On the power of a unique quantum witness

TL;DR

This paper demonstrates a deterministic method to reduce problems with polynomially bounded quantum witness spaces to problems with a unique quantum witness, advancing understanding of quantum complexity classes.

Contribution

It introduces an efficient reduction technique transforming problems with bounded quantum witnesses into ones with a single quantum witness in the QMA class.

Findings

Reduction from polynomial-bounded to unique quantum witness

Use of the Alternating subspace in tensor powers for reduction

Establishment of a deterministic procedure for quantum witness reduction

Abstract

In a celebrated paper, Valiant and Vazirani raised the question of whether the difficulty of NP-complete problems was due to the wide variation of the number of witnesses of their instances. They gave a strong negative answer by showing that distinguishing between instances having zero or one witnesses is as hard as recognizing NP, under randomized reductions. We consider the same question in the quantum setting and investigate the possibility of reducing quantum witnesses in the context of the complexity class QMA, the quantum analogue of NP. The natural way to quantify the number of quantum witnesses is the dimension of the witness subspace W in some appropriate Hilbert space H. We present an efficient deterministic procedure that reduces any problem where the dimension d of W is bounded by a polynomial to a problem with a unique quantum witness. The main idea of our reduction is to consider the Alternating subspace of the d-th tensor power of H. Indeed, the intersection of this subspace with the d-th tensor power of W is one-dimensional, and therefore can play the role of the unique quantum witness.