum Random Access Code). Reducing the number of qubits will alleviate physical noise (typically, decoherence) and, as a result, the quality of the binary solution of QRAO may be robust against noise, which is however unknown. In this paper, we numerically demonstrate that the mean approximation ratio of the (3, 1)-QRAC Hamiltonian, i.e., the Hamiltonian utilizing the encoding of 3 bits into 1 qubit by QRAC, is less affected by noise compared to the conventional Ising Hamiltonian used in quantum annealer and QAOA (Quantum Approximate Optimization Algorithm). Based on this observation, we discuss a plausible mechanism behind the robustness of QRAO under depolarizing noise. Finally, we assess the number of shots required to estimate the values of binary variables correctly under depolarizing noise and show that the (3, 1)-QRAC Hamiltonian requires less shots to achieve the same accuracy compared to the Ising Hamiltonian.
Combinatorial optimization is one of the fields where near term quantum devices are being utilized with hybrid quantum-classical algorithms to demonstrate potentially practical applications of quantum computing. One of the most well studied problems in combinatorial optimization is the Max-Cut problem. The problem is also highly relevant to quantum and other types of “post Moore” architectures due to its similarity with the Ising model and other reasons. In this paper, we introduce a scalable hybrid multilevel approach to solve large instances of Max-Cut using both classical only solvers and quantum approximate optimization algorithm (QAOA). We compare the results of our solver to existing state of the art large-scale Max-Cut solvers. We demonstrate excellent performance of both classical and hybrid quantum-classical approaches and show that using QAOA within our framework is comparable to classical approaches. Reproducibility: Our solver is publicly available at https://github.com/angone/MLMax-cut.
Abstract MaxCut is a key NP-hard combinatorial optimization problem. Quantum computing offers methods to solve such problems potentially better than classical counterparts, with the Quantum Approximate Optimization Algorithm (QAOA) being a state-of-the-art example. However, the performance of quantum methods is currently hindered by hardware noise and limited qubit volumes. We present a variational Qubit-Efficient MaxCut (QEMC) algorithm that requires only O(log⁡N)O(\log N)O(logN) O ( log N ) qubits to tackle graphs of size N , an exponential reduction compared to QAOA. We demonstrate cutting-edge performance for 32-node graph instances (5 qubits) on real superconducting hardware, and for graphs with up to 2048 nodes (11 qubits) via classical simulations. The QEMC algorithm is based on an innovative encoding scheme, with potentially broad applicability, that empowers it with strong noise resilience, but also enables its efficient classical simulation. As such, the QEMC algorithm provides a challenging benchmark for QAOA on noisy devices and offers a novel quantum-inspired approach.