From Promises to Totality: A Framework for Ruling Out Quantum Speedups

TL;DR

This paper develops a framework using promise-aware complexity measures and function completions to determine when partial Boolean functions can or cannot have superpolynomial quantum query speedups.

Contribution

It introduces promise versions of combinatorial measures and analyzes structured promises to characterize quantum speedup possibilities.

Findings

Collapse of promise and completion measures implies polynomial relation between D(f) and Q(f).

Sharp characterizations for symmetric partial functions and promises on Hamming slices.

Completion complexity captures the potential for superpolynomial quantum speedups.

Abstract

We study when partial Boolean functions can (and cannot) exhibit superpolynomial quantum query speedups, and develop a general framework for ruling out such speedups via two complementary lenses: promise-aware complexity measures and function completions. First, we introduce promise versions of standard combinatorial measures (including block sensitivity and related variants) and prove that if the relevant promise and completion measures collapse, then deterministic and quantum query complexities are necessarily polynomially related, i.e., $D(f)=poly(Q(f))$. We then analyze structured families of promises, including symmetric partial functions and promises supported on Hamming slices, obtaining sharp (up to polynomial factors) characterizations in terms of a single gap parameter for the symmetric case and refined slice-dependent bounds for $k$-slice domains. Next, we formalize completion complexity as the minimum of a measure over total completions of a partial function, and show that completability of a measure captures the possibility of superpolynomial quantum speedups. Finally, we apply this viewpoint to derive broad non-speedup criteria for some classes of functions admitting well-behaved completions, such as functions with low maximum influence on both the standard and $p$-biased hypercubes and functions with efficiently identifiable domains, and then show some hardness results for general completion techniques.