## My Speech to 500 Australian Teenage Schoolboys About Mathematics

I suppose I should start with who I am and what I do and perhaps why I am here in front of you. But I’m not going to do that, at least not yet. I don’t want to stand here and list all my accomplishments so that you may be impressed and that would convince you to listen to me. No. I don’t want to do that because I know it wouldn’t work. I know that because it wouldn’t have worked on me when I was in your place and someone else was up here.

Now, of course you can tell by my accent that I wasn’t literally down there. I was in Canada. And I sure as hell wasn’t wearing a tie. But I imagine our priorities were fairly similar: friends, getting away parents, maybe sports (in my case hockey of course and yours maybe footy), but most importantly… mathematics! No. Video games.

I don’t think there is such a thing as being innately gifted in anything. Though, I am pretty good at video games. People become very good at things they practice. A little practice leads to a small advantage, which leads to opportunities for better practice, and things snowball. The snowball effect. Is that a term you guys use in Australia? I mean, it seems like an obvious analogy for a Canadian. It’s how you make a snowman after all. You start with a small handful of snow and you start to roll it on the ground. The snow on the ground sticks to the ball and it gets bigger and bigger until you have a ball as tall as you!

Practice leads to a snowball effect. After a while, it looks like you are gifted at the thing you practiced, but it was really just the practice. Success then follows from an added sprinkling of luck and determination. That’s what I want to talk to you about today: practice.

I don’t want to use determination in the sense that I was stubbornly defiant in the face of adversity. Though, from the outside it might look that way. You can either be determined to avoid failure or determined to achieve some objective. Being determined to win is different from being determined not to lose.

There is something psychologically different between winning and not losing. You see, losing implies a winner, which is not you. But winning does not require a loser, because you can play against yourself. This was the beauty of disconnected video games of 80’s and 90’s. You played against yourself, or maybe “the computer”. That doesn’t mean it was easy. I’ll given anyone here my Nintendo if they can beat Super Mario Bros. in one go. (I’m not joking. I gave my children the same offer and they barely made it past the first level). It was hard and frustrating, but no one was calling you a loser on the other end. And when you finally beat the game, you could be proud. Proud of yourself and for yourself. Not for the fake internet points you get on social media, but for you.

I actually really did want to talk to you today about mathematics. What I want to tell you is that, when I was your age, I treated mathematics like a video game. I wanted to win. I wanted to prove to myself that I could solve every problem. Some nights I stayed up all night trying to solve a single problem. You know how they say you can’t have success without failure? This is a perfect example. The more you fail at trying to solve a maths problem, the more you understand when you finally do solve it. And what came along with failing and eventually succeeding in all those maths problems? Practice.

Well I don’t know much about the Australian education system and culture. But I’m guessing from Hollywood you know a bit about highschool in North America. I’m sure you know about prom, and of course about Prom King and Prom Queen. What you may not know is that the King and Queen’s court always has a jester. That is, along with King and Queen, each year has a Class Clown — the joker, the funny guy. I wasn’t the prom king, or queen. But I did win the honour of class clown.

When I finished highschool, I was really good at three things: video games, making people laugh, and mathematics. I promise you, there is no better combination. If there was a nutrition guide for the mind, it would contain these three things. Indeed, now more than ever before, you need to be three types of smart. You need to be quick, reactive, and adaptive — the skills needed to beat a hard video game. You need emotional intelligence, you need to know what others are thinking and feeling — how to make them laugh. And finally you need to be able to solve problems, and all real problems require maths to solve them.

There are people in the world, lots of people — billions, perhaps — who look in awe at the ever increasing complexity of systems business, government, schools, and technology, including video games. They look, and they feel lost. Perhaps you know someone that can’t stand new technology, or change in general. Perhaps they don’t even use a piece of technology because they believe they will never understand how to use it.

You all are young. But you know about driving, voting, and paying taxes, for example. Perhaps it looks complicated, but at least you believe that you can and will be able to do it when the time comes. Imagine feeling that such things were just impossible. That would be a weird feeling. You brain can’t handle such dissonance. So you would need to rationalise it in one way or another. You’d say it’s just not necessary, or worse, it’s something some “other” people do. At that point, for your brain to maintain a consistent story, it will start to reject new information and facts that aren’t consistent with your new story.

This is all sounds far fetched, but I guarantee you know many people with such attitudes. To make them sound less harmful, they call them “traditional”. How do otherwise “normal” people come to hold these views? It’s actually quite simple: they fear, not what they don’t understand, but what they have convinced themselves is unnecessarily complicated. I implore you, start today, start right now. Study maths. It is the only way to intellectually survive in a constantly changing world.

Phew that was a bit depressing. Let me give you a more fun and trivial example. Just this weekend I flew from Sydney to Bendigo. The flight was scheduled to be exactly 2 hours. I was listening to an audiobook and I wondered if I would finish it during the flight. Seems obvious right? If there was less than 2 hours left in the audiobook, then I would finish. If not, then I would not finish. But here’s the thing, audiobooks are read soooo slow. So, I listen to them at 1.25x speed. There was 3 hours left. Does anyone know the answer?

Before I tell you, let me remind you, not many people would ask themselves this question. I couldn’t say exactly why, but in some cases it’s because the person has implicitly convinced themselves that such a question is just impossible to answer. It’s too complicated. So their brain shuts that part of inquiry off. Never ask complicated questions it says. Then this happens: an entire world — no most of the entire universe — is closed off. Don’t close yourself off from the universe. Study maths.

By the way, the answer. It’s not the exact answer but here was my quick logic based on the calculation I could do in my head. If I had been listening at 1.5x speed, then every hour of flight time would get through 1.5 hours of audiobook. That’s 1 hour 30 minutes. So two hours of flight time would double that, 3 hours of audiobook. Great. Except I wasn’t listening at 1.5x speed. I was listening at a slower speed and so I would definitely get through less than 3 hours. The answer was no.

In fact, by knowing what to multiple or divide by what, I could know that I would have exactly 36 minutes left of the audiobook. Luckily or unluckily, the flight was delayed and I finished the book anyway. Was thinking about maths pointless all along? Maybe. But since flights are scheduled by mathematical algorithms, maths saved the day in the end. Maths always wins.

How about another. Who has seen a rainbow? I feel like that should be a trick question just to see who is paying attention. Of course, you have all seen a rainbow. As you are trying to think about the last time you saw a rainbow, you might also be thinking that they are rare — maybe even completely random things. But now you probably see the punchline — maths can tell you exactly where to find a rainbow.

Here is how a rainbow is formed. Notice that number there. That angle never changes. So you can use this geometric diagram to always find the rainbow. The most obvious aspect is that the rainbow exits the same general direction that the sunlight entered the raindrop. So to see a rainbow, the sun has to be behind you.

And there’s more. If the sun is low in the sky, the rainbow will be high in the sky. And if the sun is high, you might not be able to see a rainbow at all. But if you take out the garden hose to find it, make sure you are looking down. Let me tell you my favourite rainbow story. I was driving the family to Canberra. We were driving into the sunset at some point when I drove through a brief sun shower. Since the sun was shining and it was raining, one of my children said, “Maybe we’ll see a rainbow!”

Maybe. Ha. A mathematician knows no maybes. As they looked out their windows, I knew — yes — we would see a rainbow. I said, after passing through the shower, “Everyone look out the back window and look up.” Because the sun was so low, it was apparently the most wonderful rainbow ever seen. I say apparently because I couldn’t see it, on account of me driving. But no matter. I was content in knowing I could conjure such beauty with the power of mathematics.

I could have ended there, since I’m sure you are all highly convinced to catch up on all your maths lessons and homework. However, since I have time, I will tell you a little bit about what maths has enabled me to get paid to do. Namely, quantum physics and computation. Maybe you’ve heard about quantum physics? Maybe you’ve heard about uncertainty (the world is chaotic and random), or superposition (things can be in two places at once and cats can be dead and alive at the same time), or entanglement (what Einstein called spooky action at a distance).

But I couldn’t tell you more about quantum physics than that without maths. This is not meant to make it sound difficult. It should make it sound beautiful. This is quantum physics. It’s called the Schrodinger Equation. That’s about all there is to it. All that stuff about uncertainty, superposition, entanglement, multiple universes, and so on—it’s all contained in this equation. Without maths, we would not have quantum physics. And without quantum physics, we would not have GPS, lasers, MRI, or computers — no computers to play video games and no computers to look at Instagram. Thank a quantum physicist for these things.

Quantum physics also helps us understand the entire cosmos. From the very first instant of the Big Bang born out of a quantum fluctuation to the fusing of Hydrogen into Helium inside stars giving us all energy and life on Earth to the most exotic things in our universe: black holes. These all cannot be understood without quantum physics. And that can’t be understood without mathematics.

And now I use the maths of quantum physics to help create new computing devices that may allow us to create new materials and drugs. This quantum computer has nothing mysterious or special about it. It obeys an equation just as the computers you carry around in your pockets do. But the equations are different and different maths leads to different capabilities.

I don’t want to put up those equations, because if I showed them to even my 25 year-old self, I would run away screaming. But then again, I didn’t know then what I know now, and what I’m telling you today. Anyone can do this. It just takes time. Every mathematician has put in the time. There is no secret recipe beyond this. Start now.

## When will we have a quantum computer? Never, with that attitude

We are quantum drunks under the lamp post—we are only looking at stuff that we can shine photons on.

In a recently posted paper, M.I. Dyakonov outlines a simplistic argument for why quantum computing is impossible. It’s so far off the mark that it’s hard to believe that he’s even thought about math and physics before. I’ll explain why.

Find a coin. I know. Where, right? I actually had to steal one from my kid’s piggy bank. Flip it. I got heads. Flip it again. Heads. Again. Tails. Again, again, again… HHTHHTTTHHTHHTHHTTHT. Did you get the same thing? No, of course you didn’t. That feels obvious. But why?

Let’s do some math. Wait! Where are you going? Stay. It will be fun. Actually, it probably won’t. I’ll just tell you the answer then. There are about 1 million different combinations of heads and tails in a sequence of 20 coin flips. The chances that we would get the same string of H’s and T’s is 1 in a million. You might as well play the lottery if you feel that lucky. (You’re not that lucky, by the way, don’t waste your money.)

Now imagine 100 coin flips, or maybe a nice round number like 266. With just 266 coin flips, the number of possible sequences of heads and tails is just larger than the number of atoms in the entire universe. Written in plain English the number is 118 quinvigintillion 571 quattuorvigintillion 99 trevigintillion 379 duovigintillion 11 unvigintillion 784 vigintillion 113 novemdecillion 736 octodecillion 688 septendecillion 648 sexdecillion 896 quindecillion 417 quattuordecillion 641 tredecillion 748 duodecillion 464 undecillion 297 decillion 615 nonillion 937 octillion 576 septillion 404 sextillion 566 quintillion 24 quadrillion 103 trillion 44 billion 751 million 294 thousand 464. Holy fuck!

So obviously we can’t write them all down. What about if we just tried to count them one-by-one, one each second? We couldn’t do it alone, but what if all people on Earth helped us? Let’s round up and say there are 10 billion of us. That wouldn’t do it. What if each of those 10 billion people had a computer that could count 10 billion sequences per second instead? Still no. OK, let’s say, for the sake of argument, that there were 10 billion other planets like Earth in the Milky Way and we got all 10 billion people on each of the 10 billion planets to count 10 billion sequences per second. What? Still no? Alright, fine. What if there were 10 billion galaxies each with these 10 billion planets? Not yet? Oh, fuck off.

Even if there were 10 billion universes, each of which had 10 billion galaxies, which in turn had 10 billion habitable planets, which happened to have 10 billion people, all of which had 10 billion computers, which count count 10 billion sequences per second, it would still take 100 times the age of all those universes to count the number of possible sequences in just 266 coin flips. Mind. Fucking. Blown.

Why I am telling you all this? The point I want to get across is that humanity’s knack for pattern finding has given us the false impression that life, nature, the universe, or whatever, is simple. It’s not. It’s really fucking complicated. But like a drunk looking for their keys under the lamp post, we only see the simple things because that’s all we can process. The simple things, however, are the exception, not the rule.

Suppose I give you a problem: simulate the outcome of 266 coin tosses. Do you think you could solve it? Maybe you are thinking, well you just told me that I couldn’t even hope to write down all the possibilities—how the hell could I hope to choose from one of them. Fair. But, then again, you have the coin and 10 minutes to spare. As you solve the problem, you might realize that you are in fact a computer. You took an input, you are performing the steps in an algorithm, and will soon produce an output. You’ve solved the problem.

A problem you definitely could not solve is to simulate 266 coin tosses if the outcome of each toss depended on the outcome of the previous tosses in an arbitrary way, as if the coin had a memory. Now you have to keep track of the possibilities, which we just decided was impossible. Well, not impossible, just really really really time consuming. But all the ways that one toss could depend on previous tosses is yet even more difficult to count—in fact, it’s uncountable. One situation where it is not difficult is the one most familiar to us—when each coin toss is completely independent of all previous and future tosses. This seems like the only obvious situation because it is the only one we are familiar with. But we are only familiar with it because it is one we know how to solve.

Life’s complicated in general, but not so if we stay on the narrow paths of simplicity. Computers, deep down in their guts, are making sequences that look like those of coin-flips. Computers work by flipping transistors on and off. But your computer will never produce every possible sequence of bits. It stays on the simple path, or crashes. There is nothing innately special about your computer which forces it to do this. We never would have built computers that couldn’t solve problems quickly. So computers only work at solving problems that can we found can be solved because we are at the steering wheel forcing them to the problems which appear effortless.

In quantum computing it is no different. It can be in general very complicated. But we look for problems that are solvable, like flipping quantum coins. We are quantum drunks under the lamp post—we are only looking at stuff that we can shine photons on. A quantum computer will not be an all-powerful device that solves all possible problems by controlling more parameters than there are particles in the universe. It will only solve the problems we design it to solve, because those are the problems that can be solved with limited resources.

We don’t have to track (and “keep under control”) all the possibilities, as Dyakonov suggests, just as your digital computer does not need to track all its possible configurations. So next time someone tells you that quantum computing is complicated because there are so many possibilities involved, remind them that all of nature is complicated—the success of science is finding the patches of simplicity. In quantum computing, we know which path to take. It’s still full of debris and we are smelling flowers and picking the strawberries along the way, so it will take some time—but we’ll get there.

## The minimal effort explanation of quantum computing

Quantum computing is really complicated, right? Far more complicated than conventional computing, surely. But, wait. Do I even understand how my laptop works? Probably not. I don’t even understand how a doorknob works. I mean, I can use a doorknob. But don’t ask me to design one, or even draw a picture of the inner mechanism.

We have this illusion (it has the technical name in the illusion of explanatory depth) that we understand things we know how to use. We don’t. Think about it. Do you know how a toilet works? A freezer? A goddamn doorknob? If you think you do, try to explain it. Try to explain how you would build it. Use pictures if you like. Change your mind about understanding it yet?

We don’t use quantum computers so we don’t have the illusion we understand how they work. This has two side effects: (1) we think conventional computing is generally well-understood or needs no explanation, and (2) we accept the idea that quantum computing is hard to explain. This, in turn, causes us to try way too hard at explaining it.

Perhaps by now you are thinking maybe I don’t know how my own computer works. Don’t worry, I googled it for you. This was the first hit.

Imagine if a computer were a person. Suppose you have a friend who’s really good at math. She is so good that everyone she knows posts their math problems to her. Each morning, she goes to her letterbox and finds a pile of new math problems waiting for her attention. She piles them up on her desk until she gets around to looking at them. Each afternoon, she takes a letter off the top of the pile, studies the problem, works out the solution, and scribbles the answer on the back. She puts this in an envelope addressed to the person who sent her the original problem and sticks it in her out tray, ready to post. Then she moves to the next letter in the pile. You can see that your friend is working just like a computer. Her letterbox is her input; the pile on her desk is her memory; her brain is the processor that works out the solutions to the problems; and the out tray on her desk is her output.

That’s all. That’s the basic first layer understanding of how this device you use everyday works. Now google “how does a quantum computer work” and you are met right out of the gate with an explanation of theoretical computer science, Moore’s law, the physical limits of simulation, and so on. And we haven’t even gotten to the quantum part yet. There we find qubits and parallel universes, spooky action at a distance, exponential growth, and, wow, holy shit, no wonder people are confused.

What is going on here? Why do we try so hard to explain every detail of quantum physics as if it is the only path to understanding quantum computation? I don’t know the answer to that question. Maybe we should ask a sociologist. But let me try something else. Let’s answer the question how does a quantum computer work at the same level as the answer above to how does a computer work. Here we go.

How does a quantum computer work?

Imagine if a quantum computer were a person. Suppose you have a friend who’s really good at developing film. She is so good that everyone she knows posts their undeveloped photos to her. Each morning, she goes to her letterbox and finds a pile of new film waiting for her attention. She piles them up on her desk until she gets around to looking at them. Each afternoon, she takes a photo off the top of the pile, enters a dark room where she works at her perfected craft of film development. She returns with the developed photo and puts this in an envelope addressed to the person who sent her the original film and sticks it in her out tray, ready to post. Then she moves to the next photo in the pile. You can’t watch your friend developing the photos because the light would spoil the process. Your friend is working just like a quantum computer. Her letterbox is her input; the pile on her desk is her classical memory; while the film is with her in the dark room it is her quantum memory; her brain and hands are the quantum processor that develops the film; and the out tray on her desk is her output.