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

Journal name:
Nature
Volume:
534,
Pages:
516–519
Date published:
DOI:
doi:10.1038/nature18318
Received
Accepted
Published online

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.

At a glance

Figures

  1. Quantum simulation of the Schwinger mechanism.
    Figure 1: Quantum simulation of the Schwinger mechanism.

    a, The instability of the vacuum due to quantum fluctuations is one of the most fundamental effects in gauge theories. We simulate the coherent real-time dynamics of particle–antiparticle creation by realizing the Schwinger model (one-dimensional quantum electrodynamics) on a lattice, as described in the main text. b, The experimental setup for the simulation consists of a linear Paul trap, where a string of 40Ca+ ions is confined. The electronic states of each ion, depicted as horizontal lines, encode a spin |↑〉 or |↓〉. These states can be manipulated using laser beams (see Methods for details).

  2. Encoding Wilson’s lattice gauge theories in digital quantum simulators.
    Figure 2: Encoding Wilson’s lattice gauge theories in digital quantum simulators.

    Matter fields, represented by one-component fermion fields at sites n, interact via equation (1) with gauge variables defined on the links connecting the sites. a, Unoccupied odd (occupied even) sites, represented by filled (empty) circles, indicate the presence of an electron (positron). b, Gauge variables (shown as horizontal blue thick lines) are represented by operators with integer eigenvalues Ln = 0, ±1,…, ±∞. c, By mapping the fields to Pauli operators , we obtain a spin model (the spins are represented by filled/empty arrows). In this language, the Gauss law governing the interaction of fermions and gauge variables reads , where is the diagonal Pauli matrix. The realization of the Schwinger model on a small-scale device requires an optimized use of resources. We achieve this by eliminating the gauge fields at the cost of obtaining a model with long-range couplings (and additional local terms). More specifically, the Gauss law determines the gauge fields for a given matter configuration and background field ϵ0. The elimination of the operators transforms the original model with nearest-neighbour terms into a pure spin model with long-range couplings that corresponds to the Coulomb interaction between the charged particles. d, Coupling matrix of the resulting interactions for N = 10, along with the total spin Hamiltonian . For illustration, e shows the couplings involving the fifth spin. The colours (and thicknesses) of lines represent the different interaction strengths cij according to the matrix shown in d. For implementing in a scalable and efficient way, we introduce time steps of length T (f), each subdivided into three sections (g). In each of these (length not to scale), one of the three parts of is realized as explained in Methods. h, The protocol for realizing for N = 10. The ions interact according to the Mølmer–Sørensen (MS) Hamiltonian . During each short time window of length ΔtI, a different set of ions is coupled by .

  3. Time evolution of the particle number density, ν.
    Figure 3: Time evolution of the particle number density, ν.

    a, We show the ideal evolution under the Schwinger Hamiltonian shown in Fig. 2d, the ideal evolution considering time discretization errors (see Fig. 2), the expected evolution including an experimental (exp.) error model (see Methods) and the experimental data for electric field energy J = w and particle mass m = 0.5w (see equation (1)). After postselection of the experimental data (see Methods), the remaining populations are {86 ± 2, 79 ± 1, 73 ± 1, 69 ± 1}% after {1, 2, 3, 4} time steps (averaged over all data sets). Error bars correspond to standard deviations estimated from a Monte Carlo bootstrapping procedure. The insets show the initial state of the simulation (left inset), corresponding to the bare vacuum with particle number density ν = 0, as well as one example of a state containing one pair (right inset), that is, a state with ν = 0.5, represented as filled/empty arrows as in Fig. 2. b, Experimental data and c, theoretical prediction for the evolution of the particle number density ν as a function of the dimensionless time wt and the dimensionless particle mass m/w, with J = w.

  4. Time evolution of the vacuum persistence amplitude and entanglement.
    Figure 4: Time evolution of the vacuum persistence amplitude and entanglement.

    We show the square of the vacuum persistence amplitude |G(t)|2 (the Loschmidt echo), which quantifies the decay of the unstable vacuum, and the logarithmic negativity En, a measure of the entanglement between the left and the right halves of the system. a, b, The time evolution of |G(t)|2 (a) and En (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 (compare equation (1)), as a function of the dimensionless time wt. c, d, The time evolution of |G(t)|2 (c) and En (d) changes for different values of J and fixed particle mass m = 0. Circles correspond to the experimental data and squares connected by solid lines to the expected evolution assuming an experimental error model explained in Methods. Error bars correspond to standard deviations estimated from a Monte Carlo bootstrapping procedure. e, Illustration of the creation of a particle–antiparticle pair starting from the bare vacuum state.

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

  6. Finite size effects.
    Extended Data Fig. 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.

  7. Experimental pulse sequence.
    Extended Data Fig. 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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Author information

  1. These authors contributed equally to this work.

    • Esteban A. Martinez &
    • Christine A. Muschik

Affiliations

  1. Institute for Experimental Physics, University of Innsbruck, 6020 Innsbruck, Austria

    • Esteban A. Martinez,
    • Philipp Schindler,
    • Daniel Nigg,
    • Alexander Erhard,
    • Thomas Monz &
    • Rainer Blatt
  2. Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria

    • Christine A. Muschik,
    • Markus Heyl,
    • Philipp Hauke,
    • Marcello Dalmonte,
    • Peter Zoller &
    • Rainer Blatt
  3. Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria

    • Christine A. Muschik,
    • Philipp Hauke,
    • Marcello Dalmonte &
    • Peter Zoller
  4. Physics Department, Technische Universität München, 85747 Garching, Germany

    • Markus Heyl

Contributions

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.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

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

Author details

Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: Comparison of the evolutions of the particle number density ν(t) and the rate function λ(t). (159 KB)

    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.

  2. Extended Data Figure 2: Finite size effects. (206 KB)

    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.

  3. Extended Data Figure 3: Experimental pulse sequence. (319 KB)

    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.

Additional data