Beyond Quantum Advantage: Improved Classical Algorithms for the Binary Paint Shop Problem

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 objective is to find a colouring sequence such that each car model pair is 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 Quantum Approximate Optimization Algorithm (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. We provide an improved upper-bound of the BPSP by extending the QAOA to depth $p=17$, outputting an expected paint swap ratio of $0.334$ via an exact computation while numerical extrapolation suggests a further reduction to a value of $0.295$. 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 $0.329$. Given that the QAOA with logarithmic circuit depth does not exhibit a quantum advantage for sparse optimization problems such as the BPSP, this implies the existence of a classical algorithm that outperforms 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, yielding a paint swap ratio of approximately $0.280$ beating all known classical and quantum algorithms for the BPSP.