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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Extended data figures and tables
Extended Data Figures
- 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 = e−Nλ(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.
- 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.
- 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.