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.

Quantum resource theories

“…Deciding membership of SEP(H) is a notoriously difficult problem. In fact, the problem is NP-Hard (Gharibian, 2010;Gurvits, 2003). To cope with this difficulty, one strategy involves relaxing the separability constraint to encompass a more computationally manageable and experimentally verifiable set of states.…”

“…In some QRTs the optimization problem above is relatively easy, like the resource theory of coherence, and it can be solved using standard techniques in semidefinite programming. For other QRTs, like entanglement theory, the problem above is computationally hard (Gharibian, 2010;Gurvits, 2003) (particularly, the computational time is believed to grow exponentially with the dimension of H). In such cases, the condition that W ∈ F * does not have a simple form.…”

Quantum Parameterized Complexity

“…Proof. The proof follows from Theorem 30 and the fact that Ground State k-Connectivity is QCMA-complete for constant k [16].…”

Introduction to Reconfiguration

“…Work on SATISFIABILITY RECONFIGURATION was later generalized by Gharibian and Sikora to a quantum version [134] and by Wrochna to constraint satisfaction [47], where a constraint satisfaction problem consists of a set of variables and a set of constraints and a solution to the problem is an assignment of values to variables that satisfies all the constraints. Constraint satisfaction reachability was shown to be in FPT when parameterized by k + [79].…”