1.15

## Keio University

Skip to 0 minutes and 4 secondsOne of the most important concepts you must grasp to understand quantum computing is the superposition and interference of waves. Since waves are moving through space, you can have more than one wave passing through one place at the same time. When that happens, the waves add up or superimpose on one another. Imagine, we have a wave like this and another wave of a slightly different frequency like this. It doesn’t matter if they are light waves or water waves, it’s the same. If we take the two waves and we add them up, we get a pattern like this.

Skip to 0 minutes and 42 secondsYou can see that on the left, the waves are almost lined up, which we call “being in phase,” and so their peaks add up. This is called “constructive interference.” On the right, one wave is positive where the other is negative, which we call “being out of phase,” and the two waves cancel out. This is called “destructive interference.” When we let the waves run longer, you can see that the interference grows and shrinks. This growing and shrinking pattern is called “beating.” The figures show the waves moving up and down, but in sound the air moves back and forth instead but the principle is exactly the same. Let's see if we can hear that beating pattern in sound.

Skip to 1 minute and 24 secondsWelcome Shinnosuke Ozawa, one of my students, and a talented bass guitar player. I am happy to be here. Good. So with the bass guitar, you adjust the tension on the strings using these pins and that adjusts the sound or the frequency at which the strings vibrate. First, let’s hear your bass guitar in tune. Play one note for me.

Skip to 1 minute and 54 secondsOkay, that’s an A, right? Yes. I think that’s 360 hertz if I did the calculation right. Now, by placing his fingers on the fret, he can play the same note on two different strings. Let’s hear the two strings together.

Skip to 2 minutes and 15 secondsvery nice. Now, we take one of those strings and put it just slightly out of tune with the other, if we are lucky, we will be able to hear the interference between the two types of waves. Let’s try that. Put one string a little bit out of tune here and see what happens.

Skip to 2 minutes and 44 secondsYeah, that’s fantastic. Did you hear that? As the frequency of the two strings got closer and closer together the beating got longer and slower until finally it stopped and they were in tune together. Thanks. You're welcome The amount of interference we get depends on both the amplitude, how strong the wave is and the phase, whether it's near the beginning, middle or end of its cycle. When waves are propagating through space, the distance they have to travel depends on the angle and the phase in turn depends on that distance. If we have one source, we can see the waves radiating out in every direction.

Skip to 3 minutes and 29 secondsIf we have two sources, we can see the waves radiating out in every direction and in some places those waves reinforce and in others they cancel giving us constructive and destructive interference. In the applications and animations in the article accompanying this video, you can explore interference in detail.

# Superposition and Interference

One of the most important concepts you must grasp to understand quantum computing is the superposition and interference of waves. Since waves are moving through space, you can have more than one wave passing through one place at the same time. When that happens, the wave add up, or superimpose on one another.

In addition to the video, we have prepared a set of applications for you to play with, and we have made the 3-D models in the video available.

## One-dimensional interference

In the upcoming application, you will have the opportunity to see the following behaviors:

1. constructive interference
2. destructive interference
3. “beats” with different frequencies

This will be followed by a quiz with questions that can be answered using the application.

## 2-dimensional interference

Interference can happen in more than one dimension. If we have two sources that are each sending out a sine wave, it might look something like this, if the waves stay the same height as they move away:

After the 1-D interference application and quiz, you will see an application demonstrating 2-D interference, as well.

To aid the vision-impaired, or for those who just enjoy 3-D printing, we have created some 3-D printable models representing some of the key concepts. You saw these models in the video, feel free to print your own.

Most 3-D printing software can handle a file type known as STL. Here, we provide an STL file for you to print, or the source code in a language called OpenSCAD, if you would like to modify the shape.

• A single wave source radiating in two dimensions, with $1/r$ decay of amplitude
• Two wave sources radiating in two dimensions demonstrating interference, with $1/r$ decay of amplitude

## $N$-dimensional interference

Humans can’t effectively visualize more than three dimensions, but in fact a quantum computer creates interference across many different variables, which we can treat as separate dimensions. We will see this effect when we discuss quantum algorithms in Week 3.