Entanglement transition and suppression of critical phase of thermofield double state in monitored quantum circuit with unitary $R$ matrix gates

Abstract

We study quantum circuits with gates composed randomly of identity operators, projectors, or a kind of $R$ matrices which satisfy the Yang-Baxter equation and are unitary and dual-unitary. This enables us to translate the quantum circuit into a topological object with distinguished overcrossings and undercrossings. The circuit corresponds to a classical loop model and is post-selection free when an overcrossing and an undercrossing coincide. The entanglement entropy between the final state and initial state is given by the spanning number of the classical model, and they share the same phase diagram. Whenever an overcrossing and undercrossing differ, the circuit extends beyond the classical model. Considering a specific case with $R$ matrices randomly replaced by SWAP gates, we demonstrate that the topological effect originating from worldline braiding dominates, and only the area-law phase remains in the thermodynamic limit, regardless of how small the replacement probability is. We also find evidence of an altered phase diagram for non-Clifford cases.

Figures (6)

• Figure 1: Entanglement protocol. $U$ is a quantum circuit. The time direction goes upwards. The part on the right-hand side depicts an identity evolution of the right part of a thermofield double state $\ket{\Psi}$ (\ref{['eq:tfds']}). $U$ acts on the left part of $\ket{\Psi}$. We are interested in the entanglement between two parts, indicated by red circles at top.
• Figure 2: Entropy produced by a link of worldlines. (a) Link composed by two $R(c)$ matrix gates (\ref{['eq:circuit-piece']}a). The von Neumann entropy is zero at Clifford points and nonzero otherwise. (b) Link composed by one $R(c)$ matrix gate and one SWAP gate (\ref{['eq:circuit-piece']}b). The von Neumann entropy is not zero at $c=\pm 1$ as a result of different topology.
• Figure 3: (a) A worldline configuration of loops with crossings. Red: worldlines that have both ends on the same boundary, but do not entangle with the other boundary. Black: worldlines that connect two boundaries and contribute to the spanning number. Blue and orange: worldlines that are entangled by a link. (b) Phase diagram of the entanglement entropy in log-scale for $R(c)$ set at Clifford points, computed for system size $L=2^{11}$, $t=L$ and are averaged from $10240$ samples. $p$ is the probability of $R$. $q'$ is defined by $(q'-1/2) = (q-1/2)(1-p)$ where $(1-p)q$ is the probability of $P$ on odd time layers. Blue part represents a critical region, where entanglement entropy grows logarithmically with system size $L$, and corresponds to the Goldstone phase of CPLC. White parts are area-law regions where entanglement entropy decays to 0 as system size $L$ grows and correspond to two short loop phases of CPLC. The red dashed line corresponds to data shown in (c). (c) Entanglement entropy of different system size when $p=0.5$.
• Figure 4: Red dashed line is a reference line for $S=1$. All data points are averaged from 16384 samples. (a) System size scaling of average entanglement entropy at time $t=L$ for $q=0.5$, $r=0.1$, which corresponds to the critical region of CPLC. Data shows convergence to $S=1$. The dash-dot line corresponds to $p=0.8$, $r=0$. (b) Averaged entanglement for $p=0.3$, $r=0.1$, $t=L=256, 512, 1024$. Orange dash-dot line marks the transition point of CPLC.
• Figure 5: (Color online) An example for the calculation of the second order Rényi entropy. The system length $L=16$, the subsystem length $L_A = 6$ and the system evolves for 3 time steps. The 2 sites that connect to $\bar{A}$ are marked by larger dots and one of their worldlines is marked by a thickened red line. The horizontal dashed line separates the forward and backward evolution $U$ and $U^{\dagger}$.