Quantum Merlin-Arthur proof systems for synthesizing quantum states

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 $\sf{NP}$, referred to as $\sf{stateQMA}$, 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 $\sf{stateQIP}$ recently introduced by Rosenthal and Yuen (ITCS 2022) [57], 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 ($\sf{stateBQP}$) and space-uniform polynomial-space quantum circuits ($\sf{statePSPACE}$). Furthermore, we establish that the family of $\sf{UQMA}$ witnesses, considered as one of the most natural candidates for $\sf{stateQMA}$ containments, is in $\sf{stateQMA}$. Additionally, we demonstrate that $\sf{stateQCMA}$ achieves perfect completeness.