Partition function estimation with a quantum coin toss

• URL: http://arxiv.org/abs/2411.17816v1
• Date: Tue, 26 Nov 2024 19:01:19 GMT
• Title: Partition function estimation with a quantum coin toss
• Authors: Thais de Lima Silva, Lucas Borges, Leandro Aolita,
• Abstract summary: Estimating quantum partition functions is a critical task in a variety of fields. This paper introduces a quantum algorithm for estimating the partition function $Z_beta$ of a generic Hamiltonian $H$ up to multiplicative error.
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• 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.

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