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arXiv:1910.11842 · Physical Review A

Computing partition functions in the one-clean-qubit model

Anirban Ch Narayan Chowdhury, Rolando D. Somma, Yiğit Subaşı

Published Mar 17, 2021• 23 citations
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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.

DOI
10.1103/physreva.103.032422

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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/(\ensuremathϵrelZ)]2{[M/({\ensuremath{\epsilon}}_{\mathrm{rel}}\mathcal{Z})]}^{2}[M/(\ensuremathϵrel​Z)]2, where MMM is the dimension of the quantum system, Z\mathcal{Z}Z is the partition function, and \ensuremathϵrel{\ensuremath{\epsilon}}_{\mathrm{rel}}\ensuremathϵrel​ 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 Z\mathcal{Z}Z, our method may run faster than exact classical methods, whose complexities are polynomial in MMM. 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.
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Estimating quantum partition functions is a critical task in a variety of fields. However, the problem is classically intractable in general due to the exponential scaling of the Hamiltonian dimension NNN in the number of particles. This paper introduces a quantum algorithm for estimating the partition function ZβZ_βZβ​ of a generic Hamiltonian HHH up to multiplicative error based on a quantum coin toss. The coin is defined by the probability of applying the quantum imaginary-time evolution propagator fβ[H]=e−βH/2f_β[H]=e^{-βH/{2}}fβ​[H]=e−βH/2 at inverse temperature βββ to the maximally mixed state, realized by a block-encoding of fβ[H]f_β[H]fβ​[H] into a unitary quantum circuit followed by a post-selection measurement. Our algorithm does not use costly subroutines such as quantum phase estimation or amplitude amplification; and the binary nature of the coin allows us to invoke tools from Bernoulli-process analysis to prove a runtime scaling as O(N/Zβ)\mathcal{O}(N/{Z_β})O(N/Zβ​), quadratically better than previous general-purpose algorithms using similar quantum resources. Moreover, since the coin is defined by a single observable, the method lends itself well to quantum error mitigation. We test this in practice with a proof-of-concept 9-qubit experiment, where we successfully mitigate errors through a simple noise-extrapolation procedure. Our findings offer an interesting alternative for quantum partition function estimation relevant to early-fault quantum hardware.
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