Merlin-Arthur Games and Stoquastic Complexity

TL;DR

This paper introduces the MA-complete stoquastic k-SAT problem, explores the complexity of local stoquastic Hamiltonian problems, and situates average LH-MIN within the AM class, advancing understanding of stoquastic quantum complexity.

Contribution

It defines the first non-trivial MA-complete problem, introduces the StoqMA class, and analyzes the complexity of average local stoquastic Hamiltonian problems.

Findings

Stoquastic k-SAT is MA-complete.

Stoquastic LH-MIN is StoqMA-complete.

Average LH-MIN is in AM.

Abstract

MA is a class of decision problems for which `yes'-instances have a proof that can be efficiently checked by a classical randomized algorithm. We prove that MA has a natural complete problem which we call the stoquastic k-SAT problem. This is a matrix-valued analogue of the satisfiability problem in which clauses are k-qubit projectors with non-negative matrix elements, while a satisfying assignment is a vector that belongs to the space spanned by these projectors. Stoquastic k-SAT is the first non-trivial example of a MA-complete problem. We also study the minimum eigenvalue problem for local stoquastic Hamiltonians that was introduced in quant-ph/0606140, stoquastic LH-MIN. A new complexity class StoqMA is introduced so that stoquastic LH-MIN is StoqMA-complete. Lastly, we consider the average LH-MIN problem for local stoquastic Hamiltonians that depend on a random or `quenched disorder' parameter, stoquastic AV-LH-MIN. We prove that stoquastic AV-LH-MIN is contained in the complexity class \AM, the class of decision problems for which yes-instances have a randomized interactive proof with two-way communication between prover and verifier.