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

Yeongwoo Hwang and Jiaqing Jiang

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AbstractGibbs 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 $O(n^2 poly(log n))$ 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.► BibTeX data@article{Hwang2026gibbsstate, doi = {10.22331/q-2026-09-18-2209}, url = {https://doi.org/10.22331/q-2026-09-18-2209}, title = {Gibbs state preparation for commuting {H}amiltonian: {M}apping to classical {G}ibbs sampling}, author = {Hwang, Yeongwoo and Jiang, Jiaqing}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2209}, month = sep, year = {2026} }► References [1] Ángela Capel, Cambyse Rouzé, and Daniel Stilck França. ``The modified logarithmic sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions'' (2020). arXiv:2009.11817. arXiv:2009.11817 [2] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. ``Rapid thermalization of spin chain commuting hamiltonians''.

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Foundations and Trends in Theoretical Computer Science 10, 159–282 (2015). doi: 10.1561/0400000066. https://doi.org/10.1561/0400000066 [58] Dorit Aharonov and Lior Eldar. ``The commuting local hamiltonian on locally-expanding graphs is in np'' (2013). doi: 10.1007/s11128-014-0877-9. arXiv:1311.7378. https://doi.org/10.1007/s11128-014-0877-9 arXiv:1311.7378 [59] Dorit Aharonov and Lior Eldar. ``On the complexity of commuting local hamiltonians, and tight conditions for topological order in such systems''. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science. Pages 334–343. IEEE (2011). doi: 10.1109/FOCS.2011.58. arXiv:1102.0770. https://doi.org/10.1109/FOCS.2011.58 arXiv:1102.0770Cited byCould not fetch Crossref cited-by data during last attempt 2026-09-18 09:51:03: Could not fetch cited-by data for 10.22331/q-2026-09-18-2209 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-09-18 09:51:04: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractGibbs 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 $O(n^2 poly(log n))$ 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.► BibTeX data@article{Hwang2026gibbsstate, doi = {10.22331/q-2026-09-18-2209}, url = {https://doi.org/10.22331/q-2026-09-18-2209}, title = {Gibbs state preparation for commuting {H}amiltonian: {M}apping to classical {G}ibbs sampling}, author = {Hwang, Yeongwoo and Jiang, Jiaqing}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2209}, month = sep, year = {2026} }► References [1] Ángela Capel, Cambyse Rouzé, and Daniel Stilck França. ``The modified logarithmic sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions'' (2020). arXiv:2009.11817. arXiv:2009.11817 [2] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé. ``Rapid thermalization of spin chain commuting hamiltonians''.

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