Classical Simulation of High Temperature Quantum Ising Models

Elizabeth Crosson; Samuel Slezak

doi:10.22331/q-2025-07-09-1788

Classical Simulation of High Temperature Quantum Ising Models

Elizabeth Crosson¹ and Samuel Slezak^2,3

¹Phasecraft Inc., Washington DC, USA
²Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
³Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA

Published:	2025-07-09, volume 9, page 1788
Editor:	Simon Apers
Eprint:	arXiv:2002.02232v2
Doi:	https://doi.org/10.22331/q-2025-07-09-1788
Citation:	Quantum 9, 1788 (2025).

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Abstract

We consider generalized quantum Ising models, including those which could describe disordered materials or quantum annealers, and we prove that for all temperatures above a system-size independent threshold the path integral Monte Carlo method based on worldline heat-bath updates always mixes to stationarity in time $\mathcal{O}(n \log n)$ for an $n$ qubit system, and therefore provides a fully polynomial-time approximation scheme for the partition function. This result holds whenever the temperature is greater than four plus twice the maximum interaction degree (valence) over all qubits, measured in units of the local coupling strength. For example, this implies that the classical simulation of the thermal state of a superconducting device modeling a frustrated quantum Ising model with maximum valence of 6 and coupling strengths of 1 GHz is always possible at temperatures above 800 mK. Despite the quantum system being at high temperature, the classical spin system resulting from the quantum-to-classical mapping contains strong couplings which cause the single-site Glauber dynamics to mix slowly, therefore this result depends on the use of worldline updates (which are a form of cluster updates that can be implemented efficiently). This result places definite constraints on the temperatures required for a quantum advantage in analog quantum simulation with various NISQ devices based on equilibrium states of quantum Ising models.

► BibTeX data

@article{Crosson2025classicalsimulation,
  doi = {10.22331/q-2025-07-09-1788},
  url = {https://doi.org/10.22331/q-2025-07-09-1788},
  title = {Classical {S}imulation of {H}igh {T}emperature {Q}uantum {I}sing {M}odels},
  author = {Crosson, Elizabeth and Slezak, Samuel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1788},
  month = jul,
  year = {2025}
}

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[3] E. J. Crosson and D. A. Lidar, "Prospects for quantum enhancement with diabatic quantum annealing", Nature Reviews Physics 3 7, 466 (2021).

[4] Álvaro M. Alhambra, "Quantum Many-Body Systems in Thermal Equilibrium", PRX Quantum 4 4, 040201 (2023).

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[6] Tomotaka Kuwahara, Álvaro M. Alhambra, and Anurag Anshu, "Improved Thermal Area Law and Quasilinear Time Algorithm for Quantum Gibbs States", Physical Review X 11 1, 011047 (2021).

[7] M. E. Stroeks, J. Helsen, and B. M. Terhal, "Spectral estimation for Hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution", New Journal of Physics 24 10, 103024 (2022).

[8] Jacob Bringewatt and Michael Jarret, "Effective Gaps Are Not Effective: Quasipolynomial Classical Simulation of Obstructed Stoquastic Hamiltonians", Physical Review Letters 125 17, 170504 (2020).

[9] Ryan L. Mann and Tyler Helmuth, "Efficient algorithms for approximating quantum partition functions", Journal of Mathematical Physics 62 2, 022201 (2021).

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