A complexity phase transition at the EPR Hamiltonian

2026-04-14

AI Summary

This work investigates the computational complexity classification of local Hamiltonian problems generated by symmetric two-body interactions with positive weights. By analyzing the energy-level ordering of local terms and combining recursive perturbation gadgets with a novel construction based on large spin chains—augmented by the Jordan-Wigner transformation—the study reveals a three-phase structure: QMA-complete, StoqMA-complete, or reducible to a newly introduced problem termed EPR*. The EPR* problem is identified as the critical point marking the complexity phase transition between tractable and intractable regimes and is conjectured to lie in BPP. This result establishes a complete complexity classification for this class of local Hamiltonian problems, providing a theoretical foundation for understanding the boundary between computational ease and hardness in quantum many-body systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• local Hamiltonian
• computational complexity
• complexity phase transition
• EPR*
• QMA

Innovation

• complexity phase transition
• EPR*
• perturbative gadgets
• Jordan-Wigner transformation
• local Hamiltonian