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 a 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, is in stateQMA. Additionally, we demonstrate that stateQCMA achieves perfect completeness.

翻译：复杂性理论通常关注使用经典输入和输出解决计算问题的难度，即使借助量子计算机也是如此。在量子世界中，自然可以应用不同的复杂性概念，即量子态合成的复杂性。我们研究了NP类的态合成对应物，称为stateQMA，其关注的是通过一个多项式时间量子验证器，借助一个全知但不可信证明者发送的单个量子消息来制备特定量子态。这是Rosenthal和Yuen（ITCS 2022）最近引入的stateQIP类的一个子类，后者允许证明者和验证者之间进行多项式次交互。我们的主要结果包括该类及其变体的误差降低（具有指数小间隙或有界空间），以及该类与其他基本态合成类（即由均匀多项式时间量子电路生成的态（stateBQP）和空间均匀多项式空间量子电路生成的态（statePSPACE））之间的关系。此外，我们证明了被视为最自然候选之一的UQMA见证族属于stateQMA。我们还展示了stateQCMA可以实现完美完备性。