Space-Efficient Error Reduction for Unitary Quantum Computations

TL;DR

This paper introduces space-efficient error reduction techniques for unitary quantum computations, significantly reducing workspace requirements and enabling strong amplification in logarithmic-space quantum settings.

Contribution

It presents the first methods for error reduction requiring only logarithmic workspace in unitary quantum computations, improving efficiency over previous approaches.

Findings

Achieves error reduction with O(log(p/(c-s))) qubits of extra space

Enables strong amplification for logarithmic-space unitary quantum computations

Derives complexity-theoretic consequences like PSPACE upper bounds for QMA

Abstract

This paper develops general space-efficient methods for error reduction for unitary quantum computation. Consider a polynomial-time quantum computation with completeness $c$ and soundness $s$, either with or without a witness (corresponding to QMA and BQP, respectively). To convert this computation into a new computation with error at most $2^{-p}$, the most space-efficient method known requires extra workspace of ${O \bigl( p \log \frac{1}{c-s} \bigr)}$ qubits. This space requirement is too large for scenarios like logarithmic-space quantum computations. This paper presents error-reduction methods for unitary quantum computations (i.e., computations without intermediate measurements) that require extra workspace of just ${O \bigl( \log \frac{p}{c-s} \bigr)}$ qubits. This in particular gives the first methods of strong amplification for logarithmic-space unitary quantum computations with two-sided bounded error. This also leads to a number of consequences in complexity theory, such as the uselessness of quantum witnesses in bounded-error logarithmic-space unitary quantum computations, the PSPACE upper bound for QMA with exponentially-small completeness-soundness gap, and strong amplification for matchgate computations.