Optimal sampling of dynamical large deviations via matrix product states

Abstract

The large deviation (LD) statistics of dynamical observables is encoded in the spectral properties of deformed Markov generators. Recent works have shown that tensor network methods are well suited to compute the relevant leading eigenvalues and eigenvectors accurately. However, the efficient generation of the corresponding rare trajectories is a harder task. Here we show how to exploit the MPS approximation of the dominant eigenvector to implement an efficient sampling scheme which closely resembles the optimal (so-called "Doob") dynamics that realises the rare events. We demonstrate our approach on three well-studied lattice models, the Fredrickson-Andersen and East kinetically constrained models (KCMs), and the symmetric simple exclusion process (SSEP). We discuss how to generalise our approach to higher dimensions.

Dynamical Large Deviations of Two-Dimensional Kinetically Constrained Models Using a Neural-Network State Ansatz

“…We use this ansatz to represent the long-time configurational probability distributions associated with rare trajectories, inspired by its recent success within the variational optimization of quantum systems [28]. The similarities between variational energy minimization in quantum systems and finding the SCGF as the largest eigenvalue of a tilted generator have inspired the use of variational techniques for studying large deviations in dynamical systems, in particular tensor network methods [29][30][31][32][33]. However, current variational approaches to calculating largedeviation functions are usually limited to one-dimensional systems, while the flexibility of the neural-network ansatz allows for straightforward generalization to higher spatial dimensions.…”

Slow dynamics and large deviations in classical stochastic Fredkin chains

“…( 28) we can study statistical properties of the trajectory ensemble of the Fredkin model for long-time trajectories. We do this by means of numerical tensor networks along the lines of similar recent work in KCMs [78][79][80][81][82][83]. Figure 5 shows the LD statistics obtained numerically.…”

Finite time large deviations via matrix product states

“…Furthermore, we show how to use the results here to directly simulate stochastic trajectories in finite-time tilted ensembles at small computational cost, thus generalising the method of Ref. [29].…”

“…The reason is that away from the long time limit the corresponding dynamical partition sums (i.e., moment generating functions) do not obey a LD principle in time -only obeying an LD principle in space for large sizes -and as a consequence they are not determined only by the leading eigenvalue of the tilted generator, but by their whole spectrum. If time is very short, one can get away with direct sampling, but for intermediate times the usual sampling approaches fall short [29]. Here we develop a scheme to study these rare events by implementing well-developed TN techniques to simulate time evolution.…”