Quantum State Synthesis: Relation with Decision Complexity Classes and Impossibility of Synthesis Error Reduction

TL;DR

This paper explores the connections between quantum state synthesis complexity and decision complexity classes, revealing that synthesis error cannot generally be reduced and linking synthesis class collapses to decision class collapses.

Contribution

It establishes the relationship between quantum state synthesis complexity classes and decision classes, and proves the impossibility of reducing synthesis error.

Findings

Collapse of synthesis classes implies collapse of decision classes in high error regime

For reasonable errors, BQP and QCMA classes are related through synthesis complexity

Synthesis error cannot be reduced regardless of computational power

Abstract

This work investigates the relationships between quantum state synthesis complexity classes (a recent concept in computational complexity that focuses on the complexity of preparing quantum states) and traditional decision complexity classes. We especially investigate the role of the synthesis error parameter, which characterizes the quality of the synthesis in quantum state synthesis complexity classes. We first show that in the high synthesis error regime, collapse of synthesis classes implies collapse of the equivalent decision classes. For more reasonable synthesis error, we then show a similar relationships for BQP and QCMA. Finally, we show that for quantum state synthesis classes it is in general impossible to improve the quality of the synthesis: unlike the completeness and soundness parameters (which can be improved via repetition), the synthesis error cannot be reduced, even with arbitrary computational power.