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arXiv:2309.04910 · Communications in Mathematical Physics

Commuting Local Hamiltonian Problem on 2D Beyond Qubits

Sandy Irani, Jiaqing Jiang

Published Oct 3, 2025• 2 citations
arXivHTMLPDF
DOI
10.1007/s00220-025-05462-8

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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 nature of entanglement. However, unlike the general local Hamiltonian problem, the exact complexity of the commuting local Hamiltonian problem (CLH) remains unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit classical verifiers. Despite intense work, proofs placing CLH in NP rely heavily on an underlying 2D lattice structure, or a very constrained local dimension and locality. 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 for pairs of projectors and the Structure Lemma for C^* algebras. Our rounding technique is much more flexible than previous work and allows us to remove constraints on local dimension in exchange for a rank-1 assumption. Using our rounding technique, we prove the following two results: 1) 2D-CLH for rank-1 instances are contained in NP, independent of the qudit dimension. It is notable that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality of the Hamiltonian terms. 2) 3D-CLH for rank-1 instances are in NP. To our knowledge this is the first time a family of {3D} commuting local Hamiltonians has been contained in NP. Our results apply to Hamiltonians with large qudit degree and remain non-trivial despite the quantum Lovász Local Lemma. [Andris Ambainis et al., 2012]

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The commuting local-Hamiltonian (CLH) problem is a restriction of the local-Hamiltonian problem, in which the terms of the Hamiltonian are required to pairwise commute. A long line of work has shown that the problem lies in $\mathsf{NP}$ for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is $\mathsf{QMA}$-complete. CLH is complete for the complexity class $\mathsf{QIMA}$, defined through quantum verifiers whose local gates commute. Therefore, CLH is $\mathsf{QMA}$-complete if and only if $\mathsf{QIMA}=\mathsf{QMA}$. In this work, we introduce a classical-oracle analogue $\mathsf{QIMA}^{\mathcal{O}}$ and construct a classical oracle $\mathcal{O}$ such that $\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}$. Since $\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}$ for any classical oracle $\mathcal{O}$, this implies $\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}$ for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is $\mathsf{BQP}$-hard, and hence also against the possibility that it is $\mathsf{QMA}$-complete.

quant-pharXiv:2609.38957•Oct 1, 2026•0 citations
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 nature of entanglement. However, unlike the general local Hamiltonian problem, the exact complexity of the commuting local Hamiltonian problem (CLH) remains unknown. A number of works have shown that increasingly expressive families of commuting local Hamiltonians admit classical verifiers. Despite intense work, proofs placing CLH in NP rely heavily on an underlying 2D lattice structure, or a very constrained local dimension and locality. 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 for pairs of projectors and the Structure Lemma for C^* algebras. Our rounding technique is much more flexible than previous work and allows us to remove constraints on local dimension in exchange for a rank-1 assumption. Using our rounding technique, we prove the following two results: 1) 2D-CLH for rank-1 instances are contained in NP, independent of the qudit dimension. It is notable that this family of commuting local Hamiltonians has no restriction on the local dimension or the locality of the Hamiltonian terms. 2) 3D-CLH for rank-1 instances are in NP. To our knowledge this is the first time a family of {3D} commuting local Hamiltonians has been contained in NP. Our results apply to Hamiltonians with large qudit degree and remain non-trivial despite the quantum Lovász Local Lemma. [Andris Ambainis et al., 2012]
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.
The commuting local-Hamiltonian (CLH) problem is a restriction of the local-Hamiltonian problem, in which the terms of the Hamiltonian are required to pairwise commute. A long line of work has shown that the problem lies in NP\mathsf{NP}NP for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is QMA\mathsf{QMA}QMA-complete. CLH is complete for the complexity class QIMA\mathsf{QIMA}QIMA, defined through quantum verifiers whose local gates commute. Therefore, CLH is QMA\mathsf{QMA}QMA-complete if and only if QIMA=QMA\mathsf{QIMA}=\mathsf{QMA}QIMA=QMA. In this work, we introduce a classical-oracle analogue QIMAO\mathsf{QIMA}^{\mathcal{O}}QIMAO and construct a classical oracle O\mathcal{O}O such that BQPO⊈QIMAO\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}BQPO⊆QIMAO. Since BQPO⊆QMAO\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}BQPO⊆QMAO for any classical oracle O\mathcal{O}O, this implies QIMAO≠QMAO\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}QIMAO=QMAO for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is BQP\mathsf{BQP}BQP-hard, and hence also against the possibility that it is QMA\mathsf{QMA}QMA-complete.