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arXiv:2407.15499 · Quantum

Complexity of geometrically local stoquastic Hamiltonians

Asad Raza, Jens Eisert, Alex B. Grilo

Published Feb 11, 2026• 0 citations
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Abstract

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.

DOI
10.22331/q-2026-02-11-2004

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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.
Despite having an unnatural definition, StoqMA\mathsf{StoqMA}StoqMA plays a central role in Hamiltonian complexity, e.g., in the classification theorem of the complexity of Hamiltonians by Cubitt and Montanaro (SICOMP 2016). Moreover, it lies between the two randomized extensions of NP\mathsf{NP}NP, MA\mathsf{MA}MA and AM\mathsf{AM}AM. Therefore, understanding the exact power of StoqMA\mathsf{StoqMA}StoqMA (and hopefully collapsing it with more natural complexity classes) is of great interest for different reasons. In this work, we take a step further in understanding this complexity class by showing that the Stoquastic Sparse Hamiltonians problem (StoqSH\mathsf{StoqSH}StoqSH) is in StoqMA\mathsf{StoqMA}StoqMA. Since Stoquastic Local Hamiltonians are StoqMA\mathsf{StoqMA}StoqMA-hard, this implies that StoqSH\mathsf{StoqSH}StoqSH is StoqMA\mathsf{StoqMA}StoqMA-complete. We complement this result by showing that the separable version of StoqSH\mathsf{StoqSH}StoqSH is StoqMA(2)\mathsf{StoqMA}(2)StoqMA(2)-complete, where StoqMA(2)\mathsf{StoqMA}(2)StoqMA(2) is the version of StoqMA\mathsf{StoqMA}StoqMA that receives two unentangled proofs.
We continue the study of the Stoquastic Local Hamiltonian problem, a physically motivated restriction of the QMA-complete Local Hamiltonian problem (Kitaev, Shen, and Vyalyi, 2002). For the βββ-gapped, frustration-free case, Bravyi, Bessen, and Terhal (2006) showed that the problem is MA-complete when β=1/poly(n)β= 1/\mathrm{poly}(n)β=1/poly(n). Aharonov and Grilo (2019) derandomized this algorithm and proved membership in NP for constant gap β=Ω(1)β= Ω(1)β=Ω(1). We present an improved algorithm and analysis, establishing membership in NP even when β=Ω(1/(log⁡log⁡n))β= Ω(1/(\log\log n))β=Ω(1/(loglogn)). We complement our result with an explicit example demonstrating why the analysis does not extend directly to β=o(1/log⁡log⁡n)β= o(1/\log\log n)β=o(1/loglogn).
Understanding the entanglement structure of local Hamiltonian ground spaces is a physically motivated problem, with applications ranging from tensor network design to quantum error-correcting codes. To this end, we study the complexity of estimating ground state entanglement, and more generally entropy estimation for low energy states and Gibbs states. We find, in particular, that the classes qq-QAM [Kobayashi, le Gall, Nishimura, SICOMP 2019] (a quantum analogue of public-coin AM) and QMA(2) (QMA with unentangled proofs) play a crucial role for such problems, showing: (1) Detecting a high-entanglement ground state is qq-QAM-complete, (2) computing an additive error approximation to the Helmholtz free energy (equivalently, a multiplicative error approximation to the partition function) is in qq-QAM, (3) detecting a low-entanglement ground state is QMA(2)-hard, and (4) detecting low energy states which are close to product states can range from QMA-complete to QMA(2)-complete. Our results make progress on an open question of [Bravyi, Chowdhury, Gosset and Wocjan, Nature Physics 2022] on free energy, and yield the first QMA(2)-complete Hamiltonian problem using local Hamiltonians (cf. the sparse QMA(2)-complete Hamiltonian problem of [Chailloux, Sattath, CCC 2012]).
We study the complexity of Local Hamiltonian problems under the restriction that every local term is an unweighted occurrence of the same fixed interaction. In the resulting Single-Interaction Hamiltonian (SIH) model, once the interaction is fixed, the Hamiltonian is specified entirely by its ordered interaction hypergraph. Our main technical tool is an exact compiler that converts any fixed finite alphabet of local interactions into a single interaction. On a designated auxiliary subspace, the compiled Hamiltonian reproduces the source Hamiltonian exactly, while the complete spectrum below a chosen separation scale is preserved with multiplicity. Combined with a finite-alphabet normalization, this gives a universal polynomial-time reduction from kkk-Local Hamiltonian to SIH3k+3_{3k+3}3k+3​ up to a known global energy scale and inverse-polynomial approximation. Sharper constructions give QMA-completeness of SIH3_33​ and of geometrically local SIH8_88​ with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH3_33​ and QMA-completeness of 1-Pinned Stoq-SIH4_44​. We also establish hardness results for uniformly system-size-dependent interactions and derive single-interaction formulations of the Hamiltonian quantum PCP conjecture, frustration-free, exponentially precise, and guided Local Hamiltonian problems. These results show that independently chosen local matrices and independently tunable coupling strengths are not necessary for a broad range of Hamiltonian-complexity phenomena.
We study computational problems related to the Schrödinger operator H=−Δ+VH = -Δ+ VH=−Δ+V in the real space under the condition that (i) the potential function VVV is smooth and has its value and derivative bounded within some polynomial of nnn and (ii) VVV only consists of O(1)O(1)O(1)-body interactions. We prove that (i) simulating the dynamics generated by the Schrödinger operator implements universal quantum computation, i.e., it is BQP-hard, and (ii) estimating the ground energy of the Schrödinger operator is as hard as estimating that of local Hamiltonians with no sign problem (a.k.a. stoquastic Hamiltonians), i.e., it is StoqMA-complete. This result is particularly intriguing because the ground energy problem for general bosonic Hamiltonians is known to be QMA-hard and it is widely believed that StoqMA⊊QMA\texttt{StoqMA}\varsubsetneq \texttt{QMA}StoqMAQMA.