Constant-Space Quantum Interactive Proofs Against Multiple Provers

TL;DR

This paper characterizes the computational power of multi-prover quantum interactive proof systems with limited verifiers, showing they match classes like NEXP and NE under certain conditions, and are as powerful as Turing machines without time bounds.

Contribution

It establishes the computational complexity bounds of multi-prover quantum interactive proofs with finite automata verifiers, answering an open question in the field.

Findings

Multi-prover systems with polynomial time recognize NEXP.

One-way automata verifiers recognize context-free languages.

Unbounded time systems are as powerful as Turing machines.

Abstract

We present upper and lower bounds of the computational complexity of the two-way communication model of multiple-prover quantum interactive proof systems whose verifiers are limited to measure-many two-way quantum finite automata. We prove that (i) the languages recognized by those multiple-prover systems running in expected polynomial time are exactly the ones in NEXP, the nondeterministic exponential-time complexity class, (ii) if we further require verifiers to be one-way quantum automata, then their associated proof systems recognize context-free languages but not beyond languages in NE, the nondeterministic linear exponential-time complexity class, and moreover, (iii) when no time bound is imposed, the proof systems become as powerful as Turing machines. The first two results answer affirmatively an open question, posed by Nishimura and Yamakami [J. Comput. System Sci, 75, pp.255--269, 2009], of whether multiple-prover quantum interactive proof systems are more powerful than single-prover ones. Our proofs are simple and intuitive, although they heavily rely on an earlier result on multiple-prover interactive proof systems of Feige and Shamir [J. Comput. System Sci., 44, pp.259--271, 1992].