Gibbs Sampling of Continuous Potentials on a Quantum Computer

Arsalan Motamedi, Pooya Ronagh
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:36322-36371, 2024.

Abstract

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 and achieves polynomial quantum speedups in mean estimation in the Gibbs measure for generic non-convex periodic functions. At high temperatures the algorithm also allows for exponentially improved precision in sampling from Morse functions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-motamedi24a, title = {{G}ibbs Sampling of Continuous Potentials on a Quantum Computer}, author = {Motamedi, Arsalan and Ronagh, Pooya}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {36322--36371}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/motamedi24a/motamedi24a.pdf}, url = {https://proceedings.mlr.press/v235/motamedi24a.html}, abstract = {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 and achieves polynomial quantum speedups in mean estimation in the Gibbs measure for generic non-convex periodic functions. At high temperatures the algorithm also allows for exponentially improved precision in sampling from Morse functions.} }
Endnote
%0 Conference Paper %T Gibbs Sampling of Continuous Potentials on a Quantum Computer %A Arsalan Motamedi %A Pooya Ronagh %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-motamedi24a %I PMLR %P 36322--36371 %U https://proceedings.mlr.press/v235/motamedi24a.html %V 235 %X 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 and achieves polynomial quantum speedups in mean estimation in the Gibbs measure for generic non-convex periodic functions. At high temperatures the algorithm also allows for exponentially improved precision in sampling from Morse functions.
APA
Motamedi, A. & Ronagh, P.. (2024). Gibbs Sampling of Continuous Potentials on a Quantum Computer. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:36322-36371 Available from https://proceedings.mlr.press/v235/motamedi24a.html.

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