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$_1$.

Highlights

  • One of the notions at the heart of classical complexity theory is the class NP and the fact that deciding whether a boolean formula is satisfiable or not is NP-complete [6, 17]

  • We investigate the class QMA by asking the following simple, yet fundamental question: what makes a quantum witness potentially more powerful than a classical one? Is it the fact that to describe a quantum state one needs to specify an exponential number of possibly different amplitudes? Is it the different relative phases in the quantum state? Or is it something else altogether?

  • We prove that the class oSQMA admits perfect completeness, which implies a stronger lower bound for the class QMA1 than the previously known QCMA bound

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Summary

Introduction

One of the notions at the heart of classical complexity theory is the class NP and the fact that deciding whether a boolean formula is satisfiable or not is NP-complete [6, 17]. The famous PCP theorem [3, 4] provided a new, surprising description of the class NP: any language in NP can be verified efficiently by accessing probabilistically a constant number of bits of a polynomial-size witness This opened the way to showing that in many cases, approximating the solution of NP-hard problems remains as hard as solving them exactly. There have been a series of results, mostly negative, towards the goal of proving or disproving the quantum PCP theorem, but there is still no conclusive evidence [2] Another important open question about the class QMA is whether the witness really need be a quantum state or it is enough for the polynomial-time quantum verifier to receive a classical witness. Our main result shows that SQMA is, equal to QMA and the same for the two-prover case

Result
The following Basis State Check on Subset States problem is QMA-complete:
Definitions
Complexity classes and complete problems
Subset state approximations
A QMA-complete problem based on subset states
On the perfectly complete version of SQMA
Conclusions
Full Text
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