We revisit the Markov Entropy Decomposition, a classical convex relaxation
algorithm introduced by Poulin and Hastings to approximate the free energy in
quantum spin lattices. We identify a sufficient condition for its convergence,
namely the decay of the effective interaction. We prove that this condition is
satisfied for systems in 1D at any temperature as well as in the
high-temperature regime under a certain commutativity condition on the
Hamiltonian. This yields polynomial and quasi-polynomial time approximation
algorithms in these settings, rispettivamente. Inoltre, the decay of the
effective interaction implies the decay of the conditional mutual information
for the Gibbs state of the system. We then use this fact to devise a rounding
scheme that maps the solution of the convex relaxation to a global state and
show that the scheme can be efficiently implemented on a quantum computer, thus
proving efficiency of quantum Gibbs sampling under our assumption of decay of
the effective interaction.

Questo articolo esplora i giri e le loro implicazioni.

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