The Quantum Strong Exponential-Time Hypothesis

TL;DR

This paper introduces the Quantum Strong Exponential-Time Hypothesis (QSETH), a quantum analogue of SETH, enabling the derivation of quantum lower bounds for problems like Edit Distance and Proofs of Useful Work.

Contribution

It formulates QSETH as a new framework to translate quantum query lower bounds into quantum time lower bounds for problems in BQP.

Findings

Conditional quantum lower bound of Ω(n^{1.5}) for Edit Distance

QSETH-based lower bound for Proofs of Useful Work maintains quadratic gap

Framework connects quantum query complexity to time complexity lower bounds

Abstract

The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It states that CNF formulas cannot be analyzed for satisfiability with a speedup over exhaustive search. This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are hard to find. In this work, we introduce a framework of Quantum Strong Exponential-Time Hypotheses, as quantum analogues to SETH. Using the QSETH framework, we are able to translate quantum query lower bounds on black-box problems to conditional quantum time lower bounds for many problems in BQP. As an example, we illustrate the use of the QSETH by providing a conditional quantum time lower bound of $\Omega(n^{1.5})$ for the Edit Distance problem. We also show that the $n^2$ SETH-based lower bound for a recent scheme for Proofs of Useful Work, based on the Orthogonal Vectors problem holds for quantum computation assuming QSETH, maintaining a quadratic gap between verifier and prover.