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

Mark Goh, Lara Caroline Pereira dos Santos, Matthias Sperl

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.

No quantum advantage implies improved bounds and 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 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.

Paper Structure

This paper contains 17 sections, 1 theorem, 31 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Binary Paint Shop Problem
  4. BPSP as a QUBO
  5. Quantum Approximate Optimization Algorithm
  6. Mean-Field Approximate Optimization Algorithm
  7. Related Work
  8. Methods
  9. Application of the QAOA to the BPSP
  10. Quantum Annealing
  11. Simulation of MF-AOA
  12. Results
  13. QAOA
  14. Quantum Annealing
  15. MF-AOA
  16. ...and 2 more sections

Key Result

Theorem 2.3

The expected colour change $\mathbb{E}\Delta_C/n$ is lower bounded by where $H^{-1}$ is the inverse of the binary entropy function. In addition, it is upper bounded by $0.4+o(1)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: A semi-log plot of the average paint swap ratio achieved by the MF-AOA over a 1000 random generated instances for each problem size and for the D-Wave Advantage 2 Solver and the BQM solver over 50 randomly generated instances for each $n$. In order to verify the accuracy of the Monte Carlo sampling, we similarly did it for the $n=2$ case which is known to have an expected paint swap ratio of $2/3$. Averaging over a thousand instances, we found that the MF-AOA gets an average paint swap ratio of 0.6726 and a variance of 0.06 thus validating our method.

Theorems & Definitions (9)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3: Modified Theorem 1.4 of Ref. BPSP_lowerbound
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Conjecture 4.1
  • Remark 4.2