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Quantum Frontiers Journal

Comparative Study of Classical vs Quantum Algorithms for Combinatorial Problems

Daniel Popescu, Helena Petrov

Published Mar 28, 2024• 0 citations
OpenAlex

Abstract

Claims of quantum advantage for combinatorial optimisation require rigorous comparison against state-of-the-art classical algorithms under controlled experimental conditions -- yet the quantum computing literature frequently benchmarks quantum methods against weak classical baselines such as greedy heuristics or random search, inflating apparent quantum advantage. This paper presents the Quantum-Classical Combinatorial Benchmark (QCCB), a rigorous comparative study of quantum optimisation algorithms (QAOA, VQE-QUBO, hybrid QA) against nine state-of-the-art classical algorithms (Goemans-Williamson SDP, simulated annealing, tabu search, CPLEX branch-and-bound, OR-Tools CP-SAT, Gurobi MIP, Benders decomposition, extremal optimisation, and population-based iterated local search) across six canonical combinatorial problem classes (MaxCut, Vertex Cover, Graph Colouring, Knapsack, Set Cover, and Weighted Max-SAT) at problem sizes n = 12 to n = 256. QCCB finds that no quantum algorithm consistently outperforms all classical baselines across all problem classes at current NISQ scale. QAOA with QOAS adaptive depth (Bianchi, 2024) outperforms all nine classical algorithms on MaxCut at n = 48 (approximation ratio 0.924 vs. GW 0.878). However, classical Gurobi MIP and CPLEX solve MaxCut exactly for n up to n = 84 within the same time budget, eliminating the QAOA advantage at moderate scale. QCCB provides nuanced guidance on problem classes, instance sizes, and time budgets where quantum algorithms offer genuine near-term advantage.

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Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm

Danylo Lykov, Jonathan Wurtz, C. Poole +3 more

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arXiv:2206.03579•npj Quantum Information•Jul 25, 2023•35 citations
Claims of quantum advantage for combinatorial optimisation require rigorous comparison against state-of-the-art classical algorithms under controlled experimental conditions -- yet the quantum computing literature frequently benchmarks quantum methods against weak classical baselines such as greedy heuristics or random search, inflating apparent quantum advantage. This paper presents the Quantum-Classical Combinatorial Benchmark (QCCB), a rigorous comparative study of quantum optimisation algorithms (QAOA, VQE-QUBO, hybrid QA) against nine state-of-the-art classical algorithms (Goemans-Williamson SDP, simulated annealing, tabu search, CPLEX branch-and-bound, OR-Tools CP-SAT, Gurobi MIP, Benders decomposition, extremal optimisation, and population-based iterated local search) across six canonical combinatorial problem classes (MaxCut, Vertex Cover, Graph Colouring, Knapsack, Set Cover, and Weighted Max-SAT) at problem sizes n = 12 to n = 256. QCCB finds that no quantum algorithm consistently outperforms all classical baselines across all problem classes at current NISQ scale. QAOA with QOAS adaptive depth (Bianchi, 2024) outperforms all nine classical algorithms on MaxCut at n = 48 (approximation ratio 0.924 vs. GW 0.878). However, classical Gurobi MIP and CPLEX solve MaxCut exactly for n up to n = 84 within the same time budget, eliminating the QAOA advantage at moderate scale. QCCB provides nuanced guidance on problem classes, instance sizes, and time budgets where quantum algorithms offer genuine near-term advantage.
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.
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.
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.
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
Large-scale combinatorial and continuous optimisation problems remain intractable for classical algorithms as problem dimensionality scales. Quantum computing offers advantages through parallelism, amplitude amplification, and tunnelling, but practical realisation on near-term noisy intermediate-scale quantum (NISQ) devices demands careful algorithm design balancing advantage against noise resilience and circuit depth constraints. This paper presents the Quantum Optimisation Algorithm Suite (QOAS), a systematic framework across four algorithmic paradigms: Adaptive QAOA with dynamic depth scheduling; Variational Quantum Eigensolver (VQE) for QUBO problems; hybrid Quantum Annealing (QA); and Grover-enhanced Branch and Bound (GBB) for constrained integer programming. QOAS is benchmarked on five problem classes (MaxCut, TSP, Portfolio, Drug-Target, Feature Selection) across n = 12-84 qubits on IBM Quantum Eagle (127-qubit) and Falcon (27-qubit). Adaptive QAOA achieves approximation ratio 0.924 (SD = 0.018) for MaxCut at n = 48 -- outperforming standard QAOA (0.848) and Goemans-Williamson (0.878) at equivalent wall-clock time. GBB achieves 28.4x speedup over classical branch-and-bound for the n = 24 drug-target problem. QOAS provides systematic methodology for algorithm selection, circuit depth optimisation, and noise mitigation for NISQ workloads.
Abstract In this work, we compare the performance of the Quantum Approximate Optimization Algorithm (QAOA) with state-of-the-art classical solvers such as Gurobi and MQLib to solve the combinatorial optimization problem MaxCut on 3-regular graphs. The goal is to identify under which conditions QAOA can achieve “quantum advantage” over classical algorithms, in terms of both solution quality and time to solution. One might be able to achieve quantum advantage on hundreds of qubits and moderate depth p by sampling the QAOA state at a frequency of order 10 kHz. We observe, however, that classical heuristic solvers are capable of producing high-quality approximate solutions in linear time complexity. In order to match this quality for large graph sizes N, a quantum device must support depth p > 11. Otherwise, we demonstrate that the number of required samples grows exponentially with N, hindering the scalability of QAOA with p ≤ 11. These results put challenging bounds on achieving quantum advantage for QAOA MaxCut on 3-regular graphs. Other problems, such as different graphs, weighted MaxCut, maximum independent set, and 3-SAT, may be better suited for achieving quantum advantage on near-term quantum devices.