Quantum systems subject to driving and dissipation display distinctive non-equilibrium phenomena relevant to condensed matter, quantum optics, metrology and quantum error correction. An example is the emergence of phase transitions with uniquely quantum properties, which opposes the intuition that dissipation generally leads to classical behaviour. The quantum and non-equilibrium nature of such systems makes them hard to study with existing tools, such as those from equilibrium statistical mechanics, and represents a challenge for numerical simulations. Quantum computers, however, are well suited to simulating such systems, especially as hardware developments enable the controlled application of dissipative operations in a pristine quantum environment. Here we demonstrate a large-scale accurate quantum simulation of a non-equilibrium phase transition using a trapped-ion quantum computer. We simulate a quantum extension of the classical contact process that has been proposed as a description for driven gases of Rydberg atoms and has stimulated numerous attempts to determine the impact of quantum effects on the classical directed-percolation universality class. We use techniques such as qubit reuse and error avoidance based on real-time conditional logic to implement large instances of the model with 73 sites and up to 72 circuit layers and quantitatively determine the model’s critical properties. Our work demonstrates that today’s quantum computers are able to perform useful simulations of open quantum system dynamics and non-equilibrium phase transitions.
This is a preview of subscription content, access via your institution
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
The data produced by the Quantinuum devices for this work are available online in Supplementary Information. Additional classical simulation results are available upon reasonable request.
The code used to generate data for this work is available upon reasonable request.
Weimer, H., Kshetrimayum, A. & Orús, R. Simulation methods for open quantum many-body systems. Rev. Mod. Phys. 93, 015008 (2021).
Noh, K., Jiang, L. & Fefferman, B. Efficient classical simulation of noisy random quantum circuits in one dimension. Quantum 4, 318 (2020).
Verstraete, F., Wolf, M. M. & Cirac, J. I. Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5, 633 (2009).
Harris, T. E. Contact interactions on a lattice. Ann. Probab. 2, 969 (1974).
Hinrichsen, H. Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49, 815 (2000).
Marcuzzi, M., Buchhold, M., Diehl, S. & Lesanovsky, I. Absorbing state phase transition with competing quantum and classical fluctuations. Phys. Rev. Lett. 116, 245701 (2016).
Carollo, F., Gillman, E., Weimer, H. & Lesanovsky, I. Critical behavior of the quantum contact process in one dimension. Phys. Rev. Lett. 123, 100604 (2019).
Gillman, E., Carollo, F. & Lesanovsky, I. Numerical simulation of critical dissipative non-equilibrium quantum systems with an absorbing state. New J. Phys. 21, 093064 (2019).
Jo, M., Lee, J., Choi, K. & Kahng, B. Absorbing phase transition with a continuously varying exponent in a quantum contact process: a neural network approach. Phys. Rev. Res. 3, 013238 (2021).
Lesanovsky, I., Macieszczak, K. & Garrahan, J. P. Non-equilibrium absorbing state phase transitions in discrete-time quantum cellular automaton dynamics on spin lattices. Quantum Sci. Technol. 4, 02LT02 (2019).
Gillman, E., Carollo, F. & Lesanovsky, I. Nonequilibrium phase transitions in (1 + 1)-dimensional quantum cellular automata with controllable quantum correlations. Phys. Rev. Lett. 125, 100403 (2020).
Gillman, E., Carollo, F. & Lesanovsky, I. Numerical simulation of quantum nonequilibrium phase transitions without finite-size effects. Phys. Rev. A 103, L040201 (2021).
Gillman, E., Carollo, F. & Lesanovsky, I. Quantum and classical temporal correlations in (1 + 1)D quantum cellular automata. Phys. Rev. Lett. 127, 230502 (2021).
Gillman, E., Carollo, F. & Lesanovsky, I. Asynchronism and nonequilibrium phase transitions in (1 + 1)-dimensional quantum cellular automata. Phys. Rev. E 106, L032103 (2022).
Nigmatullin, R., Wagner, E. & Brennen, G. K. Directed percolation in nonunitary quantum cellular automata. Phys. Rev. Res. 3, 043167 (2021).
Henkel, M., Hinrichsen, H. & Lübeck, S. Non-Equilibrium Phase Transitions Vol. 1 (Springer, 2008).
Marro, J. & Dickman, R. Nonequilibrium Phase Transitions in Lattice Models (Cambridge Univ. Press, 1999).
Ódor, G. Universality classes in nonequilibrium lattice systems. Rev. Mod. Phys. 76, 663 (2004).
Jensen, I. Low-density series expansions for directed percolation: I. a new efficient algorithm with applications to the square lattice. J. Phys. A Math. Gen. 32, 5233 (1999).
Hinrichsen, H. Non-equilibrium phase transitions. Physica A 369, 1–28 (2006).
Ryan-Anderson, C. et al. Implementing fault-tolerant entangling gates on the five-qubit code and the color code. Preprint at https://arxiv.org/abs/2208.01863 (2022).
Pino, J. M. et al. Demonstration of the trapped-ion quantum CCD computer architecture. Nature 592, 209 (2021).
Kim, I. H. Holographic quantum simulation. Preprint at https://arxiv.org/abs/1702.02093 (2017).
Foss-Feig, M. et al. Holographic quantum algorithms for simulating correlated spin systems. Phys. Rev. Res. 3, 033002 (2021).
Barratt, F. et al. Parallel quantum simulation of large systems on small NISQ computers. npj Quantum Inf. https://doi.org/10.1038/s41534-021-00420-3 (2021).
Chertkov, E. et al. Holographic dynamics simulations with a trapped-ion quantum computer. Nat. Phys. 18, 1074 (2022).
Niu, D. et al. Holographic simulation of correlated electrons on a trapped ion quantum processor. PRX quantum 3, 030317 (2022).
Lin, S.-H., Dilip, R., Green, A. G., Smith, A. & Pollmann, F. Real- and imaginary-time evolution with compressed quantum circuits. PRX Quantum 2, 010342 (2021).
Zhang, Y., Jahanbani, S., Niu, D., Haghshenas, R. & Potter, A. C. Qubit-efficient simulation of thermal states with quantum tensor networks. Phys. Rev. B 106, 165126 (2022).
Dborin, J. et al. Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer. Nat. Commun. 13, 5977 (2022).
DeCross, M., Chertkov, E., Kohagen, M. & Foss-Feig, M. Qubit-reuse compilation with mid-circuit measurement and reset. Preprint at https://arxiv.org/abs/2210.08039 (2022).
Bonnes, L. & Läuchli, A. M. Superoperators vs. trajectories for matrix product state simulations of open quantum system: a case study. Preprint at https://arxiv.org/abs/1411.4831 (2014).
Verstraete, F., García-Ripoll, J. J. & Cirac, J. I. Matrix product density operators: simulation of finite-temperature and dissipative systems. Phys. Rev. Lett. 93, 207204 (2004).
Cui, J., Cirac, J. I. & Bañuls, M. C. Variational matrix product operators for the steady state of dissipative quantum systems. Phys. Rev. Lett. 114, 220601 (2015).
Mascarenhas, E., Flayac, H. & Savona, V. Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays. Phys. Rev. A 92, 022116 (2015).
Werner, A. H. et al. Positive tensor network approach for simulating open quantum many-body systems. Phys. Rev. Lett. 116, 237201 (2016).
White, C. D., Zaletel, M., Mong, R. S. K. & Refael, G. Quantum dynamics of thermalizing systems. Phys. Rev. B 97, 035127 (2018).
Jaschke, D., Montangero, S. & Carr, L. D. One-dimensional many-body entangled open quantum systems with tensor network methods. Quantum Sci. Technol. 4, 013001 (2018).
Cheng, S. et al. Simulating noisy quantum circuits with matrix product density operators. Phys. Rev. Res. 3, 023005 (2021).
Cheng, Z. & Potter, A. C. Matrix product operator approach to nonequilibrium Floquet steady states. Phys. Rev. B 106, L220307 (2022).
Buchhold, M., Müller, T. & Diehl, S. Revealing measurement-induced phase transitions by pre-selection. Preprint at https://arxiv.org/abs/2208.10506 (2022).
Iadecola, T., Ganeshan, S., Pixley, J. H. & Wilson, J. H. Measurement and feedback driven entanglement transition in the probabilistic control of chaos. Phys. Rev. Lett. 131, 060403 (2023).
Quantinuum System Model H1 product data sheet, version 5.00. Quantinuum https://www.quantinuum.com/products/h1 (2022).
Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).
Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).
Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations. SciPost Phys. Codebases https://doi.org/10.21468/SciPostPhysCodeb.4 (2022).
This work was made possible by a large group of people, and we would like to thank the entire Quantinuum team for their many contributions. The experiments reported in this paper were performed on the Quantinuum system model H1-1 quantum computer (https://www.quantinuum.com/), which is powered by Honeywell ion traps. Numerical calculations were performed using the ITensor library46. We thank S. Diehl, M. Buchold, K. Hemery, H. Dreyer, R. Haghshenas, N. Brown, C. Ryan-Anderson, M. DeCross, K. Mayer, C. Langlett, M. Lubasch, M. Wall, P. D. Blocher, I. Deutsch, M. Iqbal and V. Khemani for helpful discussions. This research was supported in part by the National Science Foundation under grant number PHY-1748958. A.C.P. was supported by Department of Energy DE-SC0022102 and the Alfred P. Sloan Foundation through a Sloan Research Fellowship. A.C.P. and S.G. performed this work in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. This research used resources of the Oak Ridge Leadership Computing Facility, which is a Department of Energy Office of Science User Facility supported under Contract DE-AC05-00OR22725.
The authors declare no competing interests.
Peer review information
Nature Physics thanks Kenji Toyoda and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Chertkov, E., Cheng, Z., Potter, A.C. et al. Characterizing a non-equilibrium phase transition on a quantum computer. Nat. Phys. (2023). https://doi.org/10.1038/s41567-023-02199-w
This article is cited by
Nature Physics (2023)