A simple lower bound for the complexity of estimating partition functions on a quantum computer

• URL: http://arxiv.org/abs/2404.02414v2
• Date: Tue, 9 Apr 2024 02:13:00 GMT
• Title: A simple lower bound for the complexity of estimating partition functions on a quantum computer
• Authors: Zherui Chen, Giacomo Nannicini,
• Abstract summary: We study the complexity of estimating the partition function $mathsfZ(beta)=sum_xinchi e-beta H(x)$ 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.
• Score: 0.20718016474717196
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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