Commuting Local Hamiltonians Beyond 2D

TL;DR

This work advances the study of commuting local Hamiltonians by focusing on rank-1 projector terms and introducing guided reductions coupled with a flexible rounding framework that leverages Jordan's Lemma and the Structure Lemma. It proves that rank-1 CLHs in 2D are in NP independent of local dimension, and that rank-1 CLHs in certain 3D settings with edge qudits are also in NP, achieved via triangulation and cubulation techniques that reduce higher-dimensional problems to 2-local instances. Central to the approach are new rounding schemes, a rank-1 commutation classification (singular vs reducing), and geometric constructions (triangulation/cubulation) that puncture holes and repair paths to simplify the Hamiltonian while preserving the ground-space structure. The results illuminate the boundary of NP for CLHs, provide a framework for constructively analyzing reductions, and lay groundwork for exploring whether broader CLH families can be efficiently verifiable or exhibit hardness beyond NP. The techniques have potential implications for quantum PCP investigations and the study of area laws in CLHs by enabling classical verifiers for richer quantum systems.

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.

Figures (13)

• Figure 1: An example of a guided reduction. On the left is a Hamiltonian over registers $\mathsf{R}_1 \otimes \dots \otimes \mathsf{R}_5$, with the green regions denoting individual Hamiltonian terms. A guided reduction is composed of a series of applications of a rounding schemes, which iteratively simplify the Hamiltonian. In the first round, the register $\widetilde{\mathsf{R}}_0$ is decomposed into registers $\widetilde{\mathsf{R}}_0^a$ and $\widetilde{\mathsf{R}}_0^b$ (while the non-tilde'd registers are unchanged). In the second round, $\widetilde{\mathsf{R}}_4$ is entirely removed.
• Figure 2: Puncturing holes via the Structure Lemma.
• Figure 3: A degree $4$ register $\mathsf{R}$.
• Figure 4: The process of reducing a 2D CLH instance to a 2-local CLH instance. Notice in the final figure that all terms besides those containing the center of each triangle are $2$-local. In Aharonov_AKV2018_ComplexityTwoDimensional the center terms are simply removed and corrected later.
• Figure 5: Result of \ref{['cor:classical_guided_reduction']}

Theorems & Definitions (80)

• Definition 1: Guided reduction
• Theorem 1.1: Guided reduction from rank-1 2D CLH to 2-local CLH (informal)
• Theorem 1.2: Guided reduction from rank-1 3D* CLH to 2-local CLH (informal)
• Theorem 1.3: Rounding pairs of projectors (informal version of \ref{['thm:rounding']})
• Theorem 1.4: Characterization of commuting rank $1$ projectors (informal version of \ref{['thm:rank_1_commutation_main']})
• Corollary 2.0: Structure of two commuting operators
• Lemma 2.0: Equivalence under projectors, implicit in Irani_IJ2023_CommutingLocalHamiltonian
• Lemma 2.1: Informal version of \ref{['cor:product_structure_2_local']}
• Lemma 3.1: Jordan's Lemma jordan1875essai
• Definition 2: Local Hamiltonian