Quantum Merlin-Arthur proof systems for synthesizing quantum states
Hugo Delavenne, François Le Gall, Yupan Liu, Masayuki Miyamoto
TL;DR
The work probes how to synthesize quantum states within a complexity-theoretic framework, introducing $stateQMA$ as a state-wise analogue of NP and situating it inside $stateQIP$. It develops doubly-preserving error reduction using quantum singular value transformation (QSVT) on projected-unitary encodings, enabling error reduction while preserving both the witness and the resulting state, and extends these ideas to space-bounded variants with exponential-time verification. The authors connect $stateQMA$ to other natural state-synthesizing classes such as $stateBQP$, $statePSPACE$, and $stateQCMA$, showing, among other results, that UQMA witnesses lie in $stateQMA$ and that $stateQCMA$ achieves perfect completeness. The techniques hinge on QSVT, space-efficient Hamiltonian simulation, and weighted circuit-to-Hamiltonian constructions, offering new insights and tools for the complexity of quantum state synthesis with practical implications for verification and state generation in quantum computing.
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.