Representing an Analog Clock with a Pure Quantum State

This article is meant to be relaxing—just for fun! The Bloch sphere is one of the first concepts we encounter when learning quantum computing. Each point on its surface corresponds to a pure state of a qubit and can be described by two angles, θ and φ, as shown below.

These two angles can also be mapped onto the hour and minute hands of an ordinary analog clock. If θ represents the hour hand and φ the minute hand, we get something like the illustration below.

Wouldn’t it be wonderful if someone made and sold a spherical clock like this? I doubt one exists, though.

So I created an Android mobile phone app—a “spherical clock” that displays the time on a Bloch sphere. Tap “Current Time,” and the display automatically updates every 30 or 10 seconds. Each time it updates, the yellow dot representing the quantum state briefly contracts. It is quite soothing just to sit and watch! You can also rotate the Bloch sphere and view it freely from any angle.

The current time is 4:54 p.m. Four white dots are visible in the upper-right part of the sphere. Since the clock uses a 12-hour format, they indicate four o’clock. The minute hand is moving along the path from four to five. Naturally, its apparent motion slows dramatically as it approaches the north or south pole. Depending on how you visualize it, reversing the north and south poles might feel more intuitive.

## Additional Note

Some readers may want to add a second hand. One possible approach would be to introduce another variable for the qubit: the length (r) of the line extending from the center of the sphere, where (0 < r < 1). However, points inside the Bloch sphere—rather than on its surface, where (r = 1)—represent mixed states rather than ordinary pure quantum states. This would require a different treatment, and I also find it less appealing aesthetically. So, for now, I have decided not to add this hand.