I wrote a book a while back called Quantum Entanglement for Babies. But, now all those babies are grown into toddlers! I’ve been asked what is next on their journey to quantum enlightenment. Surely they have iPads now and know how to scroll, and so I give you Quantum Entanglement for Toddlers, the infographic!
Below is a lower-res version. Here is a high-res version (5MB). Contact me for the SVG.
Desiree Vogt-Lee maintains a list of quantum computing resources called Awesome Quantum Computing. It is indeed awesome and comprehensive. Here I am looking to answer the question where do I start with quantum computing? with a more concise list of my current favourite entry points.
But, before we get started, a general piece of advice if you want to study quantum computing (or anything else for that matter): learn more maths. More? Yes. More. It doesn’t matter how much you already know. In fact, I’m going to go learn some more maths after writing out this list. (I’m not joking — the next tab in my browser is Agent-based model – Wikipedia.)
Now — in order of some sense of difficulty — here are my favourite recommendations for starting points on learning quantum computing.
The academically minded might be looking for a more traditional approach. Don’t worry. Got that covered by Quantum Computer Programming, a course lectured at Stanford University. Other standard lecture notes include those by David Mermin and John Preskill. The former is more computer-sciencey while the latter is more physicsy.
If you want to do some real quantum programming, The Quantum Katas by Microsoft Quantum is a set of tutorials on quantum programming using the Q# programming language. While it does start with the basics, there is a steep learning curve for those without a background in programming.
Quantum computing for the very curious by Andy Matuschak and Michael Nielsen is like an electronic textbook with exercises that use spaced repetition to assist in remembering key facts. This is an experimental learning tool, which at the time of writing, is still under construction.
“Thinking Quantum”: Lectures on Quantum Theory by Barak Shoshany is a set of about 16 hours worth of lecture notes which was delivered to highschool students at an international summer school. Though it is more focused on quantum physics, the first half will give you all the basic tools needed to start analysing quantum algorithms. It is quite mathematical so the reader would have to be comfortable with some mathematical abstraction. However, much of the field of quantum computing comes from a physics background and the ideas and language of quantum physics are pervasive.
The Quantum Quest by members of the QuSoft team is a web class which contains videos, lecture notes, and a pared down version of Quirk. It starts with the basics of probability and linear algebra and quickly gets you up and running with quantum circuits and algorithms.
Quirk by Craig Gidney is a quantum circuit simulator. It is incredible expressive and provides many useful visualisations. This tools is simple enough for anyone to start creating quantum circuits. However, interpreting the output does require some guidance and knowledge of probability.
If you prefer the video playlist approach, Quantum computing for the determined by Michael Nielsen is a series of short YouTube videos going over the basics of quantum information. However, if you are not putting pen to paper yourself, you are not likely to absorb the necessary mathematics to understand quantum computing.
Quantum Computing for Babies by me and whurley.
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.
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.
By now you have read many articles on quantum computing. Congratulations. You know nothing about quantum computing.
There is a magician on stage. It’s tense. Maybe it’s a primetime TV show and the production value is super high. The celebrity judges look nervous. There is epic build up music as the magician calls their assistant on stage. The assistant climbs into a box that is covered with a velvet blanket. Why a blanket? I mean, isn’t the box good enough? What a pretentious as… forget it, I’m ruining this for myself. OK, so the assistant is in the box with their head and legs sticking out. What the fuck? Who made this box, anyway? Damn it, I’m doing it again. Then—oh shit—is that a saw? What’s going to happen with that? Fuck! No! The assistant’s been cut in half! And then the quantum computer outputs the answer. Wait, what? Where did the quantum computer come from? I don’t know—quantum computing is magic like that.
By now you have read many articles on quantum computing. Congratulations. You know nothing about quantum computing. I know what you are thinking: Whoa, Chris, I wasn’t ready for these truth bombs. Take it easy on us. But I see a problem and I just need to fix it. Or, more likely, call the rental agent to fix it.
You probably think that a qubit can represent a 0 and a 1 at the same time. Or, that quantum computing takes advantage of the strange ability of subatomic particles to exist in more than one state at any time. I can hardly fault you for that. After all, we expect Scientific American and WIRED to be fairly reputable sources. And, I’m not cherry picking here—these were the first two hits after the Wikipedia entry on a Google search of “What is quantum computing?” Nearly every popular account of quantum computing has this “0 and 1 at the same time” metaphor.
I say metaphor because it is certainly not literally true that the things involved in quantum computing—those qubits mentioned above—are 0 and 1 at the same time. Why? Well, for starters, 0 and 1 are defined to be mutually exclusive (that means it’s either one OR the other). Logically, 0 is defined as [NOT 1]. Then 0 AND 1 is equal to [NOT 1] AND 1, which is a false statement. “0 and 1 at the same time” just doesn’t make sense, and it’s false anyway. Next.
OK, so what’s the big deal? We all play fast and loose with words. Surely this little… let me stop you right there, because it gets worse. Much worse.
The Scientific American article linked above then deduces that, “This lets qubits conduct vast numbers of calculations at once, massively increasing computing speed and capacity.” That’s a pretty big logical leap, but I’d say it’s a correct one. Let’s break it down. First, if a qubit can be 0 and 1 at the same time then two qubits can be 00 and 01 and 10 and 11 at the same time. And three qubits can be 000 and 001 and 010 and 011 and 100 and 101 and 110 and 111 at the same time. And… well, you get the picture. Like mold on that organic bread you bought, exponential growth!
The number of possible ways to set some number of bits, say n of them, is 2n—a big number. If n = 300, 2300 is more than the number of atoms in the universe! Think about that. Flip a coin just 300 times and the number of possible ways they could land is unfathomable. And 300 qubits could be all of them at the same time. If you believe that, then it is easy to believe that quantum computers will just calculate every possible solution to your problem at once and pick the right answer. That would be magic. Alas, this is not how quantum computers work.
Lesson 1: don’t take a bad metaphor and draw your own simplistic conclusions from it.
Try this one out from Forbes: “A bit can be at either of the two poles of the sphere, but a qubit can exist at any point on the sphere.” Spot on. This is 100% accurate. But, wait! “So, this means that a computer using qubits can store an enormous amount of information and uses less energy doing so than a classical computer.” The fuck? No. In fact, a qubit cannot be used to store and retrieve more than 1 bit of data. Again, magic, but not how quantum computers work.
Lesson 2: don’t reduce an entire field to one idea and draw your own simplistic conclusions from it.
I can just imagine what you are thinking right now. OK hotshot, how would you explain quantum computing? I’m glad you asked. After bashing a bad analogy, I’m going to use another, better analogy. I like analogies—they are my favorite method of learning. Teaching by analogy is kind of like being in two places at the same time.
Alright, I’m going to tell you the correct analogy between quantum physics and magic. Let’s think about what a magic trick looks like abstractly. The magician, who is highly trained, spends a huge amount of time choreographing a mechanism which is then hidden from the audience. The show begins, the “magic” happens, and we are returned to reality with bafflement. If you are under 20, then you also take a selfie for the Insta #fuckyeahmagic.
Now here is what happens in a quantum computation. A quantum engineer, who is highly trained, spends a huge amount of time choreographing a mechanism which is then hidden from the audience. The show begins, quantum computation happens, and we are returned the answer to our problem. Tada! Quantum computation is magic. Selfie, Insta, #fuckyeahquantum.
Let’s dig into this a bit deeper, though. Why not uncover the quantum computer—open the box—to reveal the mechanism? Well, we can’t. If we “watch” the computation happen, we expose the quantum computer to an environment and this will break the computation. The kind of things a quantum computer needs to do requires complete isolation from the environment. Just like a magician’s trick, if we reveal the mechanism, the magic doesn’t happen.
OK, fine. The “magic” will be lost, but at least I could understand the mechanism, right? Sure, that’s right. But here’s the catch: a magician spends countless hours training and preparing for the trick. Knowing the mechanism doesn’t help you understand how to actually perform the trick. Nor does seeing that the mechanism of quantum computing is some complicated math actually help you understand how it works. And don’t over simplify it—we already know that doesn’t work.
Let’s look at the example of a sword swallowing illusionist. If you don’t know what I’m talking about, it’s exactly how it sounds—a person puts a sword the length of their torso in their mouth down to the handle. How one figures out they have a proclivity for this talent, I don’t want to know. But what’s the explanation? Don’t worry, I already googled it for you, and it’s simple: “the illusionist positions their head up so that his throat and stomach make a straight line.” Oh, is that it? I’m suddenly unimpressed. So now that you too know how to swallow a sword are you going to go and do it? I fucking doubt it. That would be stupid—about as stupid as reading a few sentence description of some “explanation” of quantum computing and then declaring you understand it.
Lesson 3: don’t place your analogy at the level of explanation—place it at the level of the phenomenon. Let your analogy do the work of explanation for you.
If you like figures, I have prepared a lovely summary for you.
Well there you go. Quantum computing isn’t magic, but it can put on a good show. You can learn about how to do the tricks yourself and even perform a few with a little more effort. I suggest starting with the IBM Quantum Experience. Or, start where the real magicians do with Quantum Computing for Babies 😂
Welcome to ⟨B|raket|S⟩! The object is to close brakets, the tools of the quantum mechanic!
Created by Me, Chris Ferrie!
2 PLAYERS | AGES 10+ | 15 MINUTES
Welcome to ⟨B|raket|S⟩! The object is to close brakets, the tools of the quantum mechanic! You’ll need to create these quantum brakets to maximize your probability of winning. But, just like quantum physics, there is no complete certainty of the winner until the measurement is made!
No knowledge of quantum mechanics is require to play the game, but you will learn the calculus of the quantum as you play. Later in the rules, you’ll find out how your moves line up with the laws of quantum physics.
A deck of ⟨B|raket|S⟩ cards, a coin, and a way to keep score.
The instructions are here.
I suggest getting the cards printed professionally. All the cards images are in the cards folder. I printed the cards pictured above in Canada using https://printerstudio.ca. However, they also have a worldwide site (https://printerstudio.com).
You can print your own cards using a desktop printer with this file.
You can laser cut your own pieces using this file.
Oh, and this game is free and open source. You can find out more at the GitHub repository: https://github.com/csferrie/Brakets/.
Two new papers were recently posted on the arXiv with my first two official PhD students since becoming a faculty member! The earlier paper is titled Efficient online quantum state estimation using a matrix-exponentiated gradient method by Akram Youssry and the more recent paper is Minimax quantum state estimation under Bregman divergence by Maria Quadeer. Both papers are co-authored by Marco Tomamichel and are on the topic of quantum tomography. If you want an expert’s summary of each, look no further than the abstracts. Here, I want to give a slightly more popular summary of the work.
Efficient online quantum state estimation using a matrix-exponentiated gradient method
This work is about a practical algorithm for online quantum tomography. Let’s unpack that. First, the term work. Akram did most of that. Algorithm can be understood to be synonymous with method or approach. It’s just a way, among many possibilities, to do a thing. The thing is called quantum tomography. It’s online because it works on-the-fly as opposed to after-the-fact.
Quantum tomography refers to the problem of assigning a description to physical system that is consistent with the laws of quantum physics. The context of the problem is one of data analysis. It is assumed that experiments on this to-be-determine physical system will be made and the results of measurements are all that will be available. From those measurement results, one needs to assign a mathematical object to the physical system, called the quantum state. So, another phrase for quantum tomography is quantum state estimation.
The laws of quantum physics are painfully abstract and tricky to deal with. Usually, then, quantum state estimation proceeds in two steps: first, get a crude idea of what’s going on, and then find something nearby which satisfies the quantum constraints. The new method we propose automatically satisfies the quantum constraints and is thus more efficient. Akram proved this and performed many simulations of the algorithm doing its thing.
Minimax quantum state estimation under Bregman divergence
This work is more theoretical. You might call it mathematical quantum statistics… quantum mathematical statistics? It doesn’t yet have a name. Anyway, it definitely has those three things in it. The topic is quantum tomography again, but the focus is different. Whereas for the above paper the problem was to devise an algorithm that works fast, the goal here was to understand what the best algorithm can achieve (independent of how fast it might be).
Work along these lines in the past considered a single figure of merit, the thing the defines what “best” means. In this work Maria looked at general figures of merit called Bregman divergences. She proved several theorems about the optimal algorithm and the optimal measurement strategy. For the smallest quantum system, a qubit, a complete answer was worked out in concrete detail.
Both Maria and Akram are presenting their work next week at AQIS 2018 in Nagoya, Japan.
