Playing Pool with $|ψ\rangle$: from Bouncing Billiards to Quantum Search

Adam R. Brown

doi:10.22331/q-2020-11-02-357

Playing Pool with $|ψ\rangle$: from Bouncing Billiards to Quantum Search

Google, Mountain View, CA 94043, USA
Department of Physics, Stanford University, Stanford, CA 94305, USA

Published:	2020-11-02, volume 4, page 357
Eprint:	arXiv:1912.02207v2
Doi:	https://doi.org/10.22331/q-2020-11-02-357
Citation:	Quantum 4, 357 (2020).

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Abstract

In ``Playing Pool with $\pi$'' [1], Galperin invented an extraordinary method to learn the digits of $\pi$ by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's algorithm for quantum search. This provides an illuminating way to visualize Grover's algorithm.

Popular summary

In the 1990s, an extraordinary method was invented for generating the digits of π by counting the collisions of billiard balls. This paper shows that the billiard ball algorithm is step-for-step mathematically identical to a seemingly completely unrelated process: the proposed quantum search algorithm to be performed on quantum computers.

► BibTeX data

@article{Brown2020playingpool,
  doi = {10.22331/q-2020-11-02-357},
  url = {https://doi.org/10.22331/q-2020-11-02-357},
  title = {Playing {P}ool with {$|ψ\rangle$}: from {B}ouncing {B}illiards to {Q}uantum {S}earch},
  author = {Brown, Adam R.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {357},
  month = nov,
  year = {2020}
}

► References

[1] G. Galperin, ``Playing pool with $\pi$'', Regular and Chaotic Dynamics, v. 8, no. 4 (2003).
https://doi.org/10.1070/RD2003v008n04ABEH000252

[2] Grant Sanderson, ``The most unexpected answer to a counting puzzle'', https://youtu.be/HEfHFsfGXjs.
https://www.youtube.com/watch?v=HEfHFsfGXjs

[3] L. K. Grover, ``A Fast quantum mechanical algorithm for database search,'' Proceedings, 28th Annual ACM Symposium on the Theory of Computing (STOC), 1996, pages 212-219; arXiv:quant-ph/9605043.
https://doi.org/10.1145/237814.237866
arXiv:quant-ph/9605043

Cited by

[1] Zhiwei Chong, "Recursive approach to repeated collisions", The Mathematical Gazette 109 576, 427 (2025).

[2] Li-Yi Hsu and Ching-Hsu Chen, "Conservation Laws in Quantum Database Search", Foundations of Physics 54 5, 70 (2024).

[3] Maria Mannone and Davide Rocchesso, Quantum Computing in the Arts and Humanities 193 (2022) ISBN:978-3-030-95537-3.

[4] Yin Cai and Fu-Lin Zhang, "Hear π from quantum Galperin billiards", Canadian Journal of Physics 101 9, 491 (2023).

[5] Vishal P. Patil, Žiga Kos, and Jörn Dunkel, "Harmonic flow-field representations of quantum bits and gates", Physical Review Research 6 4, 043039 (2024).

[6] Pablo Fernández and Miguel A. Martin-Delgado, "Implementing the Grover algorithm in homomorphic encryption schemes", Physical Review Research 6 4, 043109 (2024).

[7] Keiko I. Nagao, Yuga Sakano, Takashi Shinohara, Yuji Matsuda, and Hisashi Takami, "Estimation of π via experiment", European Journal of Physics 46 4, 045006 (2025).

[8] X. M. Aretxabaleta, M. Gonchenko, N. L. Harshman, S. G. Jackson, M. Olshanii, and G. E. Astrakharchik, "The dynamics of digits: Calculating pi with Galperin's billiards", arXiv:1712.06698, (2017).

[9] Jiang Liu, "A Classical $\pi$ Machine and Grover's Algorithm", arXiv:2105.10257, (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-10 05:52:22) and SAO/NASA ADS (last updated successfully 2026-08-09 16:05:34). The list may be incomplete as not all publishers provide suitable and complete citation data.

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