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Characterizing a non-equilibrium phase transition on a quantum computer

Nature Physics volume 19, pages 1799–1804 (2023)Cite this article

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Abstract

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.

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Fig. 1: A driven-dissipative quantum circuit.
Fig. 2: Qubit reuse and real-time conditional logic.
Fig. 3: Experimental observation of critical scaling in a quantum computer.

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Data availability

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.

Code availability

The code used to generate data for this work is available upon reasonable request.

References

  1. Weimer, H., Kshetrimayum, A. & Orús, R. Simulation methods for open quantum many-body systems. Rev. Mod. Phys. 93, 015008 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  2. Noh, K., Jiang, L. & Fefferman, B. Efficient classical simulation of noisy random quantum circuits in one dimension. Quantum 4, 318 (2020).

    Article  Google Scholar 

  3. Verstraete, F., Wolf, M. M. & Cirac, J. I. Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5, 633 (2009).

    Article  Google Scholar 

  4. Harris, T. E. Contact interactions on a lattice. Ann. Probab. 2, 969 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  5. Hinrichsen, H. Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49, 815 (2000).

    Article  ADS  Google Scholar 

  6. Marcuzzi, M., Buchhold, M., Diehl, S. & Lesanovsky, I. Absorbing state phase transition with competing quantum and classical fluctuations. Phys. Rev. Lett. 116, 245701 (2016).

    Article  ADS  Google Scholar 

  7. Carollo, F., Gillman, E., Weimer, H. & Lesanovsky, I. Critical behavior of the quantum contact process in one dimension. Phys. Rev. Lett. 123, 100604 (2019).

    Article  ADS  Google Scholar 

  8. 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).

    Article  ADS  MathSciNet  Google Scholar 

  9. 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).

    Article  Google Scholar 

  10. 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).

    Article  Google Scholar 

  11. 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).

    Article  ADS  Google Scholar 

  12. Gillman, E., Carollo, F. & Lesanovsky, I. Numerical simulation of quantum nonequilibrium phase transitions without finite-size effects. Phys. Rev. A 103, L040201 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  13. Gillman, E., Carollo, F. & Lesanovsky, I. Quantum and classical temporal correlations in (1 + 1)D quantum cellular automata. Phys. Rev. Lett. 127, 230502 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  14. Gillman, E., Carollo, F. & Lesanovsky, I. Asynchronism and nonequilibrium phase transitions in (1 + 1)-dimensional quantum cellular automata. Phys. Rev. E 106, L032103 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  15. Nigmatullin, R., Wagner, E. & Brennen, G. K. Directed percolation in nonunitary quantum cellular automata. Phys. Rev. Res. 3, 043167 (2021).

    Article  Google Scholar 

  16. Henkel, M., Hinrichsen, H. & Lübeck, S. Non-Equilibrium Phase Transitions Vol. 1 (Springer, 2008).

  17. Marro, J. & Dickman, R. Nonequilibrium Phase Transitions in Lattice Models (Cambridge Univ. Press, 1999).

  18. Ódor, G. Universality classes in nonequilibrium lattice systems. Rev. Mod. Phys. 76, 663 (2004).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. 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).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Hinrichsen, H. Non-equilibrium phase transitions. Physica A 369, 1–28 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  21. 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).

  22. Pino, J. M. et al. Demonstration of the trapped-ion quantum CCD computer architecture. Nature 592, 209 (2021).

    Article  ADS  Google Scholar 

  23. Kim, I. H. Holographic quantum simulation. Preprint at https://arxiv.org/abs/1702.02093 (2017).

  24. Foss-Feig, M. et al. Holographic quantum algorithms for simulating correlated spin systems. Phys. Rev. Res. 3, 033002 (2021).

    Article  Google Scholar 

  25. 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).

  26. Chertkov, E. et al. Holographic dynamics simulations with a trapped-ion quantum computer. Nat. Phys. 18, 1074 (2022).

    Article  Google Scholar 

  27. Niu, D. et al. Holographic simulation of correlated electrons on a trapped ion quantum processor. PRX quantum 3, 030317 (2022).

    Article  ADS  Google Scholar 

  28. 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).

    Article  Google Scholar 

  29. 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).

    Article  ADS  Google Scholar 

  30. Dborin, J. et al. Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer. Nat. Commun. 13, 5977 (2022).

    Article  ADS  Google Scholar 

  31. 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).

  32. 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).

  33. 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).

    Article  ADS  Google Scholar 

  34. 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).

    Article  ADS  Google Scholar 

  35. 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).

    Article  ADS  Google Scholar 

  36. Werner, A. H. et al. Positive tensor network approach for simulating open quantum many-body systems. Phys. Rev. Lett. 116, 237201 (2016).

    Article  ADS  Google Scholar 

  37. White, C. D., Zaletel, M., Mong, R. S. K. & Refael, G. Quantum dynamics of thermalizing systems. Phys. Rev. B 97, 035127 (2018).

    Article  ADS  Google Scholar 

  38. 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).

    Article  ADS  Google Scholar 

  39. Cheng, S. et al. Simulating noisy quantum circuits with matrix product density operators. Phys. Rev. Res. 3, 023005 (2021).

    Article  ADS  Google Scholar 

  40. Cheng, Z. & Potter, A. C. Matrix product operator approach to nonequilibrium Floquet steady states. Phys. Rev. B 106, L220307 (2022).

    Article  ADS  Google Scholar 

  41. Buchhold, M., Müller, T. & Diehl, S. Revealing measurement-induced phase transitions by pre-selection. Preprint at https://arxiv.org/abs/2208.10506 (2022).

  42. 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).

    Article  ADS  MathSciNet  Google Scholar 

  43. Quantinuum System Model H1 product data sheet, version 5.00. Quantinuum https://www.quantinuum.com/products/h1 (2022).

  44. Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  45. Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).

    Google Scholar 

  46. 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).

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Acknowledgements

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.

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Authors and Affiliations

  1. Quantinuum, Broomfield, CO, USA

    Eli Chertkov, Thomas M. Gatterman, Justin A. Gerber, Kevin Gilmore, Dan Gresh, Alex Hall, Aaron Hankin, Mitchell Matheny, Tanner Mengle, David Hayes, Brian Neyenhuis, Russell Stutz & Michael Foss-Feig

  2. Department of Physics, University of Texas at Austin, Austin, TX, USA

    Zihan Cheng & Andrew C. Potter

  3. Department of Physics and Astronomy, and Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada

    Andrew C. Potter

  4. Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ, USA

    Sarang Gopalakrishnan

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Contributions

E.C., M.F.-F., D.H., A.C.P. and S.G. conceived the experiment. T.M.G., J.A.G., K.G., D.G., A. Hall, A. Hankin, M.M., T.M., B.N. and R.S. executed the experiment on the Quantinuum quantum computer. E.C. analysed the experimental data. E.C. and Z.C. performed numerical simulations. E.C., M.F.-F., Z.C., A.C.P., D.H. and S.G. wrote the paper and supplement. All authors edited the paper.

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Correspondence to Eli Chertkov.

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Nature Physics thanks Kenji Toyoda and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–21, Discussion and Table 1.

Supplementary Data

The data presented in the main paper obtained from the H1-1 quantum computer and noise-less classical simulations.

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Chertkov, E., Cheng, Z., Potter, A.C. et al. Characterizing a non-equilibrium phase transition on a quantum computer. Nat. Phys. 19, 1799–1804 (2023). https://doi.org/10.1038/s41567-023-02199-w

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