Achieving high-quality solutions faster than classical solvers on computationally hard problems is a challenge for quantum optimization to deliver utility. Using a superconducting quantum computer, we experimentally investigate the performance of a hybrid quantum-classical algorithm inspired by semidefinite programming approaches for solving the maximum-cut problem on 3-regular graphs up to several thousand variables. We leverage the structure of the input problems to address sizes beyond what current quantum machines can naively handle. We attain an average approximation ratio of 99% over a random ensemble of thousands of problem instances. We benchmark the quantum solver against similarly high-performing classical heuristics, including the gurobi optimizer, simulated annealing, and the Burer-Monteiro algorithm. A run-time analysis shows that the quantum solver on large-scale problems is competitive against gurobi but short of others on a projected 100-qubit quantum computer. We explore multiple leads to close the gap and discuss prospects for a practical quantum speedup.
Achieving high-quality solutions faster than classical solvers on computationally hard problems is a challenge for quantum optimization to deliver utility. Using a superconducting quantum computer, we experimentally investigate the performance of a hybrid quantum-classical algorithm inspired by semidefinite programming approaches for solving the maximum-cut problem on 3-regular graphs up to several thousand variables. We leverage the structure of the input problems to address sizes beyond what current quantum machines can naively handle. We attain an average approximation ratio of 99% over a random ensemble of thousands of problem instances. We benchmark the quantum solver against similarly high-performing classical heuristics, including the gurobi optimizer, simulated annealing, and the Burer-Monteiro algorithm. A run-time analysis shows that the quantum solver on large-scale problems is competitive against gurobi but short of others on a projected 100-qubit quantum computer. We explore multiple leads to close the gap and discuss prospects for a practical quantum speedup.
State-of-the-art classical optimization solvers set a high bar for quantum computers to deliver utility in this domain. Here, we introduce a quantum preconditioning approach based on the quantum approximate optimization algorithm. It transforms the input problem into a more suitable form for a solver with the level of preconditioning determined by the depth of the quantum circuit. We demonstrate that best-in-class classical heuristics such as simulated annealing and the Burer-Monteiro algorithm can converge more rapidly when given quantum preconditioned input for various problems, including Sherrington-Kirkpatrick spin glasses, random 3-regular graph maximum-cut problems, and a real-world grid energy problem. Accounting for the additional time taken for preconditioning, the benefit offered by shallow circuits translates into a practical quantum-inspired advantage for random 3-regular graph maximum-cut problems through quantum circuit emulations. We investigate why quantum preconditioning makes the problem easier and test an experimental implementation on a superconducting device. We identify challenges and discuss the prospects for a hardware-based quantum advantage in optimization via quantum preconditioning.
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.
Abstract This paper benchmarks quantum, classical, and hybrid solvers on NP-hard Max-Cut and QUBO problems, emphasizing solution quality relative to known global optima. We evaluate D-Wave’s fast annealing QPU and Hybrid solver against classical simulated annealing (SA) and Toshiba’s simulated bifurcation machine (SBM) using 139 Max-Cut instances (100 to 10,000 nodes). For small instances ( ≤ ≤ 250 nodes) with known global optima, Hybrid and SA consistently achieve optimal solutions, outperforming the QPU. For larger instances, SBM and slower SA yield superior solutions, while Hybrid and faster SA perform less effectively. Computation time varies across solvers.
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
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.
QUBO formulations of combinatorial optimization problems allow for solving them using various quantum heuristics. While large-scale quantum computations are currently still out of reach, we can already numerically test such QUBO formulations on a perhaps surprisingly large scale. In this work, we benchmark Fujitsu’s Digital Annealer (DA) on the Max-Cut problem, which captures the main complexity of the QUBO problem. We make a comprehensive benchmark against leading other heuristic algorithms on graphs with up to 53,000 variables by focusing on the wall-clock time. Moreover, we compare the DA performance against published performance results of the D-Wave hybrid quantum-classical annealer and the recently proposed QIS3 heuristic. Based on performance statistics for over 2,000 graphs from the MQLib, we find that the DA yields competitive results. We hope that this benchmark demonstrates the extent to which large QUBO instances can be heuristically solved today, yielding consistent results across different solvers.