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A simple lower bound for the complexity of estimating partition functions on a quantum computer (2404.02414v2)

Published 3 Apr 2024 in quant-ph, cs.CC, cs.DS, math.ST, and stat.TH

Abstract: We study the complexity of estimating the partition function $\mathsf{Z}(\beta)=\sum_{x\in\chi} e{-\beta H(x)}$ for a Gibbs distribution characterized by the Hamiltonian $H(x)$. We provide a simple and natural lower bound for quantum algorithms that solve this task by relying on reflections through the coherent encoding of Gibbs states. Our primary contribution is a $\varOmega(1/\epsilon)$ lower bound for the number of reflections needed to estimate the partition function with a quantum algorithm. The proof is based on a reduction from the problem of estimating the Hamming weight of an unknown binary string.

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