Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2605.26096

Rounding Almost Commuting Hamiltonians

Islam Faisal, Anand Natarajan, Alexander Poremba

quant-phcond-mat.othercs.CCSubmitted May 25, 2026• Announced May 26, 2026• 0 citations• Revised Oct 6, 2026 (v3)
v1 · May 25, 2026v2 · Jun 24, 2026v3 · Oct 6, 2026
arXivHTMLPDF

Abstract

Commuting Hamiltonians lie at the boundary between classical constraint satisfaction and quantum many-body physics, exhibiting rich quantum structure while remaining more tractable than general noncommuting models. In contrast, physical Hamiltonians are rarely exactly commuting, which naturally motivates the study of almost commuting Hamiltonians. Despite their relevance, the implications of approximate commutation are only poorly understood. In this work, we show how to efficiently approximate any almost commuting $2$-local qubit Hamiltonian by a commuting one: we give a new locality-preserving algorithmic rounding technique that maps any $2$-local Hamiltonian $H=\sum_{i=1}^m h_i$ with $\|[h_i,h_j]\| \leq ε$ to a nearby Hamiltonian $\hat{H}$ whose terms pair-wise commute, and which is within overall distance $\|H-\hat{H}\| = O(m\,ε^{1/3})$. As a consequence, we show that $δ$-approximations to the ground energy for $ε$-almost commuting $2$-local qubit Hamiltonians lie in $\mathsf{NP}$ when $δ\gg mε^{1/3}$, extending the classical containment well beyond the commuting setting. Finally, we present two applications of our rounding framework: Gibbs sampling and fast Hamiltonian simulation for almost commuting systems.

Related papers

Commuting Local Hamiltonians Beyond 2D

John Bostanci, Yeongwoo Hwang

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.

arXiv:2410.10495•Oct 14, 2024•2 citations

Commuting Local Hamiltonians Beyond 2D

Adam Bene Watts, John Bostanci

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]

arXiv:2210.09155•Leibniz international proceedings in informatics•Oct 17, 2022•1 citation

Gibbs state preparation for commuting Hamiltonian: Mapping to classical Gibbs sampling

Yeongwoo Hwang, Jiaqing Jiang

Gibbs state preparation, or Gibbs sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing Gibbs samplers for quantum Hamiltonians have largely focused on commuting local Hamiltonians (CLHs), a non-trivial subclass of Hamiltonians which include highly entangled systems such as the Toric code and quantum double model. Most previous Gibbs samplers relied on simulating the Davies generator, which is a Lindbladian associated with the thermalization process in nature.Instead of using the Davies generator, we design a different Gibbs sampler for various CLHs by giving a reduction to classical Hamiltonians, in the sense that one can efficiently prepare the Gibbs state for some CLH H on a quantum computer as long as one can efficiently do classical Gibbs sampling for the corresponding classical Hamiltonian H ( c ) . We demonstrate that our Gibbs sampler is able to replicate state-of-the-art results as well as prepare the Gibbs state in regimes which were previously unknown, such as the low temperature region, as long as there exists fast mixing Gibbs samplers for the corresponding classical Hamiltonians. Our reductions are as follows.– If H is a 2-local qudit CLH, then H ( c ) is a 2-local qudit classical Hamiltonian.– If H is a 4-local qubit CLH on 2D lattice and there are no classical qubits, then H ( c ) is a 2-local qudit classical Hamiltonian on a planar graph. As an example, our algorithm can prepare the Gibbs state for the (defected) Toric code at any non-zero temperature in O ( n 2 p o l y ( l o g n ) ) time.– If H is a 4-local qubit CLH on 2D lattice and there are classical qubits, assuming that quantum terms are uniformly correctable, then H ( c ) is a constant-local classical Hamiltonian.

arXiv:2410.04909•Quantum•Sep 18, 2026•1 citation

Complexity classification of two-qubit commuting hamiltonians

Adam Bouland, Laura Mančinska, Xue Jun Zhang

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

arXiv:1602.04145•Electronic colloquium on computational complexity•Feb 12, 2016•7 citations

Complexity Classification of Two-Qubit Commuting Hamiltonians

Adam Bouland, Laura Mančinska, Zhang, Xue

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

Leibniz international proceedings in informatics•Jan 1, 2016•5 citations

On the Complexity of Two Dimensional Commuting Local Hamiltonians

Dorit Aharonov, Kenneth, Oded, Itamar Vigdorovich

The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice but this problem too remained open, apart from some very specific cases. Here we consider a wide class of two dimensional CLH instances; these are $k$-local CLHs, for any constant $k$; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [2011] which regarded the special case of $4$-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.

arXiv:1803.02213•Leibniz international proceedings in informatics•Jan 1, 2018•5 citations
Commuting Hamiltonians lie at the boundary between classical constraint satisfaction and quantum many-body physics, exhibiting rich quantum structure while remaining more tractable than general noncommuting models. In contrast, physical Hamiltonians are rarely exactly commuting, which naturally motivates the study of almost commuting Hamiltonians. Despite their relevance, the implications of approximate commutation are only poorly understood.
In this work, we show how to efficiently approximate any almost commuting 222-local qubit Hamiltonian by a commuting one: we give a new locality-preserving algorithmic rounding technique that maps any 222-local Hamiltonian H=∑i=1mhiH=\sum_{i=1}^m h_iH=∑i=1m​hi​ with ∥[hi,hj]∥≤ε\|[h_i,h_j]\| \leq ε∥[hi​,hj​]∥≤ε to a nearby Hamiltonian H^\hat{H}H^ whose terms pair-wise commute, and which is within overall distance ∥H−H^∥=O(m ε1/3)\|H-\hat{H}\| = O(m\,ε^{1/3})∥H−H^∥=O(mε1/3). As a consequence, we show that δδδ-approximations to the ground energy for εεε-almost commuting 222-local qubit Hamiltonians lie in NP\mathsf{NP}NP when δ≫mε1/3δ\gg mε^{1/3}δ≫mε1/3, extending the classical containment well beyond the commuting setting. Finally, we present two applications of our rounding framework: Gibbs sampling and fast Hamiltonian simulation for almost commuting systems.
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.
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]
Gibbs state preparation, or Gibbs sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing Gibbs samplers for quantum Hamiltonians have largely focused on commuting local Hamiltonians (CLHs), a non-trivial subclass of Hamiltonians which include highly entangled systems such as the Toric code and quantum double model. Most previous Gibbs samplers relied on simulating the Davies generator, which is a Lindbladian associated with the thermalization process in nature.Instead of using the Davies generator, we design a different Gibbs sampler for various CLHs by giving a reduction to classical Hamiltonians, in the sense that one can efficiently prepare the Gibbs state for some CLH H on a quantum computer as long as one can efficiently do classical Gibbs sampling for the corresponding classical Hamiltonian H ( c ) . We demonstrate that our Gibbs sampler is able to replicate state-of-the-art results as well as prepare the Gibbs state in regimes which were previously unknown, such as the low temperature region, as long as there exists fast mixing Gibbs samplers for the corresponding classical Hamiltonians. Our reductions are as follows.– If H is a 2-local qudit CLH, then H ( c ) is a 2-local qudit classical Hamiltonian.– If H is a 4-local qubit CLH on 2D lattice and there are no classical qubits, then H ( c ) is a 2-local qudit classical Hamiltonian on a planar graph. As an example, our algorithm can prepare the Gibbs state for the (defected) Toric code at any non-zero temperature in O ( n 2 p o l y ( l o g n ) ) time.– If H is a 4-local qubit CLH on 2D lattice and there are classical qubits, assuming that quantum terms are uniformly correctable, then H ( c ) is a constant-local classical Hamiltonian.
We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.
We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.
The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice but this problem too remained open, apart from some very specific cases. Here we consider a wide class of two dimensional CLH instances; these are kkk-local CLHs, for any constant kkk; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being "Euclidean". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [2011] which regarded the special case of 444-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.