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Assessing fault-tolerant quantum advantage for $k$-SAT with structure

Martijn Brehm, Jordi Weggemans

TL;DR

This study assesses the practicality of fault-tolerant quantum speedups for structured $k$-SAT via hybrid benchmarking, translating quantum query bounds into run-times under surface-code overheads using both $T$-depth and $T$-count. By applying Belovs' quantum-walk backtracking detection, its binary-search-based search variant, and Grover's algorithm to structured random $k$-SAT distributions, the authors reveal four regimes of problem structure with distinct speedup landscapes. They find that, once even modest structure is present or when using $T$-count, quantum speedups largely vanish, and only Grover offers a practical edge in highly unstructured cases under optimistic assumptions; in day-long computations, quantum backtracking generally fails to beat optimized classical solvers. The results suggest limited near-term practical quantum advantage for structured $k$-SAT solving, unless future heuristics or improved quantum backtracking techniques overcome substantial overheads.

Abstract

For many problems, quantum algorithms promise speedups over their classical counterparts. However, these results predominantly rely on asymptotic worst-case analysis, which overlooks significant overheads due to error correction and the fact that real-world instances often contain exploitable structure. In this work, we employ the hybrid benchmarking method to evaluate the potential of quantum Backtracking and Grover's algorithm against the 2023 SAT competition main track winner in solving random $k$-SAT instances with tunable structure, designed to represent industry-like scenarios, using both $T$-depth and $T$-count as cost metrics to estimate quantum run times. Our findings reproduce the results of Campbell, Khurana, and Montanaro (Quantum '19) in the unstructured case using hybrid benchmarking. However, we offer a more sobering perspective in practically relevant regimes: almost all quantum speedups vanish, even asymptotically, when minimal structure is introduced or when $T$-count is considered instead of $T$-depth. Moreover, when the requirement is for the algorithm to find a solution within a single day, we find that only Grover's algorithm has the potential to outperform classical algorithms, but only in a very limited regime and only when using $T$-depth. We also discuss how more sophisticated heuristics could restore the asymptotic scaling advantage for quantum backtracking, but our findings suggest that the potential for practical quantum speedups in more structured $k$-SAT solving will remain limited.

Assessing fault-tolerant quantum advantage for $k$-SAT with structure

TL;DR

This study assesses the practicality of fault-tolerant quantum speedups for structured -SAT via hybrid benchmarking, translating quantum query bounds into run-times under surface-code overheads using both -depth and -count. By applying Belovs' quantum-walk backtracking detection, its binary-search-based search variant, and Grover's algorithm to structured random -SAT distributions, the authors reveal four regimes of problem structure with distinct speedup landscapes. They find that, once even modest structure is present or when using -count, quantum speedups largely vanish, and only Grover offers a practical edge in highly unstructured cases under optimistic assumptions; in day-long computations, quantum backtracking generally fails to beat optimized classical solvers. The results suggest limited near-term practical quantum advantage for structured -SAT solving, unless future heuristics or improved quantum backtracking techniques overcome substantial overheads.

Abstract

For many problems, quantum algorithms promise speedups over their classical counterparts. However, these results predominantly rely on asymptotic worst-case analysis, which overlooks significant overheads due to error correction and the fact that real-world instances often contain exploitable structure. In this work, we employ the hybrid benchmarking method to evaluate the potential of quantum Backtracking and Grover's algorithm against the 2023 SAT competition main track winner in solving random -SAT instances with tunable structure, designed to represent industry-like scenarios, using both -depth and -count as cost metrics to estimate quantum run times. Our findings reproduce the results of Campbell, Khurana, and Montanaro (Quantum '19) in the unstructured case using hybrid benchmarking. However, we offer a more sobering perspective in practically relevant regimes: almost all quantum speedups vanish, even asymptotically, when minimal structure is introduced or when -count is considered instead of -depth. Moreover, when the requirement is for the algorithm to find a solution within a single day, we find that only Grover's algorithm has the potential to outperform classical algorithms, but only in a very limited regime and only when using -depth. We also discuss how more sophisticated heuristics could restore the asymptotic scaling advantage for quantum backtracking, but our findings suggest that the potential for practical quantum speedups in more structured -SAT solving will remain limited.

Paper Structure

This paper contains 28 sections, 2 theorems, 31 equations, 1 figure, 17 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. Organisation of paper
  5. Algorithms, instances and cost model
  6. Quantum backtracking (detection)
  7. Quantum backtracking (search)
  8. Grover's algorithm
  9. Classical SAT solver
  10. Instances
  11. Cost model
  12. Benchmarking results
  13. The four regimes
  14. No quantum advantage potential for one day search with structure
  15. Getting an advantage is more difficult on satisfiable instances
  16. ...and 13 more sections

Key Result

Theorem 1

For any starting distribution $\sigma$ Belovs' quantum walk (configured with upper-bounds on $W$ and $R_{\sigma, M}$) solves the Detection problem with probability $>1/2$ with $\mathcal{O}\left(\sqrt{WR_{\sigma, M}}\right)$ queries to $M$.

Figures (1)

  • Figure 1: Scatter plot of median runtime for the CaDiCaL 2.0.0 solver on random $k$-SAT instances at the satisfiability threshold, see \ref{['tab:ratios']}. The dotted line indicates the exponential fit of the running time. For small $k$ you can see that we only start fitting the runtime for larger $n$, as the runtime seems to behave differently for small $n$.

Theorems & Definitions (2)

  • Theorem 1: Ref. belovs2013quantum, Theorem 4
  • Lemma 1: Adapted from Lemma 4 in Ref. cade2023quantifying