Ground State Connectivity of Local Hamiltonians

Abstract

The study of ground state energies of local Hamiltonians has played a fundamental role in quantum complexity theory. In this article, we take a new direction by introducing the physically motivated notion of “ground state connectivity” of local Hamiltonians, which captures problems in areas ranging from quantum stabilizer codes to quantum memories. Roughly, “ground state connectivity” corresponds to the natural question: Given two ground states |Ψ〉 and |ϕ〉 of a local Hamiltonian H , is there an “energy barrier” (with respect to H ) along any sequence of local operations mapping |Ψ〉 to |ϕ〉? We show that the complexity of this question can range from QCMA-complete to PSPACE-complete, as well as NEXP-complete for an appropriately defined “succinct” version of the problem. As a result, we obtain a natural QCMA-complete problem, a goal which has generally proven difficult since the conception of QCMA over a decade ago. Our proofs rely on a new technical tool, the Traversal Lemma, which analyzes the Hilbert space a local unitary evolution must traverse under certain conditions. We show that this lemma is essentially tight with respect to the length of the unitary evolution in question.