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arXiv:2604.13026

A complexity phase transition at the EPR Hamiltonian

Kunal Marwaha, James Sud

quant-phcond-mat.stat-mechcs.CCSubmitted Apr 14, 2026• Announced Apr 15, 2026• 0 citations
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Abstract

We study the computational complexity of 2-local Hamiltonian problems generated by a positive-weight symmetric interaction term, encompassing many canonical problems in statistical mechanics and optimization. We show these problems belong to one of three complexity phases: QMA-complete, StoqMA-complete, and reducible to a new problem we call EPR*. The phases are physically interpretable, corresponding to the energy level ordering of the local term. The EPR* problem is a simple generalization of the EPR problem of King. Inspired by empirically efficient algorithms for EPR, we conjecture that EPR* is in BPP. If true, this would complete the complexity classification of these problems, and imply EPR* is the transition point between easy and hard local Hamiltonians. Our proofs rely on perturbative gadgets. One simple gadget, when recursed, induces a renormalization-group-like flow on the space of local interaction terms. This gives the correct complexity picture, but does not run in polynomial time. To overcome this, we design a gadget based on a large spin chain, which we analyze via the Jordan-Wigner transformation.

Comments
47 pages, 8 figures

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We study the computational complexity of 2-local Hamiltonian problems generated by a positive-weight symmetric interaction term, encompassing many canonical problems in statistical mechanics and optimization. We show these problems belong to one of three complexity phases: QMA-complete, StoqMA-complete, and reducible to a new problem we call EPR*. The phases are physically interpretable, corresponding to the energy level ordering of the local term.
The EPR* problem is a simple generalization of the EPR problem of King. Inspired by empirically efficient algorithms for EPR, we conjecture that EPR* is in BPP. If true, this would complete the complexity classification of these problems, and imply EPR* is the transition point between easy and hard local Hamiltonians.
Our proofs rely on perturbative gadgets. One simple gadget, when recursed, induces a renormalization-group-like flow on the space of local interaction terms. This gives the correct complexity picture, but does not run in polynomial time. To overcome this, we design a gadget based on a large spin chain, which we analyze via the Jordan-Wigner transformation.
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.
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.
We prove that deciding the Berry phase for parameterised 2-local qubit Hamiltonians is BQP-complete when presented with a classical description of a guiding state, promised to overlap with the ground state of the system. Our results extend to systems with weighted Heisenberg interactions and when restricted to a 2D square or triangular lattice geometry. The techniques we develop leverage the Schrieffer--Wolff transformation, typically used in the construction of perturbative gadget reductions for local Hamiltonian problems, extending it to parameterised families of Hamiltonians. We demonstrate that there exists a choice of parameterised simulator Hamiltonians whose Berry phase well-approximates that of a parameterised target family. Using the perturbative gadget reduction framework of Oliveira and Terhal and of Schuch and Verstraete, we adapt the arguments to parameterised interactions and demonstrate the error bounds in the resulting simulation can be controlled. Additionally, we provide an explicit proof that families of 1-local Hamiltonians have a Berry phase that can be efficiently computed to inverse-polynomial precision. This establishes a complexity transition between 1-local and 2-local Hamiltonian families.
The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King (arXiv:2209.02589). We introduce a polynomial time 1+54≈0.809\frac{1+\sqrt{5}}{4}\approx 0.80941+5​​≈0.809-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of 0.720.720.72 (arXiv:2410.15544). As a special case, this also implies a 1+54\frac{1+\sqrt{5}}{4}41+5​​-approximation for Quantum Max Cut on bipartite instances, improving upon the approximation ratio of 3/43/43/4 that one can infer in a relatively straightforward manner from the work of Lee and Parekh (arXiv:2401.03616).
We consider a class of redudancy-free Hamiltonians, which are those where the trace vanishes for any product of Hamiltonian terms in which at least one term appears an odd number of times. This includes Hamiltonians which are sums of products of Paulis with no relations between them. These Hamiltonians arise naturally in Hamiltonian Decoded Quantum Interferometry, which separates the hardness of preparing quantum states into two steps: a classical decoding step, and a step preparing the thermofield double state of such a redundancy-free Hamiltonian. We focus on the hardness of these Hamiltonians, both for intrinsic interest and to understand the complexity of the state preparation step in Hamiltonian DQI. We present several results showing that the problem becomes easy at quadratically lower temperature than for a general Hamiltonian, with the complexity at given inverse temperature βββ depending on β2dβ^2 dβ2d, where ddd is the degree of the anticommutation graph. We give an efficient classical algorithm for small β2dβ^2 dβ2d to approximate the partition function, while for large β2dβ^2 dβ2d we show that approximating the partition function is NP-hard. To this end, we introduce what we call an anticommutation glass, where frustration arises purely from anticommutation relations. We give a subexponential time quantum algorithm to prepare the thermofield state for small β2dβ^2 dβ2d, and present some results toward a polynomial time algorithm based on the Feiguin-Klich Hamiltonian. As a result of possible independent interest, we show that this Hamiltonian leads to a polynomial time algorithm for preparing thermofield double states of general Hamiltonians on an interaction graph of degree ddd for β≲1/dβ\lesssim 1/dβ≲1/d. Finally, we show that estimating the ground state energy of redundancy-free Hamiltonians is QMA-complete.