Abstract
In ``Playing Pool with π'' \cite{Galperin}, Galperin invented an extraordinary method to learn the digits of π 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.
Highlights
An impractical but picturesque way to determine the digits of π is to hurl a heavy ball towards a light ball that has its back to a wall, as in Fig. 1, and count the ensuing elastic collisions
If the left ball is much heavier than the right, it is harder to slow down and reverse
The heavier the left ball, the more collisions are needed, M = 100 → #collisions = 31 M = 106 → #collisions = 3141 M = 1020 → #collisions = 31415926535 (1.2) (1.3) (1.4). These digits look familiar! In “Playing Pool with π” [1], G. Galperin proved that this algorithm really is spitting out the digits of π, since for M = 100N
Summary
An impractical but picturesque way to determine the digits of π is to hurl a heavy ball towards a light ball that has its back to a wall, as in Fig. 1, and count the ensuing elastic collisions. The left ball transfers all its momentum to the right ball. At the third and final collision, the right ball transfers all its momentum back to the left ball. If the left ball is much heavier than the right, it is harder to slow down and reverse. The heavier the left ball, the more collisions are needed,. M = 100 → #collisions = 31 M = 106 → #collisions = 3141 M = 1020 → #collisions = 31415926535 In “Playing Pool with π” [1], G Galperin proved that this algorithm really is spitting out the digits of π, since for M = 100N. I will argue that there is a precise isomorphism between bouncing billiard balls and quantum search
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