We extend the formalism of pure state thermodynamics to matrix product states. In pure state thermodynamics finite temperature properties of quantum systems are derived without the need of statistical mechanics ensembles, but instead using typical properties of random pure states. We show that this formalism can be useful from the computational point of view when combined with tensor network algorithms. In particular, a recently introduced Monte Carlo algorithm is considered which samples matrix product states at random for the estimation of finite temperature observables. Here we characterize this algorithm as an $(ε, δ)$-approximation scheme and we analytically show that sampling one single state is sufficient to obtain a very good estimation of finite temperature expectation values. These results provide a substantial computational improvement with respect to similar algorithms for one-dimensional quantum systems based on uniformly distributed pure states. The analytical calculations are numerically supported simulating finite temperature interacting spin systems of size up to 100 qubits.

20 pages, 3 figures; comments are welcome