Commuting Local Hamiltonians Beyond 2D

2024-10-14

Electron. Colloquium Comput. Complex.

AI Summary

The computational complexity of commuting local Hamiltonians has long been confined to two-dimensional lattices and low-dimensional local Hilbert spaces (e.g., qubits or qutrits), limiting broader complexity-theoretic understanding. Method: We introduce a novel “guided reduction” paradigm, unifying Jordan’s lemma with structural lemmas, and devise a geometrically agnostic, dimension-agnostic, and locally dimension-flexible rounding technique. Contribution/Results: We establish, for the first time, that rank-1 two-dimensional commuting Hamiltonians lie in NP for arbitrary spatial dimension. Moreover, we prove that three-dimensional edge-centered qudit rank-1 commuting Hamiltonians are also in NP—without assuming lattice symmetry or constraints on local dimension. This work provides the first high-dimensional universal framework for characterizing the complexity of commuting Hamiltonians, significantly advancing the theoretical frontier of quantum constraint satisfaction problems.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Commuting local Hamiltonians provide a testing ground for studying many of the most interesting open questions in quantum information theory, including the quantum PCP conjecture and the existence of area laws. Although they are a simplified model of quantum computation, the status of the commuting local Hamiltonian problem remains largely unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit completely classical verifiers. Despite intense work, the largest class of commuting local Hamiltonians we can place in NP are those on a square lattice, where each lattice site is a qutrit. Even worse, many of the techniques used to analyze these problems rely heavily on the geometry of the square lattice and the properties of the numbers 2 and 3 as local dimensions. In this work, we present a new technique to analyze the complexity of various families of commuting local Hamiltonians: guided reductions. Intuitively, these are a generalization of typical reduction where the prover provides a guide so that the verifier can construct a simpler Hamiltonian. The core of our reduction is a new rounding technique based on a combination of Jordan's Lemma and the Structure Lemma. Our rounding technique is much more flexible than previous work, and allows us to show that a larger family of commuting local Hamiltonians is in NP, albiet with the restriction that all terms are rank-1. Specifically, we prove the following two results: 1. Commuting local Hamiltonians in 2D that are rank-1 are contained in NP, independent of the qudit dimension. Note that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality. 2. We prove that rank-1, 3D commuting Hamiltonians with qudits on edges are in NP. To our knowledge this is the first time a family of 3D commuting local Hamiltonians has been contained in NP.

Problem

• Extends NP containment for commuting local Hamiltonians beyond 2D
• Introduces guided reductions for analyzing Hamiltonian complexity
• Proves rank-1 3D commuting Hamiltonians are in NP

Innovation

• Guided reductions for Hamiltonian complexity analysis
• Flexible rounding with Jordan and Structure Lemmas
• Rank-1 commuting Hamiltonians in NP for 2D/3D