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Real-time dynamics of lattice gauge theories with a few-qubit quantum computer


Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons1,2. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. This has recently stimulated theoretical effort, using Feynman’s idea of a quantum simulator3,4, to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented5,6,7. Here we report the experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realizing (1 + 1)-dimensional quantum electrodynamics (the Schwinger model8,9) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism10,11, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron–positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields12 in favour of exotic long-range interactions, which can be directly and efficiently implemented on an ion trap architecture13. We explore the Schwinger mechanism of particle–antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulation of high-energy theories using atomic physics experiments—the long-term intention is to extend this approach to real-time quantum simulations of non-Abelian lattice gauge theories.

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Figure 1: Quantum simulation of the Schwinger mechanism.
Figure 2: Encoding Wilson’s lattice gauge theories in digital quantum simulators.
Figure 3: Time evolution of the particle number density, ν.
Figure 4: Time evolution of the vacuum persistence amplitude and entanglement.


  1. Gattringer, C. & Lang, C. B. Quantum Chromodynamics on the Lattice (Springer, 2010)

  2. Calzetta, E. A. & Hu, B. L. Nonequilibrium Quantum Field Theory (Cambridge Univ. Press, 2008)

  3. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)

    MathSciNet  Article  Google Scholar 

  4. Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014)

    ADS  Article  Google Scholar 

  5. Jordan, S. P., Lee, K. S. M. & Preskill, J. Quantum algorithms for quantum field theories. Science 336, 1130–1133 (2012)

    ADS  CAS  Article  Google Scholar 

  6. Wiese, U. J. Ultracold quantum gases and lattice systems: quantum simulation of lattice gauge theories. Ann. Phys. 525, 777–796 (2013)

    MathSciNet  CAS  Article  Google Scholar 

  7. Zohar, E., Cirac, I. & Reznik, B. Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices. Rep. Prog. Phys. 79, 014401 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  8. Schwinger, J. Gauge invariance and mass. II. Phys. Rev. 128, 2425–2429 (1962)

    ADS  MathSciNet  Article  Google Scholar 

  9. Kogut, J. & Susskind, L. Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11, 395–408 (1975)

    ADS  CAS  Article  Google Scholar 

  10. Hebenstreit, F., Berges, J. & Gelfand, D. Real-time dynamics of string breaking. Phys. Rev. Lett. 111, 201601 (2013)

    ADS  CAS  Article  Google Scholar 

  11. Kasper, V., Hebenstreit, F., Oberthaler, M. & Berges, J. Schwinger pair production with ultracold atoms. Preprint at (2015)

  12. Hamer, C. J., Weihong, Z. & Oitmaa, J. Series expansions for the massive Schwinger model in Hamiltonian lattice theory. Phys. Rev. D 56, 55–67 (1997)

    ADS  CAS  Article  Google Scholar 

  13. Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277–284 (2012)

    CAS  Article  Google Scholar 

  14. Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010)

    ADS  CAS  Article  Google Scholar 

  15. Monroe, C. et al. Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Phys. Rev. A 89, 022317 (2014)

    ADS  Article  Google Scholar 

  16. Lanyon, B. P. et al. Universal digital quantum simulation with trapped ions. Science 334, 57–61 (2011)

    ADS  CAS  Article  Google Scholar 

  17. Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  18. Narozhny, N. B. & Fedotov, A. M. Creation of electron-positron plasma with superstrong laser field. Eur. Phys. J. Spec. Top. 223, 1083–1092 (2014)

    CAS  Article  Google Scholar 

  19. Barends, R. et al. Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6, 7654 (2015)

    ADS  CAS  Article  Google Scholar 

  20. Salathé, Y. et al. Digital quantum simulation of spin models with circuit quantum electrodynamics. Phys. Rev. X 5, 021027 (2015)

    Google Scholar 

  21. Zohar, E., Cirac, J. I. & Reznik, B. Simulating compact quantum electrodynamics with ultracold atoms: probing confinement and nonperturbative effects. Phys. Rev. Lett. 109, 125302 (2012)

    ADS  Article  Google Scholar 

  22. Tagliacozzo, L., Celi, A., Zamora, A. & Lewenstein, M. Optical Abelian lattice gauge theories. Ann. Phys. 330, 160–191 (2013)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  23. Banerjee, D. et al. Atomic quantum simulation of dynamical gauge fields coupled to fermionic matter: from string breaking to evolution after a quench. Phys. Rev. Lett. 109, 175302 (2012)

    ADS  CAS  Article  Google Scholar 

  24. Hauke, P., Marcos, D., Dalmonte, M. & Zoller, P. Quantum simulation of a lattice Schwinger model in a chain of trapped ions. Phys. Rev. X 3, 041018 (2013)

    CAS  Google Scholar 

  25. Bañuls, M., Cichy, K., Cirac, I. & Jansen, K. The mass spectrum of the Schwinger model with matrix product states. J. High Energy Phys. 2013, 158 (2013)

  26. Sørensen, A. & Mølmer, K. Entanglement and quantum computation with ions in thermal motion. Phys. Rev. A 62, 022311 (2000)

    ADS  Article  Google Scholar 

  27. Schindler, P. et al. A quantum information processor with trapped ions. New J. Phys. 15, 123012 (2013)

    ADS  Article  Google Scholar 

  28. Gorin, T., Prosen, T., Seligman, T. H. & Znidaric, M. Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 435, 33–156 (2006)

    ADS  Article  Google Scholar 

  29. Heyl, M., Polkovnikov, A. & Kehrein, S. Dynamical quantum phase transitions in the transverse field Ising model. Phys. Rev. Lett. 110, 135704 (2013)

    ADS  CAS  Article  Google Scholar 

  30. Plenio, M. B. & Virmani, S. An introduction to entanglement measures. Quantum Inf. Comput. 7, 1–51 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Banks, T., Susskind, L. & Kogut, J. Strong-coupling calculations of lattice gauge theories: (1 + 1)-dimensional exercises. Phys. Rev. D 13, 1043–1053 (1976)

    ADS  CAS  Article  Google Scholar 

  32. Bañuls, M. C., Cichy, K., Cirac, J. I. & Jansen, K. The mass spectrum of the Schwinger model with matrix product states. J. High Energy Phys. 2013, 158 (2013)

  33. Bañuls, M. C., Cichy, K., Cirac, J. I., Jansen, K. & Saito, H. Matrix product states for lattice field theories. Proc. 31stInt. Symp. on Lattice Field Theory 332 (2013)

  34. Graß, T., Muschik, C., Celi, A., Chhajlany, R. W. & Lewenstein, M. Synthetic magnetic fluxes and topological order in one-dimensional spin systems. Phys. Rev. A 91, 063612 (2015)

    ADS  Article  Google Scholar 

  35. Casanova, J. et al. Quantum simulation of quantum field theories in trapped ions. Phys. Rev. Lett. 107, 260501 (2011)

    ADS  CAS  Article  Google Scholar 

  36. García-Álvarez, L. et al. Fermion-fermion scattering in quantum field theory with superconducting circuits. Phys. Rev. Lett. 114, 070502 (2015)

    ADS  Article  Google Scholar 

  37. Schwinger, J. On gauge invariance and vacuum polarization. Phys. Rev. 82, 664–679 (1951)

    ADS  MathSciNet  Article  Google Scholar 

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We acknowledge discussions with C. Hempel, E. R. Ortega and U. J. Wiese. Financial support was provided by the Austrian Science Fund (FWF), through the SFB FoQuS (FWF project nos F4002-N16 and F4016-N23), by the European Commission via the integrated project SIQS and the ERC synergy grant UQUAM, by the Deutsche Akademie der Naturforscher Leopoldina (grant nos LPDS 2013-07 and LPDR 2015-01), as well as the Institut für Quantenoptik und Quanteninformation GmbH. E.A.M. is a recipient of a DOC fellowship from the Austrian Academy of Sciences. P.S. was supported by the Austrian Science Foundation (FWF) Erwin Schrödinger Stipendium 3600-N27. This research was funded by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office grant W911NF-10-1-0284. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI, or the US Government.

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Authors and Affiliations



E.A.M., C.A.M., M.D. and T.M. developed the research based on discussions with P.Z. and R.B.; E.A.M. and P.S. performed the experiments. E.A.M., C.A.M., P.S. and M.H. analysed the data and carried out numerical simulations. E.A.M., P.S., D.N., A.E. and T.M. contributed to the experimental setup. C.A.M., M.H., M.D., P.H. and P.Z. developed the theory. E.A.M., C.A.M., P.S., M.H., P.H., M.D., P.Z. and R.B. wrote the manuscript and provided revisions. All authors contributed to discussions of the results and the manuscript.

Corresponding authors

Correspondence to Esteban A. Martinez or Christine A. Muschik.

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The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks C. Wunderlich, E. Zohar and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Comparison of the evolutions of the particle number density ν(t) and the rate function λ(t).

The decay of the vacuum persistence probability is characterized by the rate function λ(t), defined by |G(t)|2 = eNλ(t). a, b, Time evolution of ν(t) (a) and λ(t) (b) for different values of the particle mass m and fixed electric field energy J = w, where w is the rate of particle–antiparticle creation and annihilation (see equation (1) in the main text). c, d, Evolution of ν(t) (c) and λ(t) (d) for different values of J and fixed particle mass m = 0 as a function of the dimensionless time wt. e, Comparison the evolutions of ν(t) and λ(t) for J = w and masses m = 0 (upper two curves) and m = w/2 (lower two curves). Error bars correspond to standard deviations estimated from a Monte Carlo bootstrapping procedure.

Source data

Extended Data Figure 2 Finite size effects.

Evolution of the particle number density (top) and the logarithmic negativity En (bottom) for different system sizes N. The logarithmic negativity is evaluated with respect to a cut in the middle of the considered spin chain and quantifies the entanglement between the two halves of the system. Both quantities are shown as a function of the dimensionless time wt for J = m = w. The shaded area corresponds to the time interval explored in the experiment.

Extended Data Figure 3 Experimental pulse sequence.

This laser pulse sequence implements the evolution described in Fig. 2f, g. The pulses are listed in the order in which they are applied, as indicated by the arrows. The pulses in the first box prepare the initial state, those in the second box implement one step of the time evolution, and those in the third box recouple the ions to the computational subspace, that is, bring back their populations to the qubit transition 4S1/2(m = −1/2) to 3D5/2(m = −1/2). The operations shown in the middle box are repeated once per evolution step, resulting in a total number of 12 + 51 × 4 + 6 = 222 pulses for 4 evolution steps. The pulses are labelled in the form Pulse(θ, ϕ, target qubit), where θ is the rotation angle (length) of the pulse, ϕ its phase, and the target qubit is an integer from 1 to 4 for addressed operations or ‘all’ for global operations. ‘R’ denotes a pulse on the qubit transition 4S1/2(m = −1/2) to 3D5/2(m = −1/2). ‘MS’ corresponds to an MS gate on the same transition. The hiding pulses ‘HidingA,B,C’ are applied on the transitions as follows: A, 4S1/2(m = −1/2) to 3D5/2(m = −5/2); B, 4S1/2(m = +1/2) to 3D5/2(m = −1/2); C, 4S1/2(m = +1/2) to 3D5/2(m = −3/2). These transitions are shown in the level scheme at the bottom right. The pulses shown in italics serve the purpose of correcting addressing crosstalk.

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Martinez, E., Muschik, C., Schindler, P. et al. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer. Nature 534, 516–519 (2016).

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