QMA variants with polynomially many provers

TL;DR

This paper explores various multi-prover quantum proof systems, establishing their equivalences and properties, including the power of polynomially many proofs and the behavior of separable measurements, advancing understanding of quantum verification complexity.

Contribution

It characterizes the power of polynomially many quantum proofs with logarithmic size, analyzes BellQMA(poly) with bounded outcomes, and proves a parallel repetition theorem for SepQMA(m).

Findings

Polynomially many logarithmic-size proofs are equivalent to QCMA.

BellQMA(poly) with polynomially bounded outcomes equals QMA.

Parallel repetition holds for SepQMA(m) with separable measurements.

Abstract

We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly MQA (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class BellQMA(poly), characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. We show that if the number of outcomes per nonadaptive measurement is a polynomially-bounded function, then the expressive power of the proof system is exactly QMA. Finally, we study a class equivalent to QMA(m), denoted SepQMA(m), where the verifier's measurement operator corresponding to outcome "accept" is a fully separable operator across the m quantum proofs. Using cone programming duality, we give an alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633--642 (2010)] that shows a perfect parallel repetition theorem for SepQMA(m) for any m.