Variational quantum simulation of U(1) lattice gauge theories with qudit systems

Lattice gauge theories are fundamental to various fields, including particle physics, condensed matter, and quantum information theory. Recent progress in the control of quantum systems allows for studying Abelian lattice gauge theories in table-top experiments. However, several challenges remain, such as implementing dynamical fermions in higher spatial dimensions and magnetic field terms. Here, we map U(1) Abelian lattice gauge theories in arbitrary spatial dimensions onto qudit systems with local interactions. We propose a variational quantum simulation scheme for the qudit system with a local Hamiltonian, that can be implemented on a universal qudit quantum device as the one developed in [Nat. Phys. 18, 1053 (2022)]. We describe how to implement the variational imaginary-time evolution protocol for ground-state preparation as well as the variational real-time evolution protocol to simulate nonequilibrium physics on universal qudit quantum computers, supplemented with numerical simulations. Our proposal can serve as a way of simulating lattice gauge theories, particularly in higher spatial dimensions, with minimal resources, regarding both system sizes and gate count.

Figure 1

Particle content on the lattice in the case of a two-dimensional spatial lattice. (Left) The hardcore bosons ${\eta }_{\mathbf{x}}$ and the fermions ${\chi }_{\mathbf{x}}$ reside on the lattice sites (orange circles), whereas the gauge fields $E_{\mathbf{x},i}$ and the fermions ${\zeta }_{\mathbf{x},i}$ reside on the links (blue ovals). (Right) The Majorana modes $c_{\mathbf{x}},{\alpha }_{\mathbf{x},i}$, and ${\beta }_{\mathbf{x},i}$ defined by Eq. (2a) and the corresponding sites/links on which they reside. Thanks to Gauss's law, this configuration permits one to replace the fermionic matter ${\psi }_{\mathbf{x}}$ residing in the original gauge theory on lattice sites, such that the final theory contains only the gauge fields $E_{\mathbf{x},i}$.

Figure 2

Two-dimensional spatial lattice with matter fields on the sites (orange circles) and gauge fields on the links (blue ovals). After performing the transformation from Eq. (19), the hopping terms between two neighboring sites (indicated by an arrow) acquire prefactors that depend on the electric fields on the adjacent links. The involved electric fields depend on the hopping direction (horizontal or vertical) and result in direction-dependent phase factors as given in Eq. (23).

Figure 3

Hadamard test for measuring unequal-time (anti)commutators. In order to measure the (anti)commutator, the ancillary qubit needs to be initialized in an equal superposition of $|0\rangle$ and $|1\rangle$ with a relative phase of $\alpha =0$ ($\frac{\pi }{2}$). We obtain the desired correlation function by measuring the probability of the ancilla to be in the state $|+\rangle$.

Figure 4

Individual layers of the variational quantum circuit for seven qudits in ($1+1$) dimensions. Each building block of the parametrized quantum circuit used for generating the trial states is built from a set of single qudit operations, given by two-level rotations, and entangling operations, given by the two-level Mølmer-Sørensen [97, 98] gate.

Figure 5

Individual layers of the variational quantum circuit for one plaquette in ($2+1$) dimensions. Each building block of the parametrized quantum circuit used for generating the trial states is built from a set of single-qudit operations, given by two-level rotations, and entangling operations, given by the CROT gate [see Eq. (77) for definition].

Figure 6

Variational imaginary-time evolution of a system of seven qudits in ($1+1$) dimensions. (Left) Fidelity ${\mathcal{F}}_{\theta }\left(\tau \right)$ of the variational state with respect to the exact ground state as a function of the imaginary time for different circuit depths ($N=1,2$, and 3 layers). All of the results converge after a finite time. The highest reached fidelities are $\sim 74\%$, $\sim 99\%$, and $>99\%$ for $N=1$, 2, and 3 layers, respectively. (Right) Expectation value of the Hamiltonian of the system in the trial state as a function of the imaginary time. The error of the ground-state energy in the final time of the simulation is below $1\%$ for $N=3$ layers.

Figure 7

Variational imaginary-time evolution of a system of one plaquette in ($2+1$) dimensions. (Left) Fidelity ${\mathcal{F}}_{\theta }\left(\tau \right)$ of the variational state with respect to the exact ground state as a function of the imaginary time for different circuit depth ($N=1,2$, and 3 layers). All of the results reach high fidelity after a finite time, around $\sim 91\%$, $\sim 99\%$, and $>99\%$ for $N=1$, 2, and 3 layers, respectively. (Right) Expectation value of the Hamiltonian of the system in the trial state as a function of the imaginary time. The error of the ground-state energy in the final time of the simulation is below $1\%$ for $N=3$ layers.

Figure 8

Variational real-time evolution of a system of five qudits in ($1+1$) dimensions. (Left) Fidelity of the variational state with respect to the exactly time-evolved state [see Eq. (79)]. The fidelity decreases with time as the correct time evolution of the initial state becomes harder and harder to approximate with fixed number of layers. (Right) Real-time dynamics after a quench of the gauge invariant fermionic numbers in the middle of the one-dimensional system. The dashed line corresponds to the exact time evolution of the fermion numbers, and the colorful lines correspond to the approximate time evolution with different number of layers $N$.

Figure 9

Evolution of the entanglement entropy of subsystem A after a quench in a system of five qudits in ($1+1$) dimensions. After initial linear growth, the entropy $\mathcal{S}\left({\rho }_A\right)$ saturates to a finite constant up to fluctuations due to finite size effects. As we increase the number of layers $N$ in the variational circuits, we observe better and better approximations to this behavior of the entropy in the trial state.

Figure 10

Variational real-time evolution of a system of one plaquette in ($2+1$) dimensions. (Left) Fidelity of the variational state with respect to the exactly time-evolved state [see Eq. (79)]. The fidelity for $N=2$ and 3 layers decreases with time, while the decrease for $N=4$ is much slower. This suggests that for a system of 1 plaquette, there might not be a need to gradually increase the circuit depth with time to approximate the time evolution reliably. (Right) Real-time dynamics after a quench of the gauge invariant fermionic numbers in the middle of the one-dimensional system. The dashed line corresponds to the exact time evolution of the fermion numbers, and the colored lines correspond to the approximate time evolution with a different number of layers $N$.

Figure 11

Evolution of the entanglement entropy of subsystem A after a quench in a system of one plaquette in ($2+1$) dimensions. We observe that the growth of entropy at very early times is not correct; however, there is a qualitative agreement with the exact diagonalization. This growth of entropy leads to small deviations from the exact result for the fermionic occupation numbers; however, these deviations do not grow with time.

Figure 12

Variational real-time evolution of a system of one plaquette in ($2+1$) dimensions with a circuit that includes a four-body gate. (Left) Fidelity of the variational state with respect to the exactly time-evolved state [(see Eq. (79)]. For $N=4$, the fidelity does not drop below $80\%$ for the whole simulation period. (Right) Real-time dynamics after a quench of the gauge invariant fermionic numbers in the bottom left corner site of the plaquette. We see virtually perfect agreement between the exactly diagonalized (dashed line) result and the result from the variational simulation for $N=4$ layers.

Figure 13

Evolution of the entanglement entropy of subsystem A after a quench in a system of 1 plaquette in ($2+1$) dimensions. The initial growth of the entropy is captured very accurately for $N=2,3,4$ layers. Furthermore, for $N=4$ layers, the time evolution of the entropy is very well approximated over the whole simulation time.

Figure 14

Variational real-time evolution of a system of seven qudits in ($1+1$) dimensions. (a) Fidelity of the variational state $|\psi (\theta \left(t\right))\rangle$ with respect to the exactly time-evolved state $|\psi \left(t\right)\rangle$. (b) Real-time dynamics after a quench of the gauge invariant fermionic numbers in the middle of the one-dimensional system. The dashed line corresponds to the exact time evolution of the fermion numbers and the colorful lines correspond to the approximate time evolution with different number of layers $N$.

Figure 15

Evolution of the entanglement entropy of subsystem A after a quench in a system of seven qudits in ($1+1$) dimensions. After initial linear growth, the entropy $\mathcal{S}\left({\rho }_A\right)$ saturates to a finite constant up to fluctuations due to finite size effects. As we increase the number of layers $N$ in the variational circuits, we observe better and better approximation to this behavior of the entropy in the trial state.