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Skip to 0 minutes and 4 seconds 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 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 seconds You 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 seconds Welcome 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 seconds Okay, 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 12 seconds Okay,

Skip to 2 minutes and 15 seconds very 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 seconds Yeah, 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 seconds If 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:

2-D wave propagation and interference of two sources, no decay

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.




重ね合わせと干渉

量子計算を理解する上で理解しなければならない最も重要な概念の一つが、波の重ね合わせと波の干渉です。ある波が空間上を伝わっていく時、同じ空間に同時に二つ以上の波が存在することも可能ですが、同じ空間に2つ以上の波が同時に存在する場合、互いの波が強め合ったり、重ねあったりするのです。

ビデオの内容に加えて、みなさんが実際に手を動かして学んでいただけるアプリや3Dモデルを用意していますので試してみてください。

一次元の波の干渉

このあとのステップで紹介するアプリでは、以下のような波の特徴を見ることができます。

  1. 波の強め合い(建設的干渉)
  2. 波の弱め合い(相殺的干渉)
  3. 異なる周波数の波による波のうなり(ビート)

これはそのアプリを使ってクイズに答えると非常にわかりやすく理解できるかと思います。

二次元の波の干渉

波の干渉は一次元以上でも同様に起こります。正弦波を出力する2つの波源があった時、その様子は以下のように、同じ高さで進んでいるように見えます。

2-D wave propagation and interference of two sources, no decay

一次元の波の干渉に関するクイズの後、二次元の波についてもアプリを使って解説していこうと思います。

視覚障害のある方や、3Dプリンターを利用する人のために、いくつかの重要な概念を説明するための、3Dプリンターで出力可能な3Dモデルを作成しました。ビデオ内で用いられているものと同じです。ご自由にぜひ3Dプリントして見てください。

多くの3Dプリントソフトウェアが使用しているのはSTLと呼ばれる形式のファイルです。以下にSTL形式のファイルと、OpenSCADと呼ばれる言語のソースコードを用意してあります。もし実際にどのような形かを確かめたい場合は、実際にプリントして見てください。

\(N\)次元の波の干渉

人間は三次元以上をうまく視覚化することができません。しかし、量子コンピュータでは、異なる次元として扱うことのできる、多くの変数の間に波の干渉を作ります。この効果については第3週で、アルゴリズムについて議論する時に見ていこうと思います。

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This video is from the free online course:

Understanding Quantum Computers

Keio University