Superposition and Interference

Two lessons back, we introduced the idea that quantum mechanics allows qubits to exhibit both wave-like and particle-like properties. In this lesson, we shall focus on the former and see how wave properties become useful in formulating accurate quantum algorithms. But before that, let’s recap wave theory.

Wave theory

As you recall from your high school or college classes, waves are disturbances or vibrations in a medium that transport energy from one point to another. There are two types of waves, transverse and longitudinal. In transverse waves, the displacement of the medium is perpendicular to the direction of wave motion. But in longitudinal waves, the displacement of the medium is parallel. We’ll limit ourselves to discussing transverse waves as they are relevant to our subject matter.

We typically define waves using their wavelength, speed, and frequency. The speed is obviously the rate at which the wave travels between two points. The frequency is the number of vibrations per unit time. The wavelength is the distance between two consecutive troughs or two consecutive crests. Places of high amplitude are called crests while places of low amplitude are called troughs. We define their amplitude as the greatest displacement from the mean position. The wavelength $\lambda$, speed $c$, and frequency $f$ are related by the following equation:

$$c = f\lambda$$

Looking at the equation above, frequency and wavelength are inversely proportional to each other, meaning that an increase in one causes a decrease in the other, and vice versa. In other words, lower frequencies mean longer wavelengths, and higher frequencies mean shorter wavelengths.