Entanglement dynamics of noisy random circuits

The process by which open quantum systems thermalize with an environment is both of fundamental interest and relevant to noisy quantum devices. As a minimal model of this process, we consider a qudit chain evolving under local random unitaries and local depolarization channels. After mapping to a statistical mechanics model, the depolarization (noise) acts like a symmetry-breaking field, and we argue that it causes the system to thermalize within a timescale independent of system size. We show that various bipartite entanglement measures—mutual information, operator entanglement, and entanglement negativity—grow at a speed proportional to the size of the bipartition boundary. As a result, these entanglement measures obey an area law: Their maximal value during the dynamics is bounded by the boundary instead of the volume. In contrast, if the depolarization only acts at the system boundary, then the maximum value of the entanglement measures obeys a volume law. We complement our analysis with scalable simulations involving Clifford gates, for both one- and two-dimensional systems.

Figure 1

The random circuit consists of random unitaries (blue blocks) and depolarizing channels (yellow strips).

Figure 2

(a) A stack of multiple copies of the circuit, decomposed (by red dashed lines) into spacetime units. (b) Haar random average via Weingarten calculus. (c) Averaged stack maps to a spin model on the honeycomb lattice. (d) Integrating out a subset of spins yields a simpler model.

Figure 3

Domain wall configurations for $I_2$ at (a, b) small $t$ and (c), (d) large $t$. Thick lines on the upper boundary indicate fixed boundary conditions. Yellow spins are $id=\left(1\right)\left(2\right)$; blue spins are (12). Domain walls in (c), (d) are horizontally directed. Here $Q=2$ and there is only one domain wall.

Figure 4

Domain wall configuration for $I_2$ at small $t$. Blue spins are (12) or (34); green spins are (12)(34); yellow spins are $id$. Two regions are separated by a domain wall. The first figure shows a configuration with $Q=4$; it has two commuting domain walls. It can be decomposed as a product of two single domain wall configurations. The equation is valid even considering the ${(1-p)}^{Q-n}$ factors.

Figure 5

The probabilistic trace setup. Trace channels (red strips) are applied with probability $p$ after random Haar or Clifford unitaries.

Figure 6

Simulated state's von-Neumann entropy $S_{\text{vN}}\left(t\right)$ for various different (a) system size $L$, (b) onsite dimension $d$.

Figure 7

(a) Mutual information for the $p=0$ unitary-only case. (b) Mutual information for the nonzero $p$ values. (c) Logarithmic entanglement negativity between $A$ and $B$ for various nonzero $p$ values.

Figure 8

(a) Maximum mutual information and (b) maximum log-negativity during the dynamics for various $p$ and $\left|A\right|$. $L=256$ in simulations above.

Figure 9

Analysis and numerics for boundary-only depolarization setting. (a) Domain wall configurations of the corresponding spin model. (b) Dynamics of the mutual information for various $\left|A\right|$ in the Clifford version. (c) Linear regression between the peak value and $\left|A\right|$. $L=128$ in simulations above.

Figure 10

Domain wall configuration for $S^{\text{op}}$ at small $t$ ($Q=4$ for illustration). The thick line on the upper boundary indicates the fixed boundary condition. In panel (a), all spins are $b=\left(12\right)\left(34\right)$. In panel (b), spins can be different, hence the domain wall. In this case, all spins have the same number of fixed points (=0), so they are colored with the same color.

Figure 11

Iteration relations used in the calculation. Each rectangular represents a summation of some configurations. Dark blue means those spins are guaranteed to be $\left(12\right)$—in other words, the domain wall is at least 1 step away from the upper boundary; light blue means the domain wall can sometimes attached to the upper boundary.

Figure 12

Relations used to calculated ${\mathcal{Z}}_{bb}$. Blue spins are $\left(12\right)$ and yellow spins are $id=\left(1\right)\left(2\right)$.

Figure 13

(a, left) An illustration of the circuit structure and (a, right) the arrangement of $A$ and $B$. See the text in Appendix pp3 for a detailed description. (b) Mutual information between $A$ and $B$ for various different $p$ and $L$.