Markov chains for probability distributions related to matrix product states and 1D Hamiltonians are introduced. With appropriate inverse temperature schedules, these chains can be combined into a random approximation scheme for ground states of such Hamiltonians. Numerical experiments suggest that a linear, i.e. fast, schedule is possible in non-trivial cases. A natural extension of these chains to 2D settings is next presented and tested. The obtained results compare well with Euclidean evolution. The proposed Markov chains are easy to implement and are inherently sign problem free (even for fermionic degrees of freedom).
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