  We introduce a classical algorithm to approximate the free energy of local,
translation-invariant, one-dimensional quantum systems in the thermodynamic
limit of infinite chain size. While the ground state problem (i.e., the free
energy at temperature $T = 0$) for these systems is expected to be
computationally hard even for quantum computers, our algorithm runs for any
fixed temperature $T > 0$ in subpolynomial time, i.e., in time
$O((\frac{1}{\varepsilon})^{c})$ for any constant $c > 0$ where $\varepsilon$
is the additive approximation error. Previously, the best known algorithm had a
runtime that is polynomial in $\frac{1}{\varepsilon}$. Our algorithm is also
particularly simple as it reduces to the computation of the spectral radius of
a linear map. This linear map has an interpretation as a noncommutative
transfer matrix and has been studied previously to prove results on the
analyticity of the free energy and the decay of correlations. We also show that
the corresponding eigenvector of this map gives an approximation of the
marginal of the Gibbs state and thereby allows for the computation of various
thermodynamic properties of the quantum system.
