Quantum Physics
[Submitted on 14 Oct 2022 (v1), last revised 20 Jul 2024 (this version, v4)]
Title:Gibbs Sampling of Continuous Potentials on a Quantum Computer
View PDFAbstract:Gibbs sampling from continuous real-valued functions is a challenging problem of interest in machine learning. Here we leverage quantum Fourier transforms to build a quantum algorithm for this task when the function is periodic. We use the quantum algorithms for solving linear ordinary differential equations to solve the Fokker--Planck equation and prepare a quantum state encoding the Gibbs distribution. We show that the efficiency of interpolation and differentiation of these functions on a quantum computer depends on the rate of decay of the Fourier coefficients of the Fourier transform of the function. We view this property as a concentration of measure in the Fourier domain, and also provide functional analytic conditions for it. Our algorithm makes zeroeth order queries to a quantum oracle of the function. Despite suffering from an exponentially long mixing time, this algorithm allows for exponentially improved precision in sampling, and polynomial quantum speedups in mean estimation in the general case, and particularly under geometric conditions we identify for the critical points of the energy function.
Submission history
From: Arsalan Motamedi [view email][v1] Fri, 14 Oct 2022 20:56:44 UTC (33 KB)
[v2] Wed, 15 Feb 2023 02:16:04 UTC (59 KB)
[v3] Wed, 31 May 2023 23:54:24 UTC (473 KB)
[v4] Sat, 20 Jul 2024 21:35:17 UTC (501 KB)
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