A slightly improved upper bound for quantum statistical zero-knowledge

TL;DR

This paper improves the upper bound for quantum statistical zero-knowledge classes by demonstrating that they can be contained within a class with a quantum linear-space honest prover, using advanced quantum measurement techniques.

Contribution

The paper introduces a slight but significant improvement to the upper bound of QSZK by employing quantum linear-space algorithms and measurement techniques.

Findings

Upper bound for QSZK refined to include quantum linear-space honest prover.

Application of quantum measurement algorithms in space-efficient quantum proof systems.

Extension of the improvement to non-interactive quantum statistical zero-knowledge.

Abstract

The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound $\mathsf{QIP(2)} \cap \text{co-}\mathsf{QIP(2)}$, which was simplified following the inclusion $\mathsf{QIP(2)} \subseteq \mathsf{PSPACE}$ established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, $\mathsf{QIP(2)}$ denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically \textit{computationally unbounded}, and $\text{co-}\mathsf{QIP(2)}$ denotes the complement of $\mathsf{QIP(2)}$. We slightly improve this upper bound to $\mathsf{QIP(2)} \cap \text{co-}\mathsf{QIP(2)}$ with a quantum linear-space honest prover. A similar improvement also applies to the upper bound for the non-interactive variant $\mathsf{NIQSZK}$. Our main techniques are an algorithmic version of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).