Quantum Query Advantage Requires Space

2026-09-04

AI Summary

This study addresses whether quantum query advantages depend on spatial resources and identifies the conditions under which such advantages vanish under space constraints. To this end, it constructs explicit Boolean functions and introduces a one-bit filtered parity construction, an oracle capacity compression technique, and novel parity-capacity inequalities to establish a space-sensitive analytical framework for quantum complexity lower bounds. The primary contribution is the first rigorous complexity separation demonstrating Q < R < Q_S, proving that quantum query complexity becomes inferior to classical randomized algorithms when workspace is severely limited. These findings reveal the necessity of spatial resources for sustaining quantum advantages and provide new theoretical foundations for understanding space-bounded quantum computation.

Abstract

Hao, Huang, and Liu (STOC'26) recently showed that optimal quantum query complexity may require large workspace even for short-output problems, and asked whether a quantum query advantage over classical computation can itself require space. We resolve this question by exhibiting an explicit total Boolean function with quantum query complexity $Q=\widetilde{\Theta}(M)$, and randomized query complexity $R=\Theta(M^{21/20})$, whereas, for every fixed $0<\eta<1/100$ and $S\le O(M^{1/100-\eta})$, its $S$-space quantum query complexity satisfies $Q_S=\omega(M^{21/20})$. Consequently, $$ Q<R<Q_S, $$ so the unrestricted quantum query advantage disappears under sufficiently small workspace. The separation is obtained through a one-bit filtered-parity construction and a space-sensitive quantum lower bound based on compressed-oracle capacity and a new parity-to-capacity inequality, which may be of independent interest.

Problem

• quantum query complexity
• space-bounded computation
• quantum advantage
• randomized query complexity
• workspace

Innovation

• quantum query complexity
• space-bounded computation
• compressed-oracle capacity
• filtered-parity construction
• query advantage separation