A quantum algorithm for the direct estimation of the steady state of open quantum systems

Nathan Ramusat and Vincenzo Savona

Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

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


Simulating the dynamics and the non-equilibrium steady state of an open quantum system are hard computational tasks on conventional computers. For the simulation of the time evolution, several efficient quantum algorithms have recently been developed. However, computing the non-equilibrium steady state as the long-time limit of the system dynamics is often not a viable solution, because of exceedingly long transient features or strong quantum correlations in the dynamics. Here, we develop an efficient quantum algorithm for the direct estimation of averaged expectation values of observables on the non-equilibrium steady state, thus bypassing the time integration of the master equation. The algorithm encodes the vectorized representation of the density matrix on a quantum register, and makes use of quantum phase estimation to approximate the eigenvector associated to the zero eigenvalue of the generator of the system dynamics. We show that the output state of the algorithm allows to estimate expectation values of observables on the steady state. Away from critical points, where the Liouvillian gap scales as a power law of the system size, the quantum algorithm performs with exponential advantage compared to exact diagonalization.

Quantum systems are always subject to the influence of their surrounding environment. This influence very often causes an irreversible evolution of the system over time, leading to a well defined steady state independently of the initial conditions. Simulating the properties of the steady state has the same computational complexity as the simulation of isolated quantum systems, which for large systems is typically intractable with classical computers.

To tackle this problem we develop a quantum algorithm, suitable to be executed on the next generation of quantum computers, that solves the equation for the steady state of an open quantum system. The algorithm brings an exponential speedup over classical computational approaches in most physically relevant cases, thus holding promise as the election tool for the study of complex open quantum systems on upcoming quantum hardware.

► BibTeX data

► References

[1] Iacopo Carusotto and Cristiano Ciuti. Quantum fluids of light. Reviews of Modern Physics, 85 (1): 299–366, feb 2013. ISSN 0034-6861. 10.1103/​RevModPhys.85.299. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.85.299.

[2] Changsuk Noh and Dimitris G Angelakis. Quantum simulations and many-body physics with light. Reports on Progress in Physics, 80 (1): 016401, jan 2017. ISSN 0034-4885. 10.1088/​0034-4885/​80/​1/​016401. URL http:/​/​stacks.iop.org/​0034-4885/​80/​i=1/​a=016401?key=crossref.309394386ceeca0e770e9c91d4731b0a.

[3] Michael J Hartmann. Quantum simulation with interacting photons. Journal of Optics, 18 (10): 104005, sep 2016. 10.1088/​2040-8978/​18/​10/​104005. URL https:/​/​doi.org/​10.1088/​2040-8978/​18/​10/​104005.

[4] Riccardo Rota, Fabrizio Minganti, Cristiano Ciuti, and Vincenzo Savona. Quantum critical regime in a quadratically driven nonlinear photonic lattice. Phys. Rev. Lett., 122: 110405, Mar 2019. 10.1103/​PhysRevLett.122.110405. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.122.110405.

[5] Riccardo Rota and Vincenzo Savona. Simulating frustrated antiferromagnets with quadratically driven qed cavities. Phys. Rev. A, 100: 013838, Jul 2019. 10.1103/​PhysRevA.100.013838. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.100.013838.

[6] Marios H. Michael, Matti Silveri, R. T. Brierley, Victor V. Albert, Juha Salmilehto, Liang Jiang, and S. M. Girvin. New class of quantum error-correcting codes for a bosonic mode. Phys. Rev. X, 6: 031006, Jul 2016. 10.1103/​PhysRevX.6.031006. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.6.031006.

[7] Hendrik Weimer, Augustine Kshetrimayum, and Román Orús. Simulation methods for open quantum many-body systems. arXiv e-prints, art. arXiv:1907.07079, July 2019. URL https:/​/​arxiv.org/​abs/​1907.07079.

[8] H. P. Breuer and F. Petruccione. The theory of open quantum systems. Oxford University Press, Great Clarendon Street, 2002. 10.1093/​acprof:oso/​9780199213900.001.0001.

[9] A. H. Werner, D. Jaschke, P. Silvi, M. Kliesch, T. Calarco, J. Eisert, and S. Montangero. Positive tensor network approach for simulating open quantum many-body systems. Phys. Rev. Lett., 116: 237201, Jun 2016. 10.1103/​PhysRevLett.116.237201. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.116.237201.

[10] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac. Matrix product density operators: Simulation of finite-temperature and dissipative systems. Phys. Rev. Lett., 93: 207204, Nov 2004. 10.1103/​PhysRevLett.93.207204. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.93.207204.

[11] Michael Zwolak and Guifré Vidal. Mixed-state dynamics in one-dimensional quantum lattice systems: A time-dependent superoperator renormalization algorithm. Phys. Rev. Lett., 93: 207205, Nov 2004. 10.1103/​PhysRevLett.93.207205. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.93.207205.

[12] Alexandra Nagy and Vincenzo Savona. Driven-dissipative quantum monte carlo method for open quantum systems. Phys. Rev. A, 97: 052129, May 2018. 10.1103/​PhysRevA.97.052129. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.97.052129.

[13] Alexandra Nagy and Vincenzo Savona. Variational quantum monte carlo method with a neural-network ansatz for open quantum systems. Phys. Rev. Lett., 122: 250501, Jun 2019. 10.1103/​PhysRevLett.122.250501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.122.250501.

[14] Michael J. Hartmann and Giuseppe Carleo. Neural-network approach to dissipative quantum many-body dynamics. Phys. Rev. Lett., 122: 250502, Jun 2019. 10.1103/​PhysRevLett.122.250502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.122.250502.

[15] Filippo Vicentini, Alberto Biella, Nicolas Regnault, and Cristiano Ciuti. Variational neural-network ansatz for steady states in open quantum systems. Phys. Rev. Lett., 122: 250503, Jun 2019. 10.1103/​PhysRevLett.122.250503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.122.250503.

[16] Nobuyuki Yoshioka and Ryusuke Hamazaki. Constructing neural stationary states for open quantum many-body systems. Phys. Rev. B, 99: 214306, Jun 2019. 10.1103/​PhysRevB.99.214306. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.99.214306.

[17] Alberto Biella, Jiasen Jin, Oscar Viyuela, Cristiano Ciuti, Rosario Fazio, and Davide Rossini. Linked cluster expansions for open quantum systems on a lattice. Phys. Rev. B, 97: 035103, Jan 2018. 10.1103/​PhysRevB.97.035103. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.97.035103.

[18] Wim Casteels, Ryan M. Wilson, and Michiel Wouters. Gutzwiller monte carlo approach for a critical dissipative spin model. Phys. Rev. A, 97: 062107, Jun 2018. 10.1103/​PhysRevA.97.062107. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.97.062107.

[19] Wouter Verstraelen, Riccardo Rota, Vincenzo Savona, and Michiel Wouters. Gaussian trajectory approach to dissipative phase transitions: The case of quadratically driven photonic lattices. Phys. Rev. Research, 2: 022037, May 2020. 10.1103/​PhysRevResearch.2.022037. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.2.022037.

[20] Wouter Verstraelen and Michiel Wouters. Gaussian quantum trajectories for the variational simulation of open quantum-optical systems. Applied Sciences, 8 (9), 2018. ISSN 2076-3417. 10.3390/​app8091427. URL https:/​/​www.mdpi.com/​2076-3417/​8/​9/​1427.

[21] Florian Lange, Zala Lenarčič, and Achim Rosch. Time-dependent generalized gibbs ensembles in open quantum systems. Phys. Rev. B, 97: 165138, Apr 2018. 10.1103/​PhysRevB.97.165138. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.97.165138.

[22] Augustine Kshetrimayum, Hendrik Weimer, and Román Orús. A simple tensor network algorithm for two-dimensional steady states. Nature Communications, 8 (1): 1291, 2017. ISSN 2041-1723. URL https:/​/​doi.org/​10.1038/​s41467-017-01511-6.

[23] Eduardo Mascarenhas, Hugo Flayac, and Vincenzo Savona. Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays. Phys. Rev. A, 92: 022116, Aug 2015. 10.1103/​PhysRevA.92.022116. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.92.022116.

[24] Jian Cui, J. Ignacio Cirac, and Mari Carmen Bañuls. Variational matrix product operators for the steady state of dissipative quantum systems. Phys. Rev. Lett., 114: 220601, Jun 2015. 10.1103/​PhysRevLett.114.220601. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.114.220601.

[25] S. Finazzi, A. Le Boité, F. Storme, A. Baksic, and C. Ciuti. Corner-space renormalization method for driven-dissipative two-dimensional correlated systems. Phys. Rev. Lett., 115: 080604, Aug 2015. 10.1103/​PhysRevLett.115.080604. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.080604.

[26] A. Chenu, M. Beau, J. Cao, and A. del Campo. Quantum simulation of generic many-body open system dynamics using classical noise. Phys. Rev. Lett., 118: 140403, Apr 2017. 10.1103/​PhysRevLett.118.140403. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.118.140403.

[27] Hefeng Wang, S. Ashhab, and Franco Nori. Quantum algorithm for simulating the dynamics of an open quantum system. Phys. Rev. A, 83: 062317, Jun 2011. 10.1103/​PhysRevA.83.062317. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.83.062317.

[28] Julio T. Barreiro, Markus Müller, Philipp Schindler, Daniel Nigg, Thomas Monz, Michael Chwalla, Markus Hennrich, Christian F. Roos, Peter Zoller, and Rainer Blatt. An open-system quantum simulator with trapped ions. Nature, 470 (7335): 486–491, Feb 2011. ISSN 1476-4687. 10.1038/​nature09801. URL https:/​/​doi.org/​10.1038/​nature09801.

[29] M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert. Dissipative quantum church-turing theorem. Phys. Rev. Lett., 107: 120501, Sep 2011. 10.1103/​PhysRevLett.107.120501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.107.120501.

[30] Guillermo García-Pérez, Matteo A. C. Rossi, and Sabrina Maniscalco. Ibm q experience as a versatile experimental testbed for simulating open quantum systems. npj Quantum Information, 6 (1): 1, 2020. ISSN 2056-6387. URL https:/​/​doi.org/​10.1038/​s41534-019-0235-y.

[31] Hong-Yi Su and Ying Li. Quantum algorithm for the simulation of open-system dynamics and thermalization. Phys. Rev. A, 101: 012328, Jan 2020. 10.1103/​PhysRevA.101.012328. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.101.012328.

[32] Zixuan Hu, Rongxin Xia, and Sabre Kais. A quantum algorithm for evolving open quantum dynamics on quantum computing devices. Scientific Reports, 10 (1): 3301, 2020. ISSN 2045-2322. URL https:/​/​doi.org/​10.1038/​s41598-020-60321-x.

[33] Ryan Sweke, Ilya Sinayskiy, and Francesco Petruccione. Simulation of single-qubit open quantum systems. Phys. Rev. A, 90: 022331, Aug 2014. 10.1103/​PhysRevA.90.022331. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.90.022331.

[34] Ryan Sweke, Ilya Sinayskiy, Denis Bernard, and Francesco Petruccione. Universal simulation of markovian open quantum systems. Phys. Rev. A, 91: 062308, Jun 2015. 10.1103/​PhysRevA.91.062308. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.91.062308.

[35] R. Di Candia, J. S. Pedernales, A. del Campo, E. Solano, and J. Casanova. Quantum simulation of dissipative processes without reservoir engineering. Scientific Reports, 5 (1): 9981, 2015. ISSN 2045-2322. URL https:/​/​doi.org/​10.1038/​srep09981.

[36] R. Sweke, M. Sanz, I. Sinayskiy, F. Petruccione, and E. Solano. Digital quantum simulation of many-body non-markovian dynamics. Phys. Rev. A, 94: 022317, Aug 2016. 10.1103/​PhysRevA.94.022317. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.94.022317.

[37] Andrew M. Childs and Tongyang Li. Efficient simulation of sparse markovian quantum dynamics. Quantum Info. Comput., 17 (11–12): 901–947, September 2017. ISSN 1533-7146. URL https:/​/​dl.acm.org/​doi/​10.5555/​3179568.3179569.

[38] Richard Cleve and Chunhao Wang. Efficient Quantum Algorithms for Simulating Lindblad Evolution. arXiv e-prints, art. arXiv:1612.09512, December 2016. URL https:/​/​arxiv.org/​abs/​1612.09512.

[39] Zidu Liu, L. M. Duan, and Dong-Ling Deng. Solving Quantum Master Equations with Deep Quantum Neural Networks. arXiv e-prints, art. arXiv:2008.05488, August 2020. URL https:/​/​arxiv.org/​abs/​2008.05488.

[40] Barbara M. Terhal and David P. DiVincenzo. Problem of equilibration and the computation of correlation functions on a quantum computer. Phys. Rev. A, 61: 022301, Jan 2000. 10.1103/​PhysRevA.61.022301. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.61.022301.

[41] Nobuyuki Yoshioka, Yuya O. Nakagawa, Kosuke Mitarai, and Keisuke Fujii. Variational quantum algorithm for nonequilibrium steady states. Phys. Rev. Research, 2: 043289, Nov 2020. 10.1103/​PhysRevResearch.2.043289. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.2.043289.

[42] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103: 150502, Oct 2009. 10.1103/​PhysRevLett.103.150502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.103.150502.

[43] Emanuel Knill, Gerardo Ortiz, and Rolando D. Somma. Optimal quantum measurements of expectation values of observables. Phys. Rev. A, 75: 012328, Jan 2007. 10.1103/​PhysRevA.75.012328. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.75.012328.

[44] Seth Lloyd. Universal quantum simulators. Science, 273 (5278): 1073–1078, 1996. ISSN 0036-8075. 10.1126/​science.273.5278.1073. URL https:/​/​science.sciencemag.org/​content/​273/​5278/​1073.

[45] 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, apr 2019. 10.1088/​1742-5468/​ab0c1c. URL https:/​/​doi.org/​10.1088.

[46] Fabrizio Minganti, Alberto Biella, Nicola Bartolo, and Cristiano Ciuti. Spectral theory of liouvillians for dissipative phase transitions. Phys. Rev. A, 98: 042118, Oct 2018. 10.1103/​PhysRevA.98.042118. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.98.042118.

[47] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information (10th Anniversary edition). Cambridge University Press, 2016. ISBN 978-1-10-700217-3. 10.1017/​CBO9780511976667.

[48] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Phys. Rev. A, 52: 3457–3467, Nov 1995. 10.1103/​PhysRevA.52.3457. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.52.3457.

[49] Nathan Lacroix, Christoph Hellings, Christian Kraglund Andersen, Agustin Di Paolo, Ants Remm, Stefania Lazar, Sebastian Krinner, Graham J. Norris, Mihai Gabureac, Johannes Heinsoo, Alexandre Blais, Christopher Eichler, and Andreas Wallraff. Improving the performance of deep quantum optimization algorithms with continuous gate sets. PRX Quantum, 1: 110304, Oct 2020. 10.1103/​PRXQuantum.1.020304. URL https:/​/​link.aps.org/​doi/​10.1103/​PRXQuantum.1.020304.

[50] Yunseong Nam, Yuan Su, and Dmitri Maslov. Approximate quantum fourier transform with o(n log(n)) t gates. npj Quantum Information, 6 (1): 26, Mar 2020. ISSN 2056-6387. 10.1038/​s41534-020-0257-5. URL https:/​/​doi.org/​10.1038/​s41534-020-0257-5.

[51] R. Cleve and J. Watrous. Fast parallel circuits for the quantum fourier transform. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 526–536, 2000. 10.1109/​SFCS.2000.892140.

[52] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. Theory of trotter error with commutator scaling. Phys. Rev. X, 11: 011020, Feb 2021. 10.1103/​PhysRevX.11.011020. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.11.011020.

[53] Francesco Tacchino, Alessandro Chiesa, Stefano Carretta, and Dario Gerace. Quantum computers as universal quantum simulators: State-of-the-art and perspectives. Advanced Quantum Technologies, 3 (3): 1900052, 2020. https:/​/​doi.org/​10.1002/​qute.201900052. URL https:/​/​onlinelibrary.wiley.com/​doi/​abs/​10.1002/​qute.201900052.

[54] Marko Žnidarič. Relaxation times of dissipative many-body quantum systems. Phys. Rev. E, 92: 042143, Oct 2015. 10.1103/​PhysRevE.92.042143. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevE.92.042143.

[55] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, feb 2016. 10.1088/​1367-2630/​18/​2/​023023. URL https:/​/​doi.org/​10.1088.

[56] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (1): 4213, 2014. ISSN 2041-1723. URL https:/​/​doi.org/​10.1038/​ncomms5213.

[57] A. Roggero and A. Baroni. Short-depth circuits for efficient expectation-value estimation. Phys. Rev. A, 101: 022328, Feb 2020. 10.1103/​PhysRevA.101.022328. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.101.022328.

[58] R. Rota, F. Storme, N. Bartolo, R. Fazio, and C. Ciuti. Critical behavior of dissipative two-dimensional spin lattices. Phys. Rev. B, 95: 134431, Apr 2017. 10.1103/​PhysRevB.95.134431. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.95.134431.

[59] Di Luo, Zhuo Chen, Juan Carrasquilla, and Bryan K. Clark. Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation. arXiv e-prints, art. arXiv:2009.05580, September 2020. URL https:/​/​arxiv.org/​abs/​2009.05580.

Cited by

[1] Elias Zapusek, Alisa Javadi, and Florentin Reiter, "Nonunitary gate operations by dissipation engineering", Quantum Science and Technology 8 1, 015001 (2023).

[2] Jonathan Wei Zhong Lau, Kian Hwee Lim, Kishor Bharti, Leong-Chuan Kwek, and Sai Vinjanampathy, "Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor", Physical Review Letters 130 24, 240601 (2023).

[3] Alexander Miessen, Pauline J. Ollitrault, Francesco Tacchino, and Ivano Tavernelli, "Quantum algorithms for quantum dynamics", Nature Computational Science 3 1, 25 (2022).

[4] Di Luo, Zhuo Chen, Juan Carrasquilla, and Bryan K. Clark, "Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation", Physical Review Letters 128 9, 090501 (2022).

[5] Alejandro Kunold, "Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians", Physica Scripta 99 6, 065111 (2024).

[6] Thomas J. Elliott and Mile Gu, "Embedding memory-efficient stochastic simulators as quantum trajectories", Physical Review A 109 2, 022434 (2024).

[7] Wouter Verstraelen, Dolf Huybrechts, Tommaso Roscilde, and Michiel Wouters, "Quantum and Classical Correlations in Open Quantum Spin Lattices via Truncated-Cumulant Trajectories", PRX Quantum 4 3, 030304 (2023).

[8] Jin-Min Liang, Qiao-Qiao Lv, Zhi-Xi Wang, and Shao-Ming Fei, "Assisted quantum simulation of open quantum systems", iScience 26 4, 106306 (2023).

[9] Wibe A. de Jong, Kyle Lee, James Mulligan, Mateusz Płoskoń, Felix Ringer, and Xiaojun Yao, "Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model", Physical Review D 106 5, 054508 (2022).

[10] Sara Santos, Xinyu Song, and Vincenzo Savona, "Low-Rank Variational Quantum Algorithm for the Dynamics of Open Quantum Systems", arXiv:2403.05908, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-07-15 10:34:54) and SAO/NASA ADS (last updated successfully 2024-07-15 10:34:55). The list may be incomplete as not all publishers provide suitable and complete citation data.