Effective Kinetic Monte Carlo for a Quantum Epidemic Process

TL;DR

This work develops the elementary Quantum Epidemic Process (eQEP), a four-level-per-site open quantum model, to study epidemic dynamics under Lindblad evolution. By exploiting weak symmetries, the authors map the quantum dynamics onto a local, time-dependent Kinetic Monte Carlo framework, enabling large-scale 2D simulations. They find that the stationary behavior matches the General Epidemic Process (GEP), while the dynamics exhibit multiple outbreaks at intermediate coherent frequencies $\Omega$, with large $\Omega$ effectively slowing the infection to a GEP with halved rates. The results clarify how quantum fluctuations and coherent oscillations influence epidemic spreading and provide a tractable platform for benchmarking against the Rydberg-inspired quantum epidemic, with potential extensions to more complex constrained dynamics. Overall, the eQEP offers a practical, scalable route to explore non-equilibrium quantum epidemics and their connections to classical stochastic models.

Abstract

Inspired by previous works on epidemic-like processes in open quantum systems, we derive an elementary quantum epidemic model that is simple enough to be studied via Quantum Jump Monte Carlo simulations at reasonably large system sizes. We show how some weak symmetries of the Lindblad equation allow us to map the dynamics onto a classical Kinetic Monte Carlo; this simplified, effective dynamics can be described via local stochastic jumps coupled with a local deterministic component. Simulations are then used to reconstruct a phase diagram which displays stationary features completely equivalent to those of completely classical epidemic processes, but richer dynamics with multiple, recurrent waves of infection.

Figures (28)

• Figure 1: Structure of the GEP: at the top, the three possible values for the local state of a site are visualised as an empty circle for S (susceptible), an orange one for I (infected) and a black one for D (dead). The two basic processes are displayed one line below, where it is understood that the two sites shown under the "Infection" label are nearest neighbors; in the example provided, the leftmost one can infect its neighbor with rate $\gamma_I$. This rate is the same in all directions (the process is isotropic). Death, on the other hand, involves a single site and occurs at rate $\gamma_D$. The panel at the bottom displays the initial condition for a two-dimensional square lattice: all sites but the central one are in the S state; the one exception, i.e. "the origin" of the infection, is set instead at time $t = 0$ in state I.
• Figure 2: (a-b) A typical long-time configuration in (a) the inactive phase and (b) the active one. The color code for the sites is taken from Fig. \ref{['fig:GEP1']}: empty or white (S), orange (I) and black (D). (c) Qualitative representation of the distribution of the final random DDS $\varrho_D$ in three different parameter regions: $\eta < \eta_c$ (blue, solid line), $\eta \gtrsim \eta_c$ (green, dashed line) and $\eta \gg \eta_c$ (orange, dotted line). The width of the peak around the origin is exaggerated for ease of visualization. The second peak progressively moves to the right as $\eta$ is increased, while also getting higher at the expense of the other one, as the probability of observing a dying trajectory also diminishes for growing $\eta$. Generally, the finite peaks are not centered around the averages $n_D$, but around larger values $n_D'$, which instead correspond to averages over the subset of "surviving" trajectories. (d) Qualitative behavior of the survival probability $P(\eta)$ (blue, solid line) and of the final average death density $n_D(\eta)$ (red, dashed line) across the transition.
• Figure 3: Visualization of the RQEP's structure: (a) level structure of a single site. The three GEP-like states $\left| S \right\rangle$, $\left| I \right\rangle$ and $\left| D \right\rangle$ are displayed as three black horizontal lines, each with its corresponding ket directly below. To the right of each, the corresponding classical state is reported in the style of Figs. \ref{['fig:GEP1']} and \ref{['fig:GEP2']}. Arrows represent dynamical processes; solid ones denote Hamiltonian terms, whereas dashed ones dissipative processes associated to jump operators. From left to right, we have "infection" (red), "dephasing" (dark yellow) and "death" (black). The shaded area highlights the only process, infection, which involves more than one site and is thus capable of producing spatial correlations. The kinetic constraint of the RQEP is illustrated in panel (b), where we use as an example five sites in a cross configuration. Due to the isotropy of the model, configurations rotated by multiples of $\pi/2$ obey analogous rules. Every case (plus rotations thereof) with a crossed-out arrow is prohibited by the constraint. The second case from the top (plus rotations thereof) is the only allowed transformation among those drawn. The rightmost column, labeled $A_{i \to f}$, shows whether or not states $\left| i \right\rangle$ and $\left| f \right\rangle$ are connected under the infection Hamiltonian \ref{['eq:conn_under_evol']}. The second entry from the top should be read as $\exists\,\, t > 0$ such that $A_{i \to f} (t) \neq 0$.
• Figure 4: This figure has been adapted from Ref. Espigares2017; the labels have been modified to match our present notation. Panel (a) displays the final average DDS $n_D$, defined in Eq. \ref{['eq:qDDS']}, as a function of the ratio $\Omega_I / \gamma_D$ for $\gamma_{deph} = 0$. The red line illustrates the numerical solution of the inhomogeneous mean-field equations; the blue and black lines constitute predictions obtained via a scaling argument (not reported here) and can for our present purposes be safely ignored. The repeating structure of jumps followed by a continuous decay in the final DDS is here apparent; from the same scaling argument the jumps across the thresholds $\tilde{\Omega}_i$$(i = 1,\,2,\,3,\,\ldots)$ are expected to be discontinuous. Panel (b) shows the corresponding patterns appearing in the dynamics for (moving horizontally) three reference values of the ratio $\Omega_I / \gamma_D$ and (moving vertically) three different times (measured in units of $\gamma_D^{-1}$). Each of the nine subpanels represents a map of the local DDSs $\left\langle \sigma_k^{DD} (t) \right\rangle$ over the entire $51\times 51$ lattice. Reading each column from left to right we observe the presence of zero, one and two outbreaks, respectively. The three colored dots at the top are used in panel (a) as a guide to the eye on where the three chosen parameter values lie on the horizontal axis.
• Figure 5: Pictorial representation of the eQEP: each site features $4$ levels, $\left| S \right\rangle$ (susceptible), $\left| I \right\rangle$ (infected), $\left| B \right\rangle$ (bedridden) and $\left| D \right\rangle$ (dead). Except for the newly-introduced bedridden state, there is a straightforward relation to the classical ones of the GEP, shown here as colored circles with the same style of Fig. \ref{['fig:GEP1']} We adopt here some of the conventions used for Fig. \ref{['fig:RQEP1']}: dashed lines denote dissipative processes, solid lines coherent (Hamiltonian) terms. There are two of the former, infection (red) and death (black), and one of the latter (blue), which is the only one connecting $\left| B \right\rangle$ to the rest. Shaded areas show where the constraint applies, i.e. just to infection. Both the Hamiltonian and death jump operators are collections of one-site, local terms. Note that in the picture above there are no arrows going left: once you move right you can never go back. This is a visualization of the $S \to I \to D$ "directionality" of the GEP, which is recovered in the eQEP.