Thermal State Preparation via Rounding Promises

Patrick Rall1, Chunhao Wang2, and Pawel Wocjan3

1IBM Quantum, MIT-IBM Watson AI Lab, Cambridge, Massachusetts 02142, USA
2Department of Computer Science and Engineering, Pennsylvania State University
3IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, New York 10598, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


A promising avenue for the preparation of Gibbs states on a quantum computer is to simulate the physical thermalization process. The Davies generator describes the dynamics of an open quantum system that is in contact with a heat bath. Crucially, it does not require simulation of the heat bath itself, only the system we hope to thermalize. Using the state-of-the-art techniques for quantum simulation of the Lindblad equation, we devise a technique for the preparation of Gibbs states via thermalization as specified by the Davies generator.
In doing so, we encounter a severe technical challenge: implementation of the Davies generator demands the ability to estimate the energy of the system unambiguously. That is, each energy of the system must be deterministically mapped to a unique estimate. Previous work showed that this is only possible if the system satisfies an unphysical 'rounding promise' assumption. We solve this problem by engineering a random ensemble of rounding promises that simultaneously solves three problems: First, each rounding promise admits preparation of a 'promised' thermal state via a Davies generator. Second, these Davies generators have a similar mixing time as the ideal Davies generator. Third, the average of these promised thermal states approximates the ideal thermal state.

► BibTeX data

► References

[1] Álvaro M Alhambra. Quantum many-body systems in thermal equilibrium. arXiv:2204.08349, 2022.

[2] Sergey Bravyi, Anirban Chowdhury, David Gosset, and Pawel Wocjan. On the complexity of quantum partition functions. arXiv:2110.15466, 2021.

[3] Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP solvers: Large speed-ups, optimality, and applications to quantum learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 132, page 27, 2019.

[4] Heinz-Peter Breuer and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002.

[5] Chi-Fang Chen and Fernando GSL Brandão. Fast thermalization from the eigenstate thermalization hypothesis. arXiv:2112.07646, 2021.

[6] Chi-Fang Chen, Michael J. Kastoryano, Fernando G. S. L. Brandão, and András Gilyén. Quantum thermal state preparation. arXiv:2303.18224, 2023.

[7] Anirban Narayan Chowdhury and Rolando D Somma. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Information & Computation, 17(1-2):41–64, 2017.

[8] Richard Cleve and Chunhao Wang. Efficient quantum algorithms for simulating Lindblad evolution. arXiv:1612.09512 Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), 2017.

[9] Edward Brian Davies. Quantum Theory of Open Systems. Academic Press, 1976.

[10] Edward Brian Davies. Generators of dynamical semigroups. Journal of Functional Analysis, 34(3):421–432, 1979.

[11] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), pages 193–204, 2019.

[12] Zoe Holmes, Gopikrishnan Muraleedharan, Rolando D. Somma, Yigit Subasi, and Burak Şahinoğlu. Quantum algorithms from fluctuation theorems: Thermal-state preparation. Quantum, 6:825, October 2022.

[13] Mária Kieferová and Nathan Wiebe. Tomography and generative training with quantum Boltzmann machines. Physical Review A, 96(6):062327, 2017.

[14] Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by uniform spectral amplification. arXiv:1707.05391, 2017.

[15] Goran Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2):119–130, 1976.

[16] Xiantao Li and Chunhao Wang. Simulating Markovian open quantum systems using higher-order series expansion. 2212.02051, 2022.

[17] John M Martyn, Zane M Rossi, Andrew K Tan, and Isaac L Chuang. Grand unification of quantum algorithms. PRX Quantum, 2(4):040203, 2021.

[18] Davide Nigro. On the uniqueness of the steady-state solution of the Lindblad–gorini–Kossakowski–Sudarshan equation. Journal of Statistical Mechanics: Theory and Experiment, 2019(4):043202, 2019.

[19] David Poulin and Pawel Wocjan. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Physical Review Letters, 103(22):220502, 2009.

[20] Patrick Rall. Faster coherent quantum algorithms for phase, energy, and amplitude estimation. Quantum, 5:566, 2021.

[21] S. Slezak and E. Crosson. Eigenstate thermalization and quantum Metropolis sampling, 2022. Presentation at QIP 2022. https:/​/​​by4rvu7RMtY.

[22] Herbert Spohn. An algebraic condition for the approach to equilibrium of an open $n$-level system. Letters in Mathematical Physics, 2(1):33–38, 1977.

[23] Kristan Temme, Tobias J Osborne, Karl G Vollbrecht, David Poulin, and Frank Verstraete. Quantum Metropolis sampling. Nature, 471(7336):87–90, 2011.

[24] Joran van Apeldoorn and András Gilyén. Improvements in quantum SDP-solving with applications. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), 2019.

[25] Joran Van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-solvers: Better upper and lower bounds. Quantum, 4:230, 2020.

[26] John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018.

[27] Pawel Wocjan and Kristan Temme. Szegedy walk unitaries for quantum maps. Commun. Math. Phys., 2023.

Cited by

[1] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[2] Chi-Fang Chen, Hsin-Yuan Huang, John Preskill, and Leo Zhou, "Local minima in quantum systems", arXiv:2309.16596, (2023).

[3] Mirko Consiglio, "Variational Quantum Algorithms for Gibbs State Preparation", arXiv:2305.17713, (2023).

[4] Xiantao Li and Chunhao Wang, "Simulating Markovian open quantum systems using higher-order series expansion", arXiv:2212.02051, (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2024-04-19 08:26:48). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2024-04-19 08:26:45).