Succinct quantum proofs for properties of finite groups

TL;DR

This paper introduces quantum proofs for properties of finite groups, demonstrating their efficiency and superiority over classical proofs in certain cases, and explores implications for complexity classes.

Contribution

It presents polynomial-length quantum proofs for group non-membership, showing they are succinct and efficiently verifiable, unlike classical proofs, and explores their impact on complexity theory.

Findings

Quantum proofs for group non-membership are polynomial-length and efficiently verifiable.

Classically, such proofs are impossible in some oracles, indicating quantum advantage.

Quantum proofs can verify properties like group order divisibility and simplicity.

Abstract

In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NP-type proof. Specifically, we consider quantum proofs for properties of black-box groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomial-length) quantum proofs for the Group Non-Membership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossible--it is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group Non-Membership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for non-membership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.