Noisy coupled qubits: Operator spreading and the Fredrickson-Andersen model

We study noise-averaged observables for a system of exchange-coupled quantum spins (qubits), each subject to a stochastic drive, by establishing mappings onto stochastic models in the strong-noise limit. Averaging over noise yields Lindbladian equations of motion; when these are subjected to a strong-noise perturbative treatment, classical master equations are found to emerge. The dynamics of noise averages of operators displays diffusive behavior or exponential relaxation, depending on whether the drive conserves one of the spin components or not. In the latter case, the second moment of operators, from which the average subsystem purity and out-of-time-order correlation functions can be extracted, is described by the Fredrickson-Andersen model, originally introduced as a model of cooperative relaxation near the glass transition. It is known that fluctuations of a ballistically propagating front in the model are asymptotically Gaussian in one dimension. We extend this by conjecturing, with strong numerical evidence, that in two dimensions the long-time fluctuations are in the Kardar-Parisi-Zhang universality class, complementing a similar observation in random unitary circuits.

Figure 1

Growth of fronts in the one-dimensional FA model with a finite average butterfly velocity $v_{\text{B}}$.

Figure 2

Top: Upward growth of a front in the 2D FA model into a region of zeros (black), starting from a row of ones. Bottom: Growth of front variance with time. The dashed line is the power law $0.13t^{2/3}$, consistent with the KPZ growth exponent $\beta =1/3$.

Figure 3

Top: Front probability distribution at $t={10}^5$ compared with the best-fit Tracy-Widom density $F_1^{\prime }\left(s\right)$ (dashed line) and Gaussian (dash-dotted line), where the width and shift of the distributions are fitting parameters. Bottom: The same plot on a logarithmic scale, which more clearly illustrates how the different behaviors of the left and right tails are well described by the Tracy-Widom distribution.

Figure 4

The averaged OTOC ${\overline{\mathcal{C}}}_{jk}\left(t\right)$ is identified with the probability for sites $j$ and $k$ to be in the same cluster. Here, we illustrate the Gaussian case, with the functional form being that of an error function and the cumulative distribution function of a Gaussian, appropriate to one dimension.

Figure 5

A purity front velocity $v_{\text{PF}}<v_{\text{B}}$ arises from the joint probability of the propagating front deviating from the velocity $v_{\text{B}}$ and an initial condition with $X$ extra zeros.

Figure 6

Top: Schematic representation of the propagation of two fronts in 1+1 space-time. White represents a region of zeros, and black dots show an active region of ones and zeros. Bottom: $-ln\overline{\gamma }\left(t\right)$ for the case when $\overline{\gamma }\left(t\right)$ is computed for a finite subsystem. The red lines at fixed spatial positions in the top panel demarcate a finite subsystem of size $\left|A\right|$. Before saturation, $-ln\overline{\gamma \left(t\right)}$ grows at a rate (shown) controlled by the purity velocity.

Figure 7

An example of exchange (left) and pair-changing (right) terms, as visualized on a square cross section of the octahedron of single-site states that span the slow subspace of model C. The solid and open circles represent the states on adjacent sites.