Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers

Khaldoon Ghanem; Alexander Schuckert; Henrik Dreyer

doi:10.22331/q-2023-11-03-1163

Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers

Khaldoon Ghanem, Alexander Schuckert, and Henrik Dreyer

Quantinuum, Leopoldstrasse 180, 80804 Munich, Germany

Published:	2023-11-03, volume 7, page 1163
Eprint:	arXiv:2305.19322v2
Doi:	https://doi.org/10.22331/q-2023-11-03-1163
Citation:	Quantum 7, 1163 (2023).

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Abstract

Simulating properties of quantum materials is one of the most promising applications of quantum computation, both near- and long-term. While real-time dynamics can be straightforwardly implemented, the finite temperature ensemble involves non-unitary operators that render an implementation on a near-term quantum computer extremely challenging. Recently, Lu, Bañuls and Cirac [28] suggested a "time-series quantum Monte Carlo method" which circumvents this problem by extracting finite temperature properties from real-time simulations via Wick's rotation and Monte Carlo sampling of easily preparable states. In this paper, we address the challenges associated with the practical applications of this method, using the two-dimensional transverse field Ising model as a testbed. We demonstrate that estimating Boltzmann weights via Wick's rotation is very sensitive to time-domain truncation and statistical shot noise. To alleviate this problem, we introduce a technique that imposes constraints on the density of states, most notably its non-negativity, and show that this way, we can reliably extract Boltzmann weights from noisy time series. In addition, we show how to reduce the statistical errors of Monte Carlo sampling via a reweighted version of the Wolff cluster algorithm. Our work enables the implementation of the time-series algorithm on present-day quantum computers to study finite temperature properties of many-body quantum systems.

Featured image: Equilibrium properties from quantum dynamics and the noise-induced infrared catastrophe.

Popular summary

Time-series quantum Monte Carlo is a hybrid classical-quantum algorithm for calculating finite temperature properties of quantum systems. While the algorithm has recently been shown to scale favourably on an ideal simulator, we show here that shot noise, accompanying realistic computations, causes the algorithm to become unstable. In this paper, we address this problem and introduce a novel technique for extracting Boltzmann weights from noisy time series, reducing errors by several orders of magnitude. Additionally, we present a sampling algorithm that significantly reduces statistical errors. These advancements make the algorithm practical on present-day quantum computers and a candidate for quantum advantage.

► BibTeX data

@article{Ghanem2023robustextractionof,
  doi = {10.22331/q-2023-11-03-1163},
  url = {https://doi.org/10.22331/q-2023-11-03-1163},
  title = {Robust {E}xtraction of {T}hermal {O}bservables from {S}tate {S}ampling and {R}eal-{T}ime {D}ynamics on {Q}uantum {C}omputers},
  author = {Ghanem, Khaldoon and Schuckert, Alexander and Dreyer, Henrik},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1163},
  month = nov,
  year = {2023}
}

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Cited by

[1] Alexander Schuckert, Or Katz, Lei Feng, Eleanor Crane, Arinjoy De, Mohammad Hafezi, Alexey V. Gorshkov, and Christopher Monroe, "Observation of a finite-energy phase transition in a one-dimensional quantum simulator", Nature Physics 21 3, 374 (2025).

[2] Kazuhiro Seki, Yuta Kikuchi, Tomoya Hayata, and Seiji Yunoki, "Simulating Floquet scrambling circuits on trapped-ion quantum computers", Physical Review Research 7 2, 023032 (2025).

[3] Joel Rajakumar and James D. Watson, "Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians", Quantum 10, 1981 (2026).

[4] Matthew L. Goh and Bálint Koczor, "Direct estimation of the density of states for fermionic systems", Physical Review Research 8 1, 013250 (2026).

[5] Chao Yin and Andrew Lucas, "Heisenberg-limited metrology with perturbing interactions", Quantum 8, 1303 (2024).

[6] Yijia Zhou, Wei Xia, Lin Li, and Weibin Li, "Diagnosing quantum many-body chaos in non-Hermitian quantum spin chain via Krylov complexity", Physical Review Research 7 3, 033281 (2025).

[7] Gianluca Morettini, Luca Capizzi, Maurizio Fagotti, and Leonardo Mazza, "Energy-Filtered Quantum States and the Emergence of Nonlocal Correlations", Physical Review Letters 133 24, 240401 (2024).

[8] Yilun Yang, Arthur Christianen, Mari Carmen Bañuls, Dominik S. Wild, and J. Ignacio Cirac, "Phase-Sensitive Quantum Measurement without Controlled Operations", Physical Review Letters 132 22, 220601 (2024).

[9] Kévin Hémery, Khaldoon Ghanem, Eleanor Crane, Sara L. Campbell, Joan M. Dreiling, Caroline Figgatt, Cameron Foltz, John P. Gaebler, Jacob Johansen, Michael Mills, Steven A. Moses, Juan M. Pino, Anthony Ransford, Mary Rowe, Peter Siegfried, Russell P. Stutz, Henrik Dreyer, Alexander Schuckert, and Ramil Nigmatullin, "Measuring the Loschmidt Amplitude for Finite-Energy Properties of the Fermi-Hubbard Model on an Ion-Trap Quantum Computer", PRX Quantum 5 3, 030323 (2024).

[10] Eric Brunner, Luuk Coopmans, Gabriel Matos, Matthias Rosenkranz, Frederic Sauvage, and Yuta Kikuchi, "Lindblad engineering for quantum Gibbs state preparation under the eigenstate thermalization hypothesis", Quantum 9, 1843 (2025).

[11] Eliot Kapit, Peter Love, Jeffrey Larson, Andrew Sornborger, Eleanor Crane, Alexander Schuckert, Teague Tomesh, Frederic Chong, and Sabre Kais, "Roadblocks and Opportunities in Quantum Algorithms -- Insights from the National Quantum Initiative Joint Algorithms Workshop, May 20--22, 2024", arXiv:2508.13973, (2025).

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