Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions

Tracking the dynamics of physical systems in real time is a prime application of quantum computers. Using a trapped-ion system with up to six qubits, we simulate the real-time dynamics of a lattice gauge theory in $1+1$ dimensions, i.e., the lattice Schwinger model, and demonstrate nonperturbative effects such as pair creation for times much longer than previously accessible. We study the gate requirement of two formulations of the model using the Suzuki-Trotter product formula, as well as the trade-off between errors from the ordering of the Hamiltonian terms, the Trotter step size, and experimental imperfections. To mitigate experimental errors, a recent symmetry-protection protocol for suppressing coherent errors and a symmetry-inspired postselection scheme are applied. This work demonstrates the integrated theoretical, algorithmic, and experimental approach that is essential for efficient simulation of lattice gauge theories and other complex physical systems.

Popular Summary

Quantum simulation is one of the most exciting prospects of quantum computers. However, the building of quantum systems with sufficiently low error rates to simulate, with high accuracy, complex physical systems in nature is a great challenge. Therefore, in addition to technological advances in hardware, one must carefully consider all theoretical and algorithmic aspects of the simulation to reduce the computational-resource requirement for the simulation. In the paper, we show how such an integrated approach to quantum simulation, along with the state-of-the-art quantum computing hardware, enables simulating a complex physical model for times much longer than previously possible.

The physical model studied is the Schwinger model, which is a low-dimensional prototype for the theory of strong force and exhibits interesting phenomena such as particle-antiparticle pair creation out of vacuum fluctuations. The observation of such nontrivial dynamics in real-time for large systems and long evolution times is hard with classical computing but is a natural problem to be studied with a quantum computer. The computer we employ works by trapping the ions and manipulating their internal quantum states by coupling them to laser beams in a controlled fashion. The system can implement universal single- and two-qubit gates that are the building blocks of any digital algorithm. By digitizing the time evolution of the Schwinger model on a computer with up to six qubits, we observe the pair-creation dynamics for much longer times than previously demonstrated. Our results are further enhanced by providing theoretical and algorithmic considerations that lead to a better understanding of the errors incurred and ways to mitigate them.

This work, therefore, is a state-of-the-art quantum simulation of a physical system on near-term quantum devices and shows how hardware technology, along with theory and algorithm research, can realize Feynman’s vision for what quantum computers may achieve one day.

Figure 1

The numerical simulation of the projection on the bare-vacuum state $P_{\mathrm{vac}}$ (upper panel) and the population of symmetry-forbidden states $P_{\overline{\mathrm{sym}}}$ (lower panel) when the system is initialized in the bare-vacuum state for $N=6$, $\mu =0.1$, and $x=0.6$. Different term orderings for the Trotterized evolution are considered: the odd-even ordering defined in Eq. (11) (blue dots) and the XYZ ordering defined in Eq. (14) (red diamonds). The blue line denotes the exact evolution.

Figure 2

The number of two-qubit gates required to simulate the time evolution under the Hamiltonian in Eq. (2) on an $N$-site lattice for time $t=N$ given an error tolerance $ϵ=0.01$, and using the second-order product formula in Eq. (8). The commutator bound on the gate complexity (green dots) is estimated from the error bound in Eq. (10). We also obtain a tighter estimate (orange dots) by exactly computing the nested commutators in Eq. (9). Finally, the empirical gate count (blue dots) is obtained through a binary search for the minimum number of time steps $t/\delta t$ such that the total error is at most $ϵ$. The straight lines are linear fits that result in the polynomial scaling given in the figure.

Figure 3

The leakage to the symmetry-forbidden sector, $P_{\overline{\mathrm{sym}}}$, defined as the population in the states with a nonvanishing total charge given a bare-vacuum initial state (upper panel), as well as the error in the bare-vacuum population (lower panel) are shown for the odd-even ordering (blue dots), the XYZ ordering before (red diamonds) and after (green squares) postselection, as well as the XYZ ordering with symmetry protection but without postselection (yellow stars). The parameters used for the plot are $\mu =0.1$, $x=0.6$, $N=4$, and $\delta t=1$. The optimal angles for the symmetry-protected simulation are provided in Appendix pp1. Sym XYZ refers to symmetry-protected XYZ ordering.

Figure 4

The circuit for the Trotterized evolution according to the Hamiltonian in Eq. (2) for $N=6$ lattice sites, with odd-even ordering of the Trotter decomposition introduced in Eq. (11). The interaction term $e^{-i\delta t{\hat{H}}^x}$ is implemented with nearest-neighbor $X_i{X}_{i+1}$ and $Y_i{Y}_{i+1}$ gates. The $e^{-i\delta t{\hat{H}}^{ZZ}}$ term is implemented with $X_i{X}_j$ and $R_i$ rotations. The $e^{-i\delta t{\hat{H}}^Z}$ term involves only $Z_i$ rotations, with the angles ${\mu }_1=-(\mu +3)\delta t$, ${\mu }_2=(\mu -2)\delta t$, ${\mu }_3=-(\mu +2)\delta t$, ${\mu }_4=(\mu -1)\delta t$, ${\mu }_5=-(\mu +1)\delta t$, and ${\mu }_6=\mu \delta t$. Qubits are initialized in the bare-vacuum state $|010101⟩$, then evolved by repeating the circuit in the parentheses $t/\delta t$ times, and measured individually in the $Z$ basis at the end.

Figure 5

Experimental results for $N=2$ and $\delta t=0.5$. (a) The upper plot shows fluctuation in the bare-vacuum population, $P_{\mathrm{vac}}$, while the lower plot shows the particle-number density, $\nu$, as a function of time, indicating the creation and annihilation of the particle-antiparticle pairs. The dashed lines are a guide to the eye. (b) The upper plot shows the local charge density $Q_n$ as measured in the experiment after postselection, while the lower plot shows its deviation from theory as a function of time.

Figure 6

Experimental results for $N=4$ and $\delta t=0.5$. (a) The upper plot shows fluctuation in the bare-vacuum population, $P_{\mathrm{vac}}(t)$, while the lower plot shows the particle-number density, $\nu (t)$. (b) The upper plot shows the local charge density $Q_n(t)$ as measured in the experiment after postselection, while the lower plot shows its deviation from theory.

Figure 7

Experimental results for $N=4$ and $\delta t=1$. (a) The upper plot shows fluctuation in the bare-vacuum population, $P_{\mathrm{vac}}(t)$, while the lower plot shows the particle-number density, $\nu (t)$. (b) The upper plot shows the local charge density $Q_n(t)$ as measured in the experiment after postselection, while the lower plot shows its deviation from theory.

Figure 8

Experimental results for $N=6$ and $\delta t=1$. (a) The upper plot shows fluctuation in the bare-vacuum population, $P_{\mathrm{vac}}(t)$, while the lower plot shows the particle-number density, $\nu (t)$. (b) The left-hand plot shows the local charge density $Q_n(t)$ as measured in the experiment after postselection, while the right-hand plot shows its deviation from theory. At $t=4$, we reach the gate-depth limit of the hardware.

Figure 9

The leakage to the symmetry-forbidden subspace (upper panel) and deviation of the experimental results from theory for the bare-vacuum population (lower panel) when different schemes to mitigate errors are applied: no mitigation (orange triangles), postselection (green squares), and symmetry protection (blue stars). Note that by definition, the leakage to the symmetry-forbidden subspace is zero after postselection. In both plots, $N=4$ and $\delta t=1.0$, the system is initiated in the bare-vacuum state and is evolved via the odd-even ordering scheme. Sym Exp refers to the experimental results with symmetry protection.

Figure 10

The population of the symmetry-forbidden subspace as a function of the angles ${\alpha }_1$ for $t=2,4,8$, for $\mu =0.1$, $x=0.6$, $N=4$, and $\delta t=1$. We choose ${\alpha }_1$ to be the smallest angle that minimizes $P_{\overline{\mathrm{sym}}}$ at each $t$.

Figure 11

The evolution of states characterized by the quantity $P_{\mathrm{\Psi }}\equiv |⟨\mathrm{\Psi }|e^{-it\hat{H}}|{\psi }_0⟩{|}^2$ in the (a)–(f) symmetry-allowed and (g) symmetry-forbidden subspace for $N=4$ and $\delta t=1.0$, starting from the bare-vacuum state $|{\psi }_0⟩$. Note that $P_{0101}\equiv P_{\mathrm{vac}}$, which is also plotted in Fig. 7 of the main text. The effectiveness of postselection in mitigating errors is both quantity and time dependent and, in some cases, it ceases to improve the agreement with theory at larger times.