Machine learning time-local generators of open quantum dynamics

In the study of closed many-body quantum systems, one is often interested in the evolution of a subset of degrees of freedom. On many occasions it is possible to approach the problem by performing an appropriate decomposition into a bath and a system. In the simplest case the evolution of the reduced state of the system is governed by a quantum master equation with a time-independent, i.e., Markovian, generator. Such evolution is typically emerging under the assumption of a weak coupling between the system and an infinitely large bath. Here we are interested in understanding to which extent a neural network function approximator can predict open quantum dynamics—described by time-local generators—from an underlying unitary dynamics. We investigate this question using a class of spin models, which is inspired by recent experimental setups. We find that indeed time-local generators can be learned. In certain situations they are even time independent and allow to extrapolate the dynamics to unseen times. This might be useful for situations in which experiments or numerical simulations do not allow to capture long-time dynamics and for exploring thermalization occurring in closed quantum systems.

Figure 1

Reduced local dynamics of a many-body quantum system. The reduced state ${\rho }_S$ describes local degrees of freedom of a quantum spin system. The remainder of the system, $B$, takes the role of a “bath” with initial state ${\rho }_B$. We consider two scenarios: (a) Model I, spin chain with open boundary conditions, where the interaction strength $V$ and range $\alpha$ are varied [see Eq. (2)]; (b) Model II, spin chain with closed boundaries and nearest-neighbor interaction, where the system-bath coupling $V^{\prime }$ and the initial temperature $\beta$ of the bath are modified. (c) The dynamics of the reduced state, ${\rho }_S\left(t\right)$, is obtained by tracing out the degrees of freedom of $B$ and to be learned by a neural network. The learned generator may be either Markovian (time independent) or time dependent.

Figure 2

Effective time-independent generator: Training and extrapolation. To obtain a time-independent generator, we use a MLP. The inputs are the time-dependent expectation values of the reduced system observables—for a single spin contained in the Bloch vector ${\mathbf{v}}_{\mathrm{b}}$— in a given training time window (blue-shaded region) and for several initial Bloch vectors (top to bottom). The MLP learns the “propagator” $M\left[{\theta }^{*}\right]$, which depends on the optimized network parameters ${\theta }^{*}$ but not on the actual time. Considering a Euler integration step of length $dt$, this provides a matrix $\overline{L}$ representing the action of an “averaged” generator on the Bloch vector. When the reduced dynamics is Markovian, this learned generator allows for the network to make predictions for unseen future times (red-shaded region).

Figure 3

Time-independent generator. (a) Model I: Comparison between the exact dynamics of the Bloch vector and the one obtained by learning an effective time-independent generator with neural networks. The density plot shows the time-averaged norm difference, $\overline{\epsilon }$ [Eq. (8) averaged over 5 randomly selected values $c$ of the initial conditions; see Eq. (7)], for various parameter choices $(V,\alpha )$. The data are calculated for $N=9$ spins with a training time of $T_{\mathrm{train}}=10/\mathrm{\Omega }$, $dt=0.01/\mathrm{\Omega }$, and a total time $T_{\mathrm{tot}}=20/\mathrm{\Omega }$ (see main text for details). We also show two examples for the Bloch vector evolution, where we choose $c=0$ and $T_{\mathrm{tot}}=20/\mathrm{\Omega }$ for illustration. Here solid curves correspond to the components of the Bloch vector propagated with the learned time-averaged generator. The dashed lines correspond to the numerically exact solution. (b) Same as (a), but for Model II and the parameter set $(V^{\prime },\beta )$.

Figure 4

Hypermodel and time-dependent generator. Hypermodels allow to encode time-dependent generators ${\mathcal{L}}_t$. The time dependence is quantified through the derivative of $L_t$ with respect to time, expressed by the positive quantity $\xi \left(t\right)$ (see text for definition). (a) Three example curves corresponding to Model II ($N=9$). The function $\xi \left(t\right)$ depends only on the parameters of the model and is independent from the initial condition chosen. Generally, the stronger the interaction $V^{\prime }$, the more pronounced the time dependence. This trend is clearly visible in (b), where we show the averaged value of $\xi \left(t\right)$ in the interval 0 to $T=10/\mathrm{\Omega }$ ($dt=0.01/\mathrm{\Omega }$), denoted as $\mathrm{\Xi }$.

Figure 5

Hypermodel accuracy: Comparison between the exact dynamics and the dynamics predicted by the hypermodel for initial state ${\rho }_0$ reported in Eq. (5) and for $N=7$. As it can be observed the agreement between the model prediction (dotted line) and the exact dynamics (full line) is excellent. This fact enables us to use the local operator learned using the hypermodel for studying the time-dependence of $L_t$.

Figure 6

Time-dependence: Behavior of the quantity $\xi \left(t\right)$ for the same parameter regimes of Fig. 5. The quantity $\mathrm{\Xi }$ grows with the interaction, meaning that the generator of the local dynamics $L_t$ becomes more time-dependent.