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arXiv:2001.08271 · Quantum Science and Technology

To quantum or not to quantum: towards algorithm selection in near-term quantum optimization

Charles Moussa, Henri Calandra, Vedran Dunjko

Published Sep 16, 2020• 50 citations
arXivHTMLPDF

Abstract

Abstract The Quantum approximate optimization algorithm (QAOA) constitutes one of the often mentioned candidates expected to yield a quantum boost in the era of near-term quantum computing. In practice, quantum optimization will have to compete with cheaper classical heuristic methods, which have the advantage of decades of empirical domain-specific enhancements. Consequently, to achieve optimal performance we will face the issue of algorithm selection, well-studied in practical computing. Here we introduce this problem to the quantum optimization domain. Specifically, we study the problem of detecting those problem instances of where QAOA is most likely to yield an advantage over a conventional algorithm. As our case study, we compare QAOA against the well-understood approximation algorithm of Goemans and Williamson on the Max-Cut problem. As exactly predicting the performance of algorithms can be intractable, we utilize machine learning (ML) to identify when to resort to the quantum algorithm. We achieve cross-validated accuracy well over 96%, which would yield a substantial practical advantage. In the process, we highlight a number of features of instances rendering them better suited for QAOA. While we work with simulated idealised algorithms, the flexibility of ML methods we employed provides confidence that our methods will be equally applicable to broader classes of classical heuristics, and to QAOA running on real-world noisy devices.

DOI
10.1088/2058-9565/abb8e5

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Abstract The Quantum approximate optimization algorithm (QAOA) constitutes one of the often mentioned candidates expected to yield a quantum boost in the era of near-term quantum computing. In practice, quantum optimization will have to compete with cheaper classical heuristic methods, which have the advantage of decades of empirical domain-specific enhancements. Consequently, to achieve optimal performance we will face the issue of algorithm selection, well-studied in practical computing. Here we introduce this problem to the quantum optimization domain. Specifically, we study the problem of detecting those problem instances of where QAOA is most likely to yield an advantage over a conventional algorithm. As our case study, we compare QAOA against the well-understood approximation algorithm of Goemans and Williamson on the Max-Cut problem. As exactly predicting the performance of algorithms can be intractable, we utilize machine learning (ML) to identify when to resort to the quantum algorithm. We achieve cross-validated accuracy well over 96%, which would yield a substantial practical advantage. In the process, we highlight a number of features of instances rendering them better suited for QAOA. While we work with simulated idealised algorithms, the flexibility of ML methods we employed provides confidence that our methods will be equally applicable to broader classes of classical heuristics, and to QAOA running on real-world noisy devices.
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This study presents a rigorous empirical evaluation of the Quantum Approximate Optimization Algorithm (QAOA) within a hybrid quantum-classical framework for solving the NP-hard Max-Cut problem. Using Python and Qiskit Aer statevector simulations, we benchmark QAOA (depths p = 2 and p = 3) against established classical algorithms, including simulated annealing, the Goemans–Williamson algorithm, and greedy heuristics, across Erdős–Rényi and regular graphs with 4 to 20 nodes (100 instances per size). Results demonstrate that low-depth QAOA suffers from significant performance degradation, achieving a 0.467 approximation ratio at 20 nodes compared to 0.856 for Goemans–Williamson, alongside a 210× runtime overhead. Convergence analysis further reveals low optimality probabilities and high iteration costs, highlighting limitations of variational optimization in NISQ-era devices. This work provides a reproducible benchmarking framework and contributes to a realistic assessment of near-term quantum advantage.
Abstract The quantum approximate optimization algorithm (QAOA) is a hybrid quantum–classical algorithm to solve binary-variable optimization problems. Due to the short circuit depth and its expected robustness to systematic errors it is a promising candidate likely to run on near-term quantum devices. We simulate the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives. When comparing solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we utilize performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. In particular for three-regular random graphs, QAOA performance shows improvement by up to two orders of magnitude compared to previous estimates, strongly reducing the performance gap with classical alternatives. This was possible by reducing the number of function evaluations per iteration and optimizing the variational parameters on small graph instances and transferring to large via training. Because QAOA’s performance guarantees are only known for limited applications and contexts, we utilize a framework for the search for quantum advantage which incorporates a large number of problem instances and all three classical solver modalities: exact, approximate, and heuristic.
The transition from theoretical quantum computing to practical real-world applications heavily relies onhybrid quantum-classical architectures designed for NoisyIntermediate-Scale Quantum (NISQ) devices. Among these approaches, the Quantum Approximate Optimization Algorithm(QAOA) is a prominent strategy for addressing complex combinatorial optimization challenges, such as the NP-hard MaxCut problem. Instead of performing the entire computationon a quantum processor, QAOA utilizes a quantum circuit toevaluate a parameterized cost function, whereas a classical optimizer refines the parameters iteratively. This study presentsa rigorous empirical evaluation comparing the performanceof QAOA against established classical solvers on small-scaleMax-Cut instances. All experiments were implemented andexecuted using Python (Qiskit Aer) on a state-vector simulator.Using Erdős–Rényi random graphs and regular graphs with4 to 20 nodes, we benchmark QAOA at depths p = 2 andp = 3 against simulated annealing, the Goemans–WilliamsonSDP-based algorithm, and a linear-time greedy heuristic. Ourexperimental results demonstrate that low-depth QAOA implementations exhibit severe performance degradation, prohibitive computational overhead (O(2^n) scaling), and low convergencerates compared to mature classical algorithms. At 20 nodes,QAOA (p = 2) achieves only a 0.467 approximation ratio versus0.856 for Goemans–Williamson, with a 210× runtime slowdownover greedy heuristics
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm to solve binary-variable optimization problems. Due to its expected robustness to systematic errors and the short circuit depth, it is one of the promising candidates likely to run on near-term quantum devices. We project the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives, both for exact or approximate solution. When comparing approximate solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we introduce performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. By selecting a suitable optimizer for the variational parameters and reducing the number of function evaluations, QAOA performance improves by up to 2 orders of magnitude compared to previous estimates. Especially for 3-regular random graphs, this setting decreases the performance gap with classical alternatives.
This study presents a rigorous empirical evaluation of the Quantum Approximate Optimization Algorithm (QAOA) within a hybrid quantum-classical framework for solving the NP-hard Max-Cut problem. Using Python and Qiskit Aer statevector simulations, we benchmark QAOA (depths p = 2 and p = 3) against established classical algorithms, including simulated annealing, the Goemans–Williamson algorithm, and greedy heuristics, across Erdős–Rényi and regular graphs with 4 to 20 nodes (100 instances per size). Results demonstrate that low-depth QAOA suffers from significant performance degradation, achieving a 0.467 approximation ratio at 20 nodes compared to 0.856 for Goemans–Williamson, alongside a 210× runtime overhead. Convergence analysis further reveals low optimality probabilities and high iteration costs, highlighting limitations of variational optimization in NISQ-era devices. This work provides a reproducible benchmarking framework and contributes to a realistic assessment of near-term quantum advantage.