Query Complexity: Worst-Case Quantum Versus Average-Case Classical

TL;DR

This paper explores the relationship between worst-case quantum query complexity and average-case classical query complexity, establishing new bounds and highlighting potential avenues to narrow the complexity gap.

Contribution

It provides improved bounds linking quantum worst-case and classical average-case query complexities, especially for monotone functions, advancing understanding of their relationship.

Findings

Classical average-case queries are bounded by O(T^5) from quantum worst-case T queries.

For monotone functions, classical average-case queries are bounded by O(T^3).

The results suggest potential methods to reduce the polynomial gap between quantum and classical complexities.

Abstract

In this note we investigate the relationship between worst-case quantum query complexity and average-case classical query complexity. Specifically, we show that if a quantum computer can evaluate a total Boolean function f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^5) queries in the average case, under a uniform distribution of inputs. If f is monotone, we show furthermore that only O(T^3) queries are needed. Previously, Beals et al. (1998) showed that if a quantum computer can evaluate f with bounded error using T queries in the worst case, then a deterministic classical computer can evaluate f using O(T^6) queries in the worst case, or O(T^4) if f is monotone. The optimal bound is conjectured to be O(T^2), but improving on O(T^6) remains an open problem. Relating worst-case quantum complexity to average-case classical complexity may suggest new ways to reduce the polynomial gap in the ordinary worst-case versus worst-case setting.