Quantum Proofs of Proximity

TL;DR

This paper introduces QMA proofs of proximity, a quantum property testing model, demonstrating exponential advantages over classical methods and providing new algorithms with quantum speedups for specific property testing problems.

Contribution

It develops the QMA proofs of proximity model, establishes quantum lower bounds via communication complexity, and presents quantum algorithms with speedups for property testing tasks.

Findings

QMAPs can be exponentially stronger than classical proofs and quantum testers.

A quantum lower bound for property testing is established via communication complexity.

Quantum algorithms are developed for testing properties like parity and bipartiteness with speedups.

Abstract

We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property $\Pi$ and reject inputs that are $\varepsilon$-far from $\Pi$, while only probing a minuscule portion of their input. We investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody, and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity. This also resolves a question raised in 2013 by Montanaro and de Wolf (cf. Theory of Computing, 2016). Our algorithmic results include a purpose an algorithmic framework that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly decomposable. A consequence of this framework is a QMA algorithm to verify the Parity of an $n$-bit string with $O(n^{2/3})$ queries and proof length. We also propose a QMA algorithm for testing graph bipartitneness, a property that lies outside of this family, for which there is a quantum speedup.