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Beyond Exact Recovery: Operational Success Probability and Sampling Cost for Approximate Quantum Optimization under Stochastic Uncertainty

Roberto Garrone

Published Oct 27, 2026• 0 citations
Publisher

Abstract

Abstract Approximate optimization algorithms are commonly evaluated through exact optimum recovery or objective-space approximation, although these criteria need not coincide with the quality of the resulting decision in its downstream operating environment. This paper develops an end-to-end evaluation framework that separates three notions of solver success: exact recovery of the mathematical optimum, near-optimality with respect to the optimization surrogate, and operational acceptability under stochastic downstream evaluation. We study the framework using a small binary inventory-control benchmark represented by a calibrated and held-out-validated QUBO surrogate and solved with exact enumeration, greedy search, simulated annealing, uniform feasible sampling, and ideal QAOA at depths p=1,2,3. The results show that mathematical near-optimality and downstream operational acceptability define distinct solution sets. For QAOA, the probability mass assigned to operationally acceptable solutions can differ materially from both exact-recovery and QUBO-near-optimal probability, producing substantial differences in the number of measurements required to observe an acceptable solution. For n=8 at p=2, the median number of shots required for 95% observation probability decreases from 392 under exact recovery to 84 under QUBO near-optimality and 31 under the operational criterion. However, operational success alone can be misleading when acceptable states are abundant. We therefore use a descriptive enrichment comparison against uniform feasible sampling. Statistically supported enrichment is observed only in selected regimes, while classical greedy and simulated-annealing baselines outperform QAOA on these small instances. The results therefore do not indicate quantum advantage; rather, they show that solver evaluation can change materially when success is defined in terms of downstream decision consequences rather than surrogate-objective recovery alone. The QAOA statevector implementation and selected frozen confirmatory cases were independently reproduced with Qiskit and Qiskit Aer to floating-point precision.

DOI
10.21203/rs.3.rs-11499298/v1

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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.
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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.
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.