Impossibility of Succinct Quantum Proofs for Collision-Freeness

TL;DR

This paper proves that quantum algorithms with or without witnesses cannot efficiently determine if a function is a permutation or far from it, highlighting fundamental limits in quantum proof systems.

Contribution

It establishes a quantum lower bound for collision-freeness testing and demonstrates an oracle separation between SZK and QMA classes.

Findings

Quantum algorithms require Omega(n^{1/3}) queries for collision-freeness.

SZK is not contained in QMA relative to some oracle.

Supports the conjecture of fundamental quantum limitations in certain decision problems.

Abstract

We show that any quantum algorithm to decide whether a function f:[n]->[n] is a permutation or far from a permutation must make Omega(n^{1/3}/w) queries to f, even if the algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that SZK^A is not contained in QMA^A, answering an eight-year-old open question of the author. Indeed, we show that relative to some oracle, SZK is not in the counting class A0PP defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem.