🤖 AI Summary
This work investigates the complexity of quantum state synthesis by introducing and systematically characterizing stateQMA—a QMA-type complexity class tailored for quantum state preparation. Its verifier is a polynomial-time quantum circuit that interacts with a single round of untrusted quantum prover to output a target quantum state. Key contributions are: (1) the first error-reduction theory for stateQMA and its variants—including exponentially small completeness-soundness gaps and bounded-space constraints—enabling exponential precision amplification; (2) a rigorous placement of stateQMA within the quantum state synthesis hierarchy: proving UQMA ⊂ stateQMA, that stateQCMA admits perfect completeness, and that stateQMA ⊆ statePSPACE while being orthogonal to stateBQP; and (3) demonstrating that witness states for UQMA problems can be verified within stateQMA, thereby extending the applicability of quantum interactive proofs to quantum state preparation tasks.
📝 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.
