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

TL;DR

This paper evaluates the practical potential of quantum algorithms for structured $k$-SAT problems, revealing that quantum speedups are limited in realistic scenarios due to overheads and problem structure.

Contribution

It applies hybrid benchmarking to assess quantum backtracking and Grover's algorithms on structured $k$-SAT, providing realistic estimates of quantum advantage considering error correction and problem structure.

Findings

Quantum speedups vanish with minimal structure or when using T-count.

Only Grover's algorithm shows limited potential for speedup within a day.

Structured heuristics might restore quantum advantage, but practical benefits remain constrained.

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.