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). Additionally, we demonstrate that stateQCMA is closed under perfect completeness. Our proof techniques are based on the quantum singular value transformation introduced by Gily\'en, Su, Low, and Wiebe (STOC 2019), and its adaption to achieve exponential precision with a bounded space.

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，即使是在量子计算机上也是如此。在量子世界中，自然可以采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类在态合成方面的对应类，称为stateQMA，其关注通过一个多项式时间量子验证器，在来自全能但不可信证明者的单个量子消息的帮助下，制备某些量子态。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多次交互。我们的主要结果包括该类及其变体的错误约简（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的关系，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）。此外，我们证明了stateQCMA在完全完备性下是封闭的。我们的证明技术基于Gilyén、Su、Low和Wiebe（STOC 2019）引入的量子奇异值变换，以及其在有界空间下实现指数精度的适应性调整。