General methods for digital quantum simulation of gauge theories

A general scheme is presented for simulating gauge theories, with matter fields, on a digital quantum computer. A Trotterized time-evolution operator that respects gauge symmetry is constructed, and a procedure for obtaining time-separated, gauge-invariant correlators is detailed. We demonstrate the procedure on small lattices, including the simulation of a $2+1\mathrm{D}$ non-Abelian gauge theory.

Figure 1

Circuits for the propagation of a pure-gauge lattice field theory. The first circuit implements ${\mathcal{U}}_K^{(1)}$ on four links (in general, $L$ links are needed). The second circuit shows the application of ${\mathcal{U}}_V^{(1)}$ to a single plaquette $\mathrm{Re}\mathrm{Tr}{U}_{13}^{†}{U}_{34}^{†}{U}_{24}{U}_{12}$, and must be applied to every plaquette in the theory. Note that in these circuits, we use a doubled line to represent a $G$-register, rather than a single qubit.

Figure 2

The lattice geometry used for the $D_4$ gauge simulation. The plaquettes are given by $U_2^{†}{U}_0^{†}{U}_3{U}_0$ and $U_3^{†}{U}_1^{†}{U}_2{U}_1$. Dash lines are used to indicate repeated links due to the periodic boundary conditions.

Figure 3

Simulation of a $D_4$ gauge theory with two plaquettes. The expectation value of one of the plaquettes is shown as a function of $t$. The exact result is shown in black; sampled data with $\mathrm{\Delta }t=0.2$ ($\mathrm{\Delta }t=0.5$) are shown in red (blue). Differences between the simulated quantum calculation are due to sampling error (estimated with error bars) and Trotterization.

Figure 4

Expectation value of a temporal Wilson loop in $D_4$ gauge theory, as a function of the time extent of the loop. For each data point, a total of $2\times {10}^5$ measurements were collected. The systematic errors (not shown) present are Trotterization and finite differencing.

Figure 5

Simulation of ${\mathbb{Z}}_2$ gauge theory with staggered fermions on four sites. The expectation value of the number density at the leftmost site as a function of $t$. A simulated quantum calculation is shown in red. Differences with the exact result are due to sampling error (100 samples per data point) and Trotterization.

Figure 6

Classical-inspired quantum circuits for $D_4$ inversion (top) and multiplication (bottom). The least-significant qubits are shown on the top. For the multiplication circuit, the target register is in the bottom three qubits.

Figure 7

Quantum circuit implementing the Fourier transform on a $D_4$ register. The least-significant qubits are on the top. Here, $H$ and $T$ are the Hadamard gate and $\frac{\pi }{8}$ gate, respectively.