Partition function estimation with a quantum coin toss

TL;DR

This paper presents a quantum algorithm for estimating partition functions efficiently using a quantum coin toss approach, avoiding complex subroutines, and demonstrates its practical potential with a 9-qubit experiment.

Contribution

The paper introduces a novel quantum algorithm for partition function estimation based on a quantum coin toss, improving runtime scaling and error mitigation.

Findings

Achieves runtime scaling as O(N/Z_β), better than previous methods.

Does not require quantum phase estimation or amplitude amplification.

Successfully mitigates errors in a 9-qubit experimental demonstration.

Abstract

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 $N$ in the number of particles. This paper introduces a quantum algorithm for estimating the partition function $Z_\beta$ of a generic Hamiltonian $H$ 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 $f_\beta[H]=e^{-\beta H/{2}}$ at inverse temperature $\beta$ to the maximally mixed state, realized by a block-encoding of $f_\beta[H]$ 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 $\mathcal{O}(N/{Z_\beta})$, 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.