From Quantum Circuits with Ultraslow Dynamics to Classical Plaquette Models

Abstract

We introduce a family of hybrid quantum circuits involving unitary gates and projective measurements that display a measurement-induced phase transition. Remarkably, the volume-law phase featuring logarithmic entanglement growth for certain initial states. We attribute this slow entanglement growth to the similarly slow growth of the participation entropy, which bounds the entanglement. Furthermore, the quantum circuit can be mapped to a classical spin model with real positive Boltzmann weights which involves local multi-spin interactions and displays glassy dynamics at finite temperature. We trace the origin of both the slow quantum dynamics and the classical glassiness to the presence of large, non-local symmetry operators. Our work establishes a novel connection between quantum entanglement dynamics and classical glassy behavior, offering a new geometric perspective on entanglement phase transitions.

Figures (5)

• Figure 1: (Left) A single time-step of the circuit with 3 unit cells. Each cell consists of two qubits, labelled $a$ and $b$. First, both $a$ and $b$ in a given cell are measured (yellow circles) in the $X$ and $Z$ directions, at a rate $p$ per unit cell. CNOT gates are then applied such that for each qubit $b$ ($\bullet$) located in unit cell $j$, the targets ($\oplus$) are the $a$ qubits at $j-1, j$ and $j+1$. Finally, $a$ and $b$ are swapped within each unit cell. (Right) A schematic of the entanglement phases in the quantum circuit and their corresponding phases in the 2D classical spin model.
• Figure 2: The time evolution of the half-chain entanglement entropy $S_{\frac{L}{2}}$, shown on a logarithmic scale, beginning from a uniform initial state of $L=100$ sites (the stabilizers are all single-site $X$), for various $p<p_c$. The entanglement growth is logarithmic in time, and non-monotonic with increasing $p$. A fit yields a universal slope $S_{\frac{L}{2}}\sim 3.7\log_{10} t + c(p)$. (Inset) The participation entropy, in this case given by $N_Z$, also grows logarithmically, with the same slope, as predicted.
• Figure 3: The boundary conditions that are imposed at rows $T-1$ and $T$ across multiple copies of $H_c(\qty{q})$, to calculate $\mathcal{Z}^{(4/2)}$. Spins in regions of the same colour are constrained to be equal. $\mathcal{Z}^{(4)}$ requires different constraints for $A$ and $\bar{A}$, while they are identical for $\mathcal{Z}^{(2)}$.
• Figure 4: The ground state configurations (left) and the boundary excitations (right) which contribute to $S^{(2)}_A$. When $\beta=\infty$, the contributions come from $\overline{X}$ that span subsystems $A$ (red) and$\overline{A}$ (blue), and are identical across subsystems that are connected by shaded lines. When $\beta<\infty$, boundary excitations begin to contribute. This involves choosing $\overline{X}$s independently on (1) (blue) and (3) (gold). An $\mathcal{O}(L)$ energy cost (red, dashed lines) is incurred from the mismatched boundaries (blue shading) on (2) and (4).
• Figure 5: Insights from the classical mapping. (Left) Scaling of entanglement entropy in the volume-law (green) and area-law (red) phases, as obtained from $H_c$ at $\beta=\infty$. $L$ is the linear size of the system. (Right) $S^{(2)}_\frac{L}{2} (t)$ at fixed $p=0.1$ for $\beta=2.0$ (solid lines) and $\beta=0.5$ (dashed lines), describing the volume- and area-law phases, respectively.