Testing Quantum Satisfiability

Quantum k-SAT (the problem of determining whether a k-local Hamiltonian is frustration-free) is known to be QMA $_1^{}$ -complete for $k\ge 3$ , and hence likely hard for quantum computers to solve. Building on a classical result of Alon and Shapira, we show that quantum k-SAT can be solved in randomised polynomial time given the 'property testing' promise that the instance is either satisfiable (by any state) or far from satisfiable by a product state; by 'far from satisfiable by a product state' we mean that $ϵn^k$ constraints must be removed before a product state solution exists, for some fixed $ϵ>0$ . The proof has two steps: we first show that for a satisfiable instance of quantum k-SAT, most subproblems on a constant number of qubits are satisfiable by a product state. We then show that for an instance of quantum k-SAT which is far from satisfiable by a product state, most subproblems are unsatisfiable by a product state. Given the promise, quantum k-SAT may therefore be solved by checking satisfiability by a product state on randomly chosen subsystems of constant size.