Quantum Merlin-Arthur proof systems for synthesizing quantum states

TL;DR

This paper explores quantum state synthesis complexity classes, introduces error reduction techniques for stateQMA, and establishes relationships among various quantum state generation classes, advancing understanding of quantum proof systems.

Contribution

It introduces error reduction methods for stateQMA and its variants, and clarifies their relationships with classes like stateBQP and statePSPACE, advancing quantum complexity theory.

Findings

Error reduction for stateQMA and variants with small gaps or bounded space.

UQMA witnesses are contained within stateQMA.

stateQCMA achieves perfect completeness.

Abstract

Complexity theory typically focuses on the difficulty of solving computational problems using classical inputs and outputs, even with a quantum computer. In the quantum world, it is natural to apply a different notion of complexity, namely the complexity of synthesizing quantum states. We investigate a state-synthesizing counterpart of the class NP, referred to as stateQMA, which is concerned with preparing certain quantum states through a polynomial-time quantum verifier with the aid of a single quantum message from an all-powerful but untrusted prover. This is a subclass of the class stateQIP recently introduced by Rosenthal and Yuen (ITCS 2022), which permits polynomially many interactions between the prover and the verifier. Our main result consists of error reduction of this class and its variants with an exponentially small gap or bounded space, as well as how this class relates to other fundamental state synthesizing classes, i.e., states generated by uniform polynomial-time quantum circuits (stateBQP) and space-uniform polynomial-space quantum circuits (statePSPACE). Furthermore, we establish that the family of UQMA witnesses, considered as one of the most natural candidates for stateQMA containments, is in stateQMA. Additionally, we demonstrate that stateQCMA achieves perfect completeness.