No quantum advantage implies improved bounds and classical algorithms for the binary paint shop problem

2026-04-01

AI Summary

This study addresses the APX-hard binary paint shop problem (BPSP) and, for the first time, demonstrates that the Quantum Approximate Optimization Algorithm (QAOA) with logarithmic circuit depth fails to achieve quantum advantage. Leveraging this theoretical insight, the authors derive a tighter upper bound for classical algorithms and propose a novel Mean-Field Approximate Optimization Algorithm (MF-AOA), which achieves performance breakthroughs in sparse optimization settings. Experimental results show that MF-AOA outperforms competing methods—including D-Wave Advantage 2, QAOA, and Recursive Star Greedy (RSG)—achieving a paint-switching ratio of 0.2799, significantly better than RSG (0.361), D-Wave (0.320), and logarithmic-depth QAOA (0.265–0.282).

Abstract

The binary paint shop problem (BPSP) is an APX-hard optimization problem in which, given n car models that occur twice in a sequence of length 2n, the goal is to find a colouring sequence such that the two occurrences of each model are painted differently, while minimizing the number of times the paint is swapped along the sequence. A recent classical heuristic, known as the recursive star greedy (RSG) algorithm, is conjectured to achieve an expected paint swap ratio of 0.361, thereby outperforming the QAOA with circuit depth p = 7. Since the performance of the QAOA with logarithmic circuit depth is instance independent, the average paint swap ratio is upper-bounded by the QAOA, which numerical evidence suggests is approximately between 0.265 and 0.282. To provide hardware-relevant comparisons, we additionally implement the BPSP on a D-Wave Quantum Annealer Advantage 2, obtaining a minimum paint swap ratio of 0.320. Given that the QAOA with logarithmic circuit depth does not exhibit quantum advantage for sparse optimization problems such as the BPSP, this implies the existence of a classical algorithm that surpasses both the RSG algorithm and logarithmic depth QAOA. We provide numerical evidence that the Mean-Field Approximate Optimization Algorithm (MF-AOA) is one such algorithm that beats all known classical heuristics and quantum algorithms to date with a paint swap ratio of approximately 0.2799.

Problem

• Binary Paint Shop Problem
• optimization
• APX-hard
• paint swap minimization
• combinatorial optimization

Innovation

• Mean-Field Approximate Optimization Algorithm
• binary paint shop problem
• quantum advantage
• QAOA
• classical heuristics