Complexity of geometrically local stoquastic Hamiltonians

The QMA-completeness of the local Hamiltonian problem is a landmark result of the field of Hamiltonian complexity that studies the computational complexity of problems in quantum many-body physics. Since its proposal, substantial effort has been invested in better understanding the problem for physically motivated important families of Hamiltonians. In particular, the QMA-completeness of approximating the ground state energy of local Hamiltonians has been extended to the case where the Hamiltonians are geometrically local in one and two spatial dimensions. Among those physically motivated Hamiltonians, stoquastic Hamiltonians play a particularly crucial role, as they constitute the manifestly sign-free Hamiltonians in Monte Carlo approaches. Interestingly, for such Hamiltonians, the problem at hand becomes more ‘‘classical’’, being hard for the class MA (the randomized version of NP) and its complexity has tight connections with derandomization. In this work, we prove that both the two- and one-dimensional geometrically local analogues remain MA-hard with high enough qudit dimension. Moreover, we show that related problems are StoqMA-complete.

💡 Research Summary

The paper investigates the computational complexity of the local Hamiltonian problem when the Hamiltonians are both stoquastic (all off‑diagonal matrix elements are non‑positive in the computational basis) and geometrically local, i.e., their interaction terms act only on neighboring particles arranged on a lattice. Building on the seminal result of Bravyi and Terhal that stoquastic Hamiltonian ground‑state energy estimation (LH‑MIN) is MA‑complete, the authors ask whether this hardness persists when the Hamiltonians are constrained to be geometrically local in one‑ and two‑dimensional lattices, a setting that matches most physically realistic many‑body models.

The main contributions are three theorems. Theorem 1 shows that a frustration‑free stoquastic Hamiltonian defined on a two‑dimensional grid of 14‑dimensional qudits with only two‑local nearest‑neighbour interactions is MA‑complete. Theorem 2 establishes the analogous result for a one‑dimensional chain of 19‑dimensional qudits, again with two‑local interactions. Both proofs consist of a reduction from any MA verification circuit to a Hamiltonian that encodes the circuit’s history state. The reduction must satisfy two stringent requirements: (i) every term of the Hamiltonian must be stoquastic, and (ii) all terms must be geometrically two‑local. To meet (i) the authors restrict the universal gate set to classically reversible gates (Toffoli and X), whose matrix representations have only non‑positive off‑diagonal entries when inserted as –U in the Hamiltonian. To meet (ii) they increase the local dimension of each particle so that three logical qubits can be encoded within a single high‑dimensional qudit; this allows a three‑qubit Toffoli gate to be simulated using only interactions between two neighboring qudits.

In the two‑dimensional construction, the authors follow the earlier circuit‑to‑Hamiltonian layout of Aharonov et al. but replace the six‑dimensional qudits with 14‑dimensional ones, arranging the logical data in a “grid of grids” where two of the sub‑spaces serve as a clock and the remaining encode the computational bits. The Toffoli gate is applied by a sequence of two‑local stoquastic terms that act on adjacent rows or columns, while the clock tracks whether a gate has already acted on a given qubit. The completeness proof uses the standard history‑state argument: if the MA verifier accepts with high probability, the constructed Hamiltonian has a ground state energy ≤ a (the low threshold). The soundness proof shows that any state violating the verification must incur an energy penalty of at least 1/poly(n), because illegal clock configurations or gate violations are penalized by stoquastic projectors that are themselves two‑local.

The one‑dimensional construction is more delicate because only a linear chain of nearest‑neighbour interactions is available. The authors again encode three logical qubits per 19‑dimensional particle, but they must introduce a polynomial‑length “re‑initialization” sub‑routine that propagates the clock and restores qubits to a clean state before the next block of gates. This introduces configurations that are not locally checkable; however, the authors prove that the subspace of such “forbidden” states still has an energy gap comparable to that of the legal subspace, preserving soundness. Completeness follows as before from the existence of a valid history state.

Finally, Theorem 3 lifts the frustration‑free restriction and shows that the general stoquastic local Hamiltonian problem on a two‑dimensional lattice of 14‑state qudits (and similarly on a one‑dimensional chain of 19‑state qudits) is StoqMA‑complete. StoqMA is a complexity class sandwiched between MA and QMA; it captures problems where a quantum Merlin can send a stoquastic witness that a classical Arthur can verify using only stoquastic measurements. The proof adapts the standard circuit‑to‑Hamiltonian construction, inserting the same stoquastic two‑local terms, and shows that the resulting Hamiltonian’s ground‑state energy decision problem is exactly as hard as any problem in StoqMA.

Overall, the paper demonstrates that even under the physically motivated constraints of geometric locality and stoquasticity, the local Hamiltonian problem remains computationally intractable: it is MA‑hard (and MA‑complete when frustration‑free) in both one and two dimensions, and StoqMA‑complete without the frustration‑free assumption. These results enrich the relatively short list of natural MA‑complete problems, provide new tools for studying derandomization conjectures (e.g., MA = NP), and clarify the boundary between quantum and classical computational power in many‑body physics.