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 (LHP) with a succinct ground state. We prove that this problem is <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mtext mathvariant="bold">MA</a:mtext></a:math>-<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><e:mtext mathvariant="bold">complete</e:mtext></e:math>. The Hamiltonian that we consider is general and might not be stoquastic. The <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><i:mtext mathvariant="bold">MA</i:mtext></i:math> verification protocol is based on the fixed-node quantum Monte Carlo method, particularly the variant of the continuous-time Markov chain introduced by Bravyi, Carleo, Gosset, and Liu. Based on our work, we also introduce a notion of strong guided states and conjecture that the LHP with a strong guided state is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><m:mtext mathvariant="bold">MA</m:mtext></m:math>-<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><q:mtext mathvariant="bold">complete</q:mtext></q:math>, which will be in contrast with the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><u:mtext mathvariant="bold">QCMA</u:mtext></u:math>-<y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><y:mtext mathvariant="bold">complete</y:mtext></y:math> result of the LHP with standard guided states.