A complexity phase transition at the EPR Hamiltonian

TL;DR

This paper investigates the computational complexity phases of 2-local Hamiltonian problems, identifying a transition point called EPR* that may separate easy from hard instances, with implications for quantum complexity theory.

Contribution

It introduces the EPR* problem as a new complexity class transition point and provides a physical interpretation of complexity phases in local Hamiltonians.

Findings

Problems are classified into QMA-complete, StoqMA-complete, or reducible to EPR*.

EPR* is conjectured to be in BPP, potentially marking the easy-hard transition.

Perturbative gadgets and a spin chain gadget are used to analyze the complexity landscape.

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.