On the power of nonstandard quantum oracles

TL;DR

This paper investigates how different quantum oracle models influence the complexity of property testing problems, revealing that certain oracle choices can significantly hinder quantum verification even with unlimited witnesses.

Contribution

It introduces new quantum oracle models for graph property testing and demonstrates how oracle design impacts quantum verifier efficiency, highlighting limitations of standard oracles.

Findings

A one-query QMA protocol for testing small disconnected subsets in graphs.

Classical witnesses do not aid quantum verification in certain oracle models.

Modifying the oracle can prevent efficient quantum verification despite unbounded witnesses.

Abstract

We study how the choices made when designing an oracle affect the complexity of quantum property testing problems defined relative to this oracle. We encode a regular graph of even degree as an invertible function $f$, and present $f$ in different oracle models. We first give a one-query QMA protocol to test if a graph encoded in $f$ has a small disconnected subset. We then use representation theory to show that no classical witness can help a quantum verifier efficiently decide this problem relative to an in-place oracle. Perhaps surprisingly, a simple modification to the standard oracle prevents a quantum verifier from efficiently deciding this problem, even with access to an unbounded witness.