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We present a method to approximate partition functions of quantum systemsusing mixed-state quantum computation. For positive semi-definite Hamiltonians,our method has expected running-time that is almost linear in$(M/(\epsilon_{\rm rel}\mathcal{Z} ))^2$, where $M$ is the dimension of thequantum system, $\mathcal{Z}$ is the partition function, and $\epsilon_{\rmrel}$ is the relative precision. It is based on approximations of theexponential operator as linear combinations of certain operators related toblock-encoding of Hamiltonians or Hamiltonian evolutions. The trace of eachoperator is estimated using a standard algorithm in the one clean qubit model.For large values of $\mathcal{Z}$, our method may run faster than exactclassical methods, whose complexities are polynomial in $M$. We also prove thata version of the partition function estimation problem within additive error iscomplete for the so-called DQC1 complexity class, suggesting that our methodprovides a super-polynomial speedup for certain parameter values. To attain adesired relative precision, we develop a classical procedure based on asequence of approximations within predetermined additive errors that may be ofindependent interest.

Computing partition functions in the one clean qubit model