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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.

Recent work has shown the effectiveness of tensor network methods for computing large deviation functions in constrained stochastic models in the infinite time limit. Here we show that these methods can also be used to study the statistics of dynamical observables at arbitrary finite time. This is a harder problem because, in contrast to the infinite time case where only the extremal eigenstate of a tilted Markov generator is relevant, for finite time the whole spectrum plays a role. We show that finite time dynamical partition sums can be computed efficiently and accurately in one dimension using matrix product states, and describe how to use such results to generate rare event trajectories on demand. We apply our methods to the Fredrickson-Andersen (FA) and East kinetically constrained models, and to the symmetric simple exclusion process (SSEP), unveiling dynamical phase diagrams in terms of counting field and trajectory time. We also discuss extensions of this method to higher dimensions.

The Fredkin spin chain serves as an interesting theoretical example of a quantum Hamiltonian whose ground state exhibits a phase transition between three distinct phases, one of which violates the area law. Here we consider a classical stochastic version of the Fredkin model, which can be thought of as a simple exclusion process subject to additional kinetic constraints, and study its classical stochastic dynamics. The ground state phase transition of the quantum chain implies an equilibrium phase transition in the stochastic problem, whose properties we quantify in terms of numerical matrix product states (MPS). The stochastic model displays slow dynamics, including power law decaying autocorrelation functions and hierarchical relaxation processes due to exponential localization. Like in other kinetically constrained models, the Fredkin chain has a rich structure in its dynamical large deviations -which we compute accurately via numerical MPS -including an active-inactive phase transition, and a hierarchy of trajectory phases connected to particular equilibrium states of the model. We also propose, via its height field representation, a generalization of the Fredkin model to two dimensions in terms of constrained dimer coverings of the honeycomb lattice.

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