Zero-knowledge proof systems for QMA

TL;DR

This paper extends zero-knowledge proof systems to all problems in QMA, a quantum complexity class, assuming certain cryptographic primitives, and introduces a new QMA-complete problem variant.

Contribution

It proves that every QMA problem has a quantum zero-knowledge proof system under specific cryptographic assumptions, generalizing classical results.

Findings

Quantum zero-knowledge proofs for QMA problems.

Introduction of a new QMA-complete local Hamiltonian problem variant.

Potential applications of the new problem in quantum complexity theory.

Abstract

Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.