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Beating classical heuristics for the binary paint shop problem with the quantum approximate optimization algorithm

Michael Streif, Sheir Yarkoni, Andrea Skolik, Florian Neukart, Martin Leib

TL;DR

The binary paint shop problem (BPSP) is APX-hard, making traditional polynomial-time approximations unlikely. The authors map BPSP to an Ising spin-glass Hamiltonian and apply the Quantum Approximate Optimization Algorithm (QAOA) at fixed depth, using a tree-based parameter strategy to avoid instance-specific training. They show that constant-depth QAOA can beat classical greedy heuristics on average in the infinite-size limit $n \to \infty$, supported by numerical results and preliminary trapped-ion experiments that reveal hardware-noise constraints. The work suggests a path toward quantum advantage for industry-scale sequencing problems on near-term devices and motivates further algorithmic and hardware refinements, including adaptive QAOA and lattice-gauge encodings.

Abstract

The binary paint shop problem (BPSP) is an APX-hard optimization problem of the automotive industry. In this work, we show how to use the Quantum Approximate Optimization Algorithm (QAOA) to find solutions of the BPSP and demonstrate that QAOA with constant depth is able to beat classical heuristics on average in the infinite size limit $n\rightarrow\infty$. For the BPSP, it is known that no classical algorithm can exist which approximates the problem in polynomial runtime. We introduce a BPSP instance which is hard to solve with QAOA, and numerically investigate its performance and discuss QAOA's ability to generate approximate solutions. We complete our studies by running first experiments of small-sized instances on a trapped-ion quantum computer through AWS Braket.

Beating classical heuristics for the binary paint shop problem with the quantum approximate optimization algorithm

TL;DR

The binary paint shop problem (BPSP) is APX-hard, making traditional polynomial-time approximations unlikely. The authors map BPSP to an Ising spin-glass Hamiltonian and apply the Quantum Approximate Optimization Algorithm (QAOA) at fixed depth, using a tree-based parameter strategy to avoid instance-specific training. They show that constant-depth QAOA can beat classical greedy heuristics on average in the infinite-size limit , supported by numerical results and preliminary trapped-ion experiments that reveal hardware-noise constraints. The work suggests a path toward quantum advantage for industry-scale sequencing problems on near-term devices and motivates further algorithmic and hardware refinements, including adaptive QAOA and lattice-gauge encodings.

Abstract

The binary paint shop problem (BPSP) is an APX-hard optimization problem of the automotive industry. In this work, we show how to use the Quantum Approximate Optimization Algorithm (QAOA) to find solutions of the BPSP and demonstrate that QAOA with constant depth is able to beat classical heuristics on average in the infinite size limit . For the BPSP, it is known that no classical algorithm can exist which approximates the problem in polynomial runtime. We introduce a BPSP instance which is hard to solve with QAOA, and numerically investigate its performance and discuss QAOA's ability to generate approximate solutions. We complete our studies by running first experiments of small-sized instances on a trapped-ion quantum computer through AWS Braket.

Paper Structure

This paper contains 23 sections, 23 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Review of QAOA
  3. The binary paint shop problem
  4. Reformulating the BPSP as a spin glass
  5. Properties of the Ising Hamiltonian
  6. Average degree
  7. Coupling strengths $J_{ij}$
  8. Solving the Binary Paint Shop problem with QAOA
  9. Numerical results
  10. Beating the greedy algorithms for large instances
  11. Experimental results
  12. Conclusion & Outlook
  13. Classical heuristics
  14. Red-first heuristic
  15. Recursive greedy heuristic
  16. ...and 8 more sections

Figures (7)

  • Figure 1: (a) A binary paint shop instance with $n=3$ cars $\{c_1,c_2,c_3\}$. (b) A valid but sub-optimal coloring with $\Delta_C=3$ color changes. (c) An optimal coloring which only requires $\Delta_C=2$ color changes to paint the sequence.
  • Figure 2: Greedy algorithm
  • Figure 3: Mapping of the binary paint shop problem onto a spin glass
  • Figure 4: Numerical results for the binary paint shop problem. The classical greedy algorithm is compared to QAOA with different levels $p$. Each data point is averaged over $100$ randomly generated instances.
  • Figure 5: The data from Tab. \ref{['tab:energies']} shown in a log-log plot together with a fit to the function $f(p)=10^b p^a$. The fit parameters are $a=(-0.279\pm 0.005)$ and $b=(-0.168\pm 0.003)$ with a coefficient of determination of $R^2=0.999$draper1998applied.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Definition 3.1