QMA with subset state witnesses

TL;DR

This paper introduces the class SQMA, restricting quantum witnesses to subset states, and proves it is equivalent to QMA, offering a new perspective on quantum proof complexity and its relation to classical witnesses.

Contribution

It defines SQMA with subset state witnesses and proves its equivalence to QMA, providing a novel characterization of quantum proof classes.

Findings

SQMA equals QMA, offering a new characterization.

Proves analogous results for QMA(2).

Introduces a new complete problem for QMA.

Abstract

The class QMA plays a fundamental role in quantum complexity theory and it has found surprising connections to condensed matter physics and in particular in the study of the minimum energy of quantum systems. In this paper, we further investigate the class QMA and its related class QCMA by asking what makes quantum witnesses potentially more powerful than classical ones. We provide a definition of a new class, SQMA, where we restrict the possible quantum witnesses to the "simpler" subset states, i.e. a uniform superposition over the elements of a subset of n-bit strings. Surprisingly, we prove that this class is equal to QMA, hence providing a new characterisation of the class QMA. We also prove the analogous result for QMA(2) and describe a new complete problem for QMA and a stronger lower bound for the class QMA$_1$.