Computing partition functions in the one-clean-qubit model

Abstract

We present a method to approximate partition functions of quantum systems using mixed-state quantum computation. For positive-semidefinite Hamiltonians, our method has an expected running-time that is almost linear in ${[M/({\ensuremath{\epsilon}}_{\mathrm{rel}}\mathcal{Z})]}^{2}$, where $M$ is the dimension of the quantum system, $\mathcal{Z}$ is the partition function, and ${\ensuremath{\epsilon}}_{\mathrm{rel}}$ is the relative precision. It is based on approximations of the exponential operator as linear combinations of certain operators related to block-encoding of Hamiltonians or Hamiltonian evolutions. The trace of each operator is estimated using a standard algorithm in the one-clean-qubit model. For large values of $\mathcal{Z}$, our method may run faster than exact classical methods, whose complexities are polynomial in $M$. We also prove that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that our method provides a superpolynomial speedup for certain parameter values. To attain a desired relative precision, we develop a classical procedure based on a sequence of approximations within predetermined additive errors that may be of independent interest.