academia – Chris Ferrie

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 point of physics

Something I lost sight of for a long time is the reason I study physics, or the reason I started studying it anyway. I got into it for no reason other than it was an exciting application of mathematics. I was in awe, not of science, but of the power of mathematics.

Now there are competing pressures. Sometimes I find myself “doing physics” for reasons that can only best be seen as practical. Fine—I’m a pragmatic person after all. But practicality here is often relative to a set of arbitrarily imposed constraints, such as requiring a CV full of publications in journals with the highest rank in order to be a good academic boi.

You may say that’s life. We all start with naive enthusiasm and end up doing monotonous things we don’t enjoy. But then we tell ourselves, and each other, lies about it being in service of some higher purpose. Scientists see it stated so often that they start to repeat it, and even start to believe it. I know I’ve written and repeated thoughtless platitudes about science many times. It’s almost necessary to convince yourself of these myths as you struggle through your school or your job. Why am I doing this, you wonder, because it certainly doesn’t feel rewarding in those moments.

On the other hand, many people are comfortable decoupling their passion from their job. Do the job to earn money which funds your true passions. Not all passions provide the immediate monetary returns one needs to live a comfortable life after all. So you can study science to learn the skills that someone will pay you to employ. There are many purely practical reasons to study physics, for example, which have nothing to do with answering to some higher calling. This certainly seems more honest than having to lie to yourself when expectations fail.

(I should point out that if you are one of those people currently struggling through graduate school, academia is not the only way—maybe not even the best way—to sate your hunger for knowledge, or just solve cool maths problems.)

A lot of scientists, teachers, and university recruiters get this wrong. There is a huge difference between being curious about nature and reality and suggesting it is morally good to devote one’s life to playing a small part in answering specific questions about such.

Einstein did not develop general relativity to usher in a new era of gravitational wave astronomy, as cool as that is. He did it because he was obsessed with answering his own questions driven by his insatiable imagination. Even the roots of the now enormous collaboration of scientists which detected gravitational waves started in a water cooler conversation among a few physicists, which is best summarized by this tweet:

Your scientists were so preoccupied with whether or not they could, they didn't stop to think if they should… https://t.co/oBKYZ9Ilsh

— Jeff Goldblum (@jeffreygoldbIum) March 9, 2019

In other words, we don’t actually do things through a consensual agreement about its potential value to a higher power called science. We think about doing certain things because we are curious, because we want to see what will happen, or because we can.

Like all other myths scientists and their adoring followers like to deride, science as a moral imperative is just that—a myth. Might we not get further with honesty, by telling ourselves and others that we are just people—people trying to do cool shit. The great things will come as they always have, emerging from complex interactions—not by everyone collectively following a blinding light at the end of tunnel, but by lighting the tunnel itself with millions of unique candles.

New papers dance!

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.

Quantum computing worst case scenario: we are Lovelace and Babbage

As we approach the peak of the second hype cycle of quantum computing, I thought it might be useful to consider the possible analogies to other technological timelines of the past. Here are three.

Considered Realism

We look most like Lovelace and Babbage, historical figures before their time. That is, many conceptual, technological, and societal shifts need to happen before—hundreds of years from now—future scientists say “hey, they were on to something”.

Charles Babbage is often described as the “father of the computer” and he is credited with inventing the digital computer. You might be forgiven, then, if you thought he actually built one. The Analytical Engine, Babbage’s proposed general purpose computer, was never built. Ada Lovelace is credited with creating the first computer program. But, again, the computer didn’t exist. So the program is not what you are currently imagining—probably, like, Microsoft Excel, but with parchment?

By the time computing began in earnest, Lovelace and Babbage were essentially forgotten. Eventually, historians restored them to their former glory—and rightfully so as they were indeed visionaries. Lovelace anticipated many ideas in theoretical computer science. However, the academic atmosphere at the time lacked the language and understanding to appreciate it.

Perhaps the same is true of quantum computation? After all, we love to tout the mystery of it all. Does this point to a lack of understanding comparable to that in computing 200 years ago?

This I see as the worst case scenario for quantum computation. We are missing several conceptual—and possibly societal—ideas to articulate this thing which obviously has merit. Eventually, humanity will have a quantum computer. But, will that future civilisation look at us as their contemporaries or a bunch of idiots mostly interested in killing each other while a few of our social elite played with ideas of quantum information?

Cautious Optimism

We are on the cusp off a quantum AI winter. We’re in for a long calm before any storm.

This is probably where most academic quantum scientists sit. We’ve seen 10-year roadmaps, 20-year roadmaps, even 50-year roadmaps. The truth is that every “scalable” proposal for quantum technology contains a little magic. We really don’t know what secret sauce is going to suddenly scale us up to a quantum computer.

On the other hand, very very few scientists believe quantum computing to be impossible—it’s going to happen eventually. At the same time, most would also not bet their own money on it happening any time soon. And, if most scientists are correct, the hype doesn’t match reality and we’re headed for a crash—a crash in funding, a crash in interest, and—worst of all—a crash in trust.

Some would argue that there are too many basic science questions unanswered before we harness the full potential of this theory that even its practitioners continue to call strange, weird, and counterintuitive. The science will march on anyway, though. Memes with truth and merit have a habit of slow and steady longevity. The ideas will evolve and—much like AI—eventually become mainstream, probably in our lifetime.

Unabated Opportunism

We will follow the same steady forward march that digital computers did the past 50 years.

If you are involved with a start-up company with an awkwardly placed “Q” in its name, this is where you sit. You believe our current devices are “quantum ENIAC machines”. Following the historical trajectory of classical computers, we just need some competitive players making a steady stream of breakthroughs and—voila!—quantum iPads for your alchemy simulations in no time. Along the way, we will reap continuing benefits from the spin-offs of quantum tech.

This is the quantum tech party line: quantum supremacy (yep, that’s a term of art now) is near. We are on the precipice of a technological—no, societal—revolution. It’s a new space race with equally high stakes. Get your Series A while the gettin’s good.

Like it or not, this is the best case scenario for the field. Scientists like to argue about what the “true” resource for quantum computation is. Turns out, it was money all along. Perhaps the hype will create a self-fulfilling prophecy that draws the the hobbyists and tinkerers that fueled much of the digital revolution. Can we engineer such a situation? I think we better find that out sooner rather than later.

Estimation… with quantum technology… using machine learning… on the blockchain

A snarky academic joke which might actually be interesting (but still a snarky joke).

Abstract

A device verification protocol using quantum technology, machine learning, and blockchain is outlined. The self-learning protocol, SKYNET, uses quantum resources to adaptively come to know itself. The data integrity is guaranteed with blockchain technology using the FelixBlochChain.

Introduction

You may have a problem. Maybe you’re interested in leveraging the new economy to maximize your B2B ROI in the mission-critical logistic sector. Maybe, like some of the administration at an unnamed university, you like to annoy your faculty with bullshit about innovation mindshare in the enterprise market. Or, maybe like me, you’d like to solve the problem of verifying the operation of a physical device. Whatever your problem, you know about the new tech hype: quantum, machine learning, and blockchain. Could one of these solve your problem? Could you really impress your boss by suggesting the use of one of these buzzwords? Yes. Yes, you can.

Here I will solve my problem using all the hype. This is the ultimate evolution of disruptive tech. Synergy of quantum and machine learning is already a hot topic¹. But this is all in-the-box. Now maybe you thought I was going outside-the-box to quantum agent-based learning or quantum artificial intelligence—but, no! We go even deeper, looking into the box that was outside the box—the meta-box, as it were. This is where quantum self-learning sits. Self-learning is protocol wherein the quantum device itself comes to learn its own description. The protocol is called Self Knowing Yielding Nearly Extremal Targets (SKYNET). If that was hard to follow, it is depicted below.

hypebox — Inside the box is where the low hanging fruit lies—**pip install tensorflow** type stuff. Outside the box is true quantum learning, where a “quantum agent” lives. But even further outside-the-meta-box is this work, *quantum self-learning*—SKYNET.

Blockchain is the technology behind bitcoin² and many internet scams. The core protocol was quickly realised to be applicable beyond digital currency and has been suggested to solve problems in health, logistics, bananas, and more. Here I introduce FelixBlochChain—a data ledger which stores runs of experimental outcomes (transactions) in blocks. The data chain is an immutable database and can easily be delocalised. As a way to solve the data integrity problem, this could be one of the few legitimate, non-scammy uses of blockchain. So, if you want to give me money for that, consider this the whitepaper.

Problem

99probs — Above: the conceptual problem. Below: the problem cast in its purest form using the formalism of quantum mechanics.

The problem is succinctly described above. Naively, it seems we desire a description of an unknown process. A complete description of such a process using traditional means is known as quantum process tomography in the physics community³. However, by applying some higher-order thinking, the envelope can be pushed and a quantum solution can be sought. Quantum process tomography is data-intensive and not scalable afterall.

The solution proposed is shown below. The paradigm shift is a reverse-datafication which breaks through the clutter of the data-overloaded quantum process tomography.

fuckyeahquantum — The proposed quantum-centric approach, called *self-learning*, wherein the device itself learns to know itself. Whoa.

It might seem like performing a measurement of $\{|\psi\rangle\!\langle \psi|, \mathbb I - |\psi\rangle\!\langle \psi|\}$ is the correct choice since this would certainly produce a deterministic outcome when $V = U$ . However, there are many other unitaries which would do the same for a fixed choice of $|\psi\rangle$ . One solution is to turn to repeating the experiment many times with a complete set of input states. However, this gets us nearly back to quantum process tomography—killing any advantage that might have been had with our quantum resource.

Solution

quantumintensifies — Schematic of the self-learning protocol, SKYNET. Notice me, Senpai!

This is addressed by drawing inspiration from ancilla-assisted quantum process tomography⁴. This is depicted above. Now the naive looking measurement, $\{|\mathbb I\rangle\!\langle\mathbb I |, \mathbb I - |\mathbb I\rangle\!\langle \mathbb I|\}$ , is a viable choice as

Though $|\langle V | U\rangle|$ is not accessible directly, it can be approximated with the estimator $P(V) = \frac{n}{N}$ , where $N$ is the number of trials and $n$ is the number of successes. Clearly, $\mathbb E[P(V)] = |\langle V | U\rangle|^2.$

Thus, we are left with the following optimisation problem:
$\min_{V} \mathbb E[P(V)] \label{eq:opt},$

subject to $V^\dagger V= \mathbb I$ . This is exactly the type of problem suitable for the gradient-free cousin of stochastic gradient ascent (of deep learning fame), called simultaneous perturbation stochastic approximation⁶. I’ll skip to the conclusion and give you the protocol. Each epoch consists of two experiments and a update rule:

$V_{k+1} = V_{k} + \frac12\alpha_k \beta_k^{-1} (P(V+\beta_k \triangle_k) - P(V-\beta_k \triangle_k))\triangle_k.$

Here $V_0$ is some arbitrary starting unitary (I chose $\mathbb I$ ). The gain sequences $\alpha_k, \beta_k$ are chosen as prescribed by Spall⁶. The main advantage of this protocol is $\triangle_k$ , which is a random direction in unitary-space. Each epoch, a random direction is chosen which guarantees an unbiased estimation of the gradient and avoids all the measurements necessary to estimation the exact gradient. As applied to the estimation of quantum gates, this can be seen as a generalisation of Self-guided quantum tomography⁷ beyond pure quantum states.

To ensure integrity of the data—to make sure I’m not lying, fudging the data, p-hacking, or post-selecting—a blochchain-based solution is implemented. In analogy with the original bitcoin proposal, each experimental datum is a transaction. After a set number of epochs, a block is added to the datachain. Since this is not implemented in a peer-to-peer network, I have the datachain—called FelixBlochChain—tweet the block hashes at @FelixBlochChain. This provides a timestamp and validation that the data taken was that used to produce the final result.

Results

SKYNET finds a description of its own process. Each $N$ is a different number of bits per epoch. The shaded region is the interquartile range over 100 trials using a randomly selected “true” gate. The solid black lines are fits which suggest the expected $1/\sqrt{N}$ performance.

Speaking of final result, it seems SKYNET works quite well, as shown above. There is still much to do—but now that SKYNET is online, maybe that’s the least of our worries. In any case, go download the source⁸ and have fun!

Acknowledgements

The author thanks the quantum technology start-up community for inspiring this work. I probably shouldn’t say this was financially supported by ARC DE170100421.

V. Dunjko and H. J. Briegel, Machine learning and artificial intelligence in the quantum domain, arXiv:1709.02779 (2017). ↩
N. Satoshi, Bitcoin: A peer-to-peer electronic cash system, (2008), bitcoin.org. ↩
I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44, 2455 (1997). ↩
J. B. Altepeter, D. Branning, E. Jerey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Ancilla-assisted quantum process tomography, Phys. Rev. Lett. 90, 193601 (2003). ↩
B. Schumacher, Sending quantum entanglement through noisy channels, arXiv:quant-ph/9604023 (1996). ↩
J. C. Spall, Multivariate stochastic approximation using a simultaneous perturbation gradient approximation, IEEE Transactions on Automatic Control 37, 332 (1992). ↩ ↩
C. Ferrie, Self-guided quantum tomography, Physical Review Letters 113, 190404 (2014). ↩
The source code for this work is available at https://gist.github.com/csferrie/1414515793de359744712c07584c6990. ↩

Why are there so many symbols in math?

“Mathematics is the language of the universe.” — every science popularizer ever

I am a mathematician. In fact, I have a PhD in mathematics. But, I am terrible at arithmetic. Confused? I certainly would have been if a self-proclaimed mathematician told me that 15 years ago.

The answer to this riddle is simple: math is not numbers. Whenever a glimpse of my research is seen by nearly anyone but another mathematician, they ask where the numbers are. It’s just a bunch of gibberish symbols, they say.

IMG_20171212_090428 — The whiteboard in my office on 12 December 2017.

Well, they are right. Without speaking the language, it is just gibberish. But why—why all these symbols?

The symbols are necessary because communicating the ideas requires it. A simple analogy is common human language.

Mandarin Chinese, for example, has many more like-sounding syllables than English. This has led to a great number of visual puns, which have become a large part of Chinese culture. For example, the phrase 福到了(“fortune has arrived”) sounds the same as 福倒了(“fortune is upside down”). Often you will see the character 福 (“fortune”, fú, which you pronounce as ‘foo’ with an ascending pitch) upside down. While, like most puns, this has no literal meaning, it denotes fortune has arrived.

Fudao, CC BY-SA 3.0, https://en.wikipedia.org/w/index.php?curid=30725479

Not laughing? OK, well, not even English jokes are funny when they have to be explained, but you get the idea. This pun just doesn’t translate to English. (Amusingly, there is also no simple common word for pun in Chinese.)

The point here is that upside down 福, with its intended emotional response, is not something you can even convey in English. The same is true in mathematics. Ideas can be explained in long-winded and confusing English sentences, but it is much easier if symbols are used.

And, there really is a sense in which the symbols are necessary. Much like the example of 福, most mathematicians use symbols in a way that is just impossible to translate to English, or any other language, without losing most of the meaning.

Here is a small example. In the picture above you will see p(x|θ). First, why θ? (theta, the eighth letter of the Greek alphabet, by the way). That’s just convention—mathematicians love Greek letters. So, you could replace all the θ’s by another symbol and the meaning wouldn’t change. It’s like the difference between writing Chinese using characters or pinyin: 拼音 = pīnyīn.

You might think that it is weird to mix symbols, such as Roman and Greek, but it now very common in many languages, particularly in online conversations. For example, Chinese write 三Q to mean “thank you”, because 三 is 3 and, in English, 3Q sounds like ‘thank you”. In English, and probably all languages now, emojis are mixed with the usual characters to great effect. You could easily write, “Have a nice day. By the way, my mood is happy and I am trying to convey warmth while saying this.” But, “Have a nice day :)” is much easier, and actually better at conveying the message.

OK, so we are cool with Greek letters now, how about p(x|θ)? That turns out to be easy to translate—it means “the probability of x given θ.” Unfortunately, much like any statement, context is everything. In this case, not even a mathematician could tell you exactly what p(x|θ) means since they have not been told what x or θ means. It like saying “Bob went to place to get thing that she asked for.” An English speaker recognises this as a grammatically correct sentence, but who is “she”, what is the “thing”, and what is the “place”? No one can know without context.

What the English speaker knows is that (probably) a man, named Bob, went to store to purchase something for a woman, whose name we don’t know. The amazing thing is that many more sentences could follow this and an English speaker could easily understand without the context. Have you ever read or listened to a story in which the characters are never named or described? You probably filled in your own context to make the story understandable for you. Maybe that invented context is fluid and changes as you hear more of the story.

The important point is that such actions are not taught. They come from experience—from being immersed in the language and a culture built from it. The same is true in mathematics. A mathematician with experience in probability theory could follow most of what is written on that whiteboard, or at least get the gist of it, without knowing the context. This isn’t something innate or magical—it’s just experience.