Local Hamiltonian Problem with succinct ground state is MA-Complete

Finding the ground energy of a quantum system is a fundamental problem in condensed matter physics and quantum chemistry. Existing classical algorithms for tackling this problem often assume that the ground state has a succinct classical description, i.e. a poly-size classical circuit for computing the amplitude. Notable examples of succinct states encompass matrix product states, contractible projected entangled pair states, and states that can be represented by classical neural networks. We study the complexity of the local Hamiltonian problem with succinct ground state. We prove this problem is MA-Complete. The Hamiltonian we consider is general and might not be stoquastic. The MA verification protocol is based on the fixed node quantum Monte Carlo method, particularly the variant of the continuous-time Markov chain introduced by Bravyi et.al. [BCGL22]. Based on our work, we also introduce a notion of strong guided states, and conjecture that the local Hamiltonian problem with strong guided state is MA-Complete, which will be in contrast with the QCMA-Complete result of the local Hamiltonian problem with standard guided states [WFC23,GLG22].