Observing quantum many-body scars in random quantum circuits

The Schwinger model describes quantum electrodynamics in $1+1$ dimensions, it is a prototype for quantum chromodynamics, and its lattice version allows for a quantum link model description that can be simulated using modern quantum devices. In this work, we devise quantum simulations to investigate the dynamics of this model in its low-dimensional form, where the gauge field degrees of freedom are described by spin-$\frac{1}{2}$ operators. We apply Trotterization to write quantum circuits that effectively generate the evolution under the Schwinger model Hamiltonian. We consider both sequential circuits, with a fixed-gate sequence, and randomized ones. Utilizing the correspondence between the Schwinger model and the PXP model, known for its quantum scars, we investigate the presence of quantum scar states in the Schwinger model by identifying states exhibiting extended thermalization times in our circuit evolutions. Our comparison of sequential and randomized circuit dynamics shows that the nonthermal sector of the Hilbert space, including the scars, is more sensitive to randomization.

Figure 1

Example of quantum circuit corresponding to a sequential Trotterized evolution with 12 qubits. Unitaries are defined as $U_{2j-1}(T/N)≔e^{-i{H}_{j}T/N}$, and $H_j={\sigma }_{2j-1}^{+}{\sigma }_{2j}^{+}{\sigma }_{2j+1}^{+}+\mathrm{H}.\mathrm{c}.$ This circuit is expected to reproduce the evolution under the full Hamiltonian $e^{-i{H}_{\text{QLM}}T}$ in the limit $N\to \infty$.

Figure 2

Evolution of expectation value of the local magnetization at the center of the spin chain, at site $j=21$, generated using exact diagonalization. The purple (dashed, dark gray) curve has the vacuum as initial state, while the green (solid, light gray) curve starts at the fully filled state. The constant curve at $\langle {\sigma }_{21}^z\rangle =0.105$ indicates the thermal value of the local magnetization for both vacuum and fully filled states.

Figure 3

(a) Entanglement entropy of energy eigenstates. Scar states in purple (dark gray), and some typical states, arbitrarily chosen, in green (light gray). (b) Overlap of the energy eigenstates with the vacuum state. Eigenstates with overlap with the vacuum state smaller than ${10}^{-12}$ are not shown. (c) Expectation value of local magnetization for each energy eigenstate. The curve is the thermal value of the local magnetization. (d) Entanglement entropy of some energy eigenstates as a function of the subsystem size $l_A$. In these plots, some eigenstates $E_n$ are labeled by their energy index $n$, a number that runs from 0 to 765, and the scar states are shown in purple (dark gray).

Figure 4

Evolution of local magnetization for the vacuum, purple (dark gray) curves, and the fully filled state, green (light gray) curves, using exact diagonalization (solid lines) and sequential quantum circuit with time step $\tau =0.1$ (dashed lines). Final time standard deviations are provided in the second column of Table 1.

Figure 5

Standard deviation of local magnetization $\langle {\sigma }_{21}^z\rangle$ and Loschmidt echo for different initial states as function of the projection of the initial state in the quantum many-body scars sector. This is the standard deviation of the random quantum circuit evolution with respect to the sequential quantum circuit, with time step size $\tau =0.1$ and duration $T=10$. The lines are error bars calculated using 1000 circuit runs separated in 10 groups of 100 runs, and the markers are the mean values.

Figure 6

Evolution of the Loschmidt echo for different initial states, computed using sequential quantum circuits with time step size $\tau =0.1$. The Loschmidt echo is the overlap between the evolved state and the initial state.

Figure 7

Evolution of the standard deviation of the Loschmidt echo for different initial states. This is the standard deviation of the random quantum circuit evolution with respect to the sequential quantum circuit, with time step size $\tau =0.1$, duration $T=10$. The random quantum circuits were averaged over $K=1000$ runs for each state.

Figure 8

Evolution of the entanglement entropy for different initial states, computed using sequential quantum circuits with time step size $\tau =0.1$. This is the half-chain entanglement entropy: The Hilbert space partition used is ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$, where ${\mathcal{H}}_A$ consists on the first $L/2$ spins, and ${\mathcal{H}}_B$ on the other $L/2$ spins.

Figure 9

Evolution of the local magnetization $\langle {\sigma }_{21}^z\rangle$ for different initial states, computed using sequential quantum circuits with time step size $\tau =0.1$. The constant curve at $\langle {\sigma }_{21}^z\rangle =0.105$ represents the thermal value of the local magnetization for these initial states.