From Classical Stochastic to Monitored Quantum Dynamics: Dynamical Phase Coexistence in East Circuit Models

Abstract

Kinetically constrained models have been widely studied in the context of glass formers and non-equilibrium statistical mechanics. Although their simple local rules often result in structureless static properties, their dynamics exhibit intricate emergent phenomena. In this work, we investigate monitored quantum circuit models that interpolate between classical stochastic and unitary quantum dynamics. For any finite measurement strength, the measurement records provide an experimentally accessible probe of the emergence of dynamical phases. By interpreting space-time resolved records as microstates of a fictitious 1+1D spin system, we employ thermodynamic concepts that allow us to investigate the dynamical coexistence between an active and inactive phase. We combine insights from classical stochastic dynamics and numerical simulations of monitored quantum dynamics to investigate different signatures of this dynamical phase coexistence as the measurement strength is varied. Our results shed light on the persistence of dynamical phase coexistence in the quantum regime, offering insights into future experimental studies of complex many-body dynamics in quantum simulators.

Figures (10)

• Figure 1: Dynamical phase coexistence in monitored quantum East circuits. (a) The discrete-time dynamics consists of unitary gates in a brickwork structure (blue squares with control qubits below) and local monitoring (two-colored diamonds). (b) Unitary gates implement qubit-qubit dynamics in terms of a conditioned rotation according to the East constraint if the control qubit is in state $\ket{1} \equiv \ket{\uparrow}$. (c) Monitoring is achieved by ancilla-assisted operations that weakly measure the system according to Eq. \ref{['eq:weak_measurements']}. Their output $k_i(t) \in \{0, 1\}$ enters the so-called space-time measurement record $\eta$. (d) Quantum expectation values $\expval{n_i(t)}$ of the occupation in state $\ket{1}$ and corresponding measurement history in a stochastic realization for $L=64$, $\omega = 0.1$ and $\gamma = 0.3$. Remarkably, we observe distinct active and inactive space-time clusters that are characterized by their respective number of $k = 1$ measurement outcomes. The dynamical phase coexistence can hence be accessed via the statistical properties of the time-integrated activity $A_{L, T} = \sum_{i,t} k_i(t)$ (cf. Fig. \ref{['fig:fig3']}), and of $\ell \times \tau$-sized inactive clusters (cf. Fig. \ref{['fig:fig4']}).
• Figure 2: Activity of the classical stochastic Floquet-East model for $\gamma = \pi/2$ and $\omega = 0.1$. (a) The activity density $a_L(s)$ is displayed for various system sizes $L$. As $L$ grows, we observe an increasingly sharp crossover (marked by dotted lines) between active and inactive phases. For comparison, we also show $a_L(s)$ for non-interacting (NI) spins, (indicated by the dashed line), where no sharp crossover exists. (b) Inset showing the scaling of $s^* = \mathrm{argmax}_s\{a'(s)\}$ with system size $L$. The crossover $s^*$ approaches $0$ as the system size increases (see dotted line with $\sim L^{-3/2}$ as a guide for the eye).
• Figure 3: Dynamical phase diagram. (a) Normalized activity density $a_{L, T}(s) / \sin^2\!\gamma$ for $L=8$, $\omega = 0.1$. As the counting field $s$ is varied, a sharp crossover emerges between active and inactive dynamical phases. The position of this crossover depends on the measurement strength $\gamma$, but generally shifts toward $s = 0$ as the system size $L$ increases (colored symbols: $L=8$ to $L=48$). (b) Cut at $\gamma = 0.5$. The crossover sharpens significantly with increasing $L$, approaching a discontinuity characteristic of a first-order phase transition. Dotted lines indicate the crossover positions $s^*$. (c) Scaling analysis of the crossover. The crossover $s^*$ approaches $s = 0$ as $L$ increases (see dotted line $\sim 1/L$ for reference), hereby confirming the observation of dynamical phase coexistence in Fig. \ref{['fig:fig1']}(d).
• Figure 4: Crossover in the statistics of inactive clusters. (a) Negative log-probability $F_{\ell \times \tau}=-\log p_{\ell \times \tau}$ of inactive $\ell \times \tau$ clusters in the ancilla records for fixed $\ell=2,4,6$ (solid, dashed, dotted) at $L=64$ and $\omega=0.1$. For small $\tau$, $F_{\ell \times \tau}$ grows proportionally to the area $\ell \tau$, consistent with nearly independent outcomes at average activity density $a_{L, T}(0)=\sin^2\!\gamma/2$. Around a $\gamma$-dependent crossover time $\tau^*$, the slope decreases and the growth becomes perimeter-like, signaling a collective nature of the inactive region. Rescaling time as $\tan^2(\gamma/2)\tau$ approximately collapses the crossover positions across $\gamma$ (see Supplemental Material SM for the perturbative derivation). (b) Crossover time $\tau^*$ versus measurement strength $\gamma$. The crossover persists for all tested $\gamma$, indicating that active–inactive dynamical phase coexistence remains in the quantum regime. Notably, the unscaled $\tau^*$ grows as $\gamma$ decreases: this is because longer observation times are needed to separate collective inactive domains from inactivity caused only by lower average activity.
• Figure S1: Dynamical phase coexistence in the classical stochastic Floquet-East model. (a) Circuit representation of the stochastic dynamics. (b) Stochastic East gates controlled by spin-flip probability $p$. (i) spin-flip allowed; (ii) spin-flip forbidden. (c) Stochastic realization. Black = spin up = 1; white = spin down = 0. The trajectory shows a typical stochastic realization starting from a random initial state for $L = 64$ spins for $p = \sin^2\omega$, with $\omega = 0.1$ as in the main text.