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.

翻译：我们研究由正权重对称相互作用项生成的2-局域哈密顿量问题的计算复杂性，这类问题涵盖统计力学和优化中的多个经典问题。我们证明这些问题属于三个复杂度相之一：QMA-完备、StoqMA-完备以及可归约到我们称为EPR*的新问题。这些相具有物理可解释性，对应于局域项的能级排序。EPR*问题是King提出的EPR问题的简单推广。受EPR经验高效算法的启发，我们推测EPR*属于BPP类。若该推测成立，将完成这些问题的复杂性分类，并意味着EPR*是易解与难解局域哈密顿量之间的转变点。我们的证明依赖于微扰技巧。其中一种简单技巧在递归时会在局域相互作用项空间上诱导类似重正化群流的演化。这一方法能给出正确的复杂性图景，但无法在多项式时间内运行。为克服此限制，我们设计了一种基于大自旋链的技巧，并通过Jordan-Wigner变换对其进行分析。