Steve Simon on Knots, World-Lines, and Quantum Computation (Full Transcript)

Well, the physicists would look at this and would want to treat time more similarly as space, and the way we do this is we plot space on an axis, and we plot time on another axis. Einstein’s so called “World Line” is this dark red line which tells you where in space is he at any given time. It’s called his World Line because it tells you where in the world is he at any given time.

Now we can go through the day, and keep your eye on the dark red ball. The ball goes up one step every hour as we go through the day. It goes back and forth in space, tracing Einstein’s position. So the world line is just a convenient way of keeping track of where Einstein is at any point in the day.

We can do the same thing with a more complicated world. So here we imagine looking down on Einstein’s neighborhood from a helicopter above. So Einstein starts his day at home, he goes to work, he goes to the cinema, he goes back home. A student on the same day starts at home, goes to the library, goes to the pub and goes home.

Now if we follow them both on the same day, Einstein goes to work, the student goes to the library, Einstein goes to the cinema, the student goes to the pub, Einstein goes home, the student goes home, it starts to look pretty complicated. But we can simplify it by looking at the space-time diagram of what happened. We do that by turning the neighborhood sideways, plotting time vertically and notice I’ve drawn a blue vertical line at the position of every object in the neighborhood that doesn’t move, such as the library or the pub; they stay fixed in space and they move through time.

Einstein and student’s world lines move around in the neighborhood as they go through time. Now you can kind of see where I am going with this. Einstein and the student’s world lines have wrapped around with each other. If you pull those tight, you’ll discover you have it knotted.

We need one more thing from the theory of relativity. We need E = mc². Again, this is a thing that I am not going to explain to you in much depth. But roughly what it means is that energy and mass are the same thing. So if we have a particle in our world like an electron — that’s a particle of matter. Now each particle of matter has an opposite particle of antimatter. In the case of electron, the antimatter particle is called the positron. Both the electron and the positron have mass. If you bring them together, however, they can annihilate each other, giving off their mass as energy, usually as light energy. The process works in reverse just as well. You can put in the energy and get out the mass of the particles.

Now, we are going to do the same thing we did with Einstein’s neighborhood. We are looking down on the neighborhood, we put in energy to create a particle and antiparticle. We put in energy to create a particle and antiparticle. Then maybe we move one particle around another, and we bring them back together, we re-annihilate them, we annihilate them; releasing the energy again.

Now if we’d look at that in the space-time diagram, it looks a little bit like this; time running vertically, we put in the energy, we put in the energy, we wrap one particle around the other, and we annihilate them, and we annihilate them again. And you can see quite clearly here that the world lines have knotted around with each other.

Do the same thing with more particles, by putting in more energy, move them around in some very complicated way and bring them back together. The space-time diagram would look a little bit like this, making a very complicated knot.

Now here’s the amazing fact upon which the rest of my talk relies. Certain particles, called Anyons exist in 2+1 dimensions. Now I should probably say what I mean by 2+1 dimensions. Two dimensions mean we’re talking about a flat surface, so these particles live on flat surfaces. We say +1 dimensions, we mean also time. So we are just saying that the particles on the flat surfaces move around in time. So, these particles called Anyons exist with the properties at the end depend on the space-time knot that their world lines have formed. So you can kind of see now why I am so concerned with whether two knots are the same or whether they are different. We can conduct an experiment by which we create some particles, move them around to form a knot and then they have some property at the end of the knot. Let me do the experiment again, create the particles, make another knot, and I want to know whether the properties of the particles at the end of the knot are the same as the properties of the particles at the end of a different knot. This is why I am concerned with whether the knots are the same or whether they are different.

Now just looking at these two simple knots, it may not be obvious, it sometimes is hard to tell if two knots are the same or whether they are different. Is there some way to unravel one and turn it into the other without cutting? Fortunately mathematicians have been thinking about this problem over 100 years, and they’ve cooked up some important tools to help us distinguish knots from each other. The most important tool is known as a Knot Invariant.

A Knot Invariant is an algorithm that takes as an input a picture of a knot and gives us an output some mathematical quantity: a number, a polynomial, some mathematical expressions, some mathematical symbols. The important thing about a Knot Invariant is that equivalent knots, 2 knots that can be deformed into each other without cutting have to give the same output. So if I have two knots — I don’t know if they are the same or not — I put them into the algorithm, and if they give two different outputs, I know immediately that they can’t be deformed into each other without cutting, they are fundamentally different knots.

Now, in order to show you how these things work, I’m actually going to show you how to calculate a Knot invariant. The problem here is I have to give you a warning that there’s going to be math. Now, I’ve given this talk at high schools before, and nobody died. So, I suspect most people can handle the amount of math that we’re going to do, but I know some people are very math-phobic like the person in that slide. If that’s you, just close your eyes when you get scared, open them up later, and everything will be OK, you won’t miss too much.

So, the Knot Invariant we are going to consider is known as Kauffman Invariant or the Jones Invariant. We start with a number which we call “A”. A stands for a number in this case. So the first rule of the Kauffman Invariant is if you ever have a loop of a string, a simple loop with nothing going through it, we can replace that loop with the algebraic combination −A² − 1/A². That combination occurs frequently, so we call it “d”.

So anyway the first rule is then, if you ever have a loop of string with nothing going through it, you can replace that loop with just the number “d”.

The second rule is a harder rule. This rule says if you have two strings that cross over each other, you can replace the picture with the two strings crossing over each other with the sum of two pictures. In the first picture, the strings go vertically, in the second picture, the strings go horizontally. The first picture gets a coefficient of A on front, the second picture gets a coefficient of 1/A on front. This may look very puzzling because you’ve replaced a picture with a sum of two pictures and we’ve put numbers in front of those pictures. So now we’re talking about adding pictures together as well as putting numbers in front of our pictures.

But all we’re doing is we’re doing math with pictures. I’ll show you it’s not that hard by actually doing a calculation. What we’re going to do is we’re going to take the rules, and we’re going to apply them to a very simple knot. This very simple knot is a figure 8 looking thing. Well, secretly we know it’s actually just a loop of string, and we folded it over to make it look like a figure 8. But suppose we didn’t know that, suppose we weren’t so clever to figure out that we could just unfold it and make it into a simple loop. We would go ahead and try to calculate the Kauffman knot invariant by following the algorithm.

So what you do is you look at the knot, and you discover the two strings are crossing over each other so I’ve circled that in that red box. Now within that red box, we have two strings crossing over each other so we can apply the rule and replace those two strings crossing over each other with a sum of two pictures. In the first the two strings go vertical, in the second two strings go horizontal. In the first you have a coefficient of A on front, in the second you have a coefficient of 1/A on front.

Now we just fill in the rest of the knot exactly like it is over on the left. So now we replaced one picture by a sum of two pictures with appropriate coefficients. Now in these pictures there are still crossings in the knot down below where I’ve now indicated them in blue, and we have to apply the Kauffman rule to these crossings as well which we do exactly the same way. Now we have a sum of 4 diagrams with appropriate coefficients.

At this point, we’ve gotten rid of all the crossings, and we’re left with only simple loops, and simple loops by the first rule get a value of “d”. So each time we have a loop we replace it by a factor of “d”. So in the first picture, for example, there’re two loops, so we get d², the second picture is just one big loop, a factor of “d” and so forth. At this point, we are now down to only symbols, and no pictures are left, so have “A”s and “d”s, so it’s just some algebra at this point, so you combine together some terms, then we use the definition of “d” being −A² − 1/A² to replace this by −d, we get a d³ canceling a −d³ and at the end of the day we get “d”. Yay!

Why does this get Yay? This is exciting for two reasons: first of all, it’s exciting because it’s the end of the math, the second reason it’s exciting, it’s because of the result giving us “d”. The reason it’s interesting that we get “d” is because at the beginning, what we actually started with was just a simple loop. We folded it over to make it look like a figure 8, but it was a simple loop and the Kauffman invariant of a simple loop is just “d”. Even though we folded it over to make it a lot more complicated when we went through this algorithm at the end of the day we get “d”. That’s how the knot invariants work. We could’ve folded over a hundred times and made it look incredibly complicated but still it would have given us “d”.

So if we have these two knots here and we want to know if they’re the same or different, we put them into the algorithm, and we get out two different algebraic results. These results don’t equal each other, and so we know immediately these two knots are fundamentally different, they cannot be turned into each other without cutting the strands. So if someone gives you this knot, you might say, “Go ahead, follow the algorithm and see what comes out.” Unfortunately, Kauffman invariants are exponentially hard to calculate. What do I mean by that? Well, in this picture here, we had 2 crossings, we ended up with 4 diagrams. Each time we had to evaluate a crossing, we doubled the number of diagrams. If we had had 3 crossings, we would have 8 diagrams, 4 crossings, we would have had 16 diagrams, and so forth and so on.

In this knot, we have about 100 crossings which would be 2¹⁰⁰ diagrams, and that number is so enormous that the world’s largest computer would take over 100 years to be able to evaluate the Kauffman invariant of this knot. So you might think, if I have a complicated knot, evaluating the Kauffman invariant maybe isn’t that interesting after all.

Well let’s go back to this amazing fact that these particles called Anyons exist where the properties at the end depend on the space-time knot that their world lines have formed. What does that mean? Well precisely the probability that the particles will annihilate at the end of the knot is proportional to the Kauffman invariant of the knot squared.

So if I had these Anyons by measuring whether they annihilate, I can estimate, I can measure the Kauffman invariant of a complicated knot. The way you do it is you produce your Anyons, you move them around to make this complicated knot, and then you try to annihilate, and you see if they annihilate. You do it many many times, so you get a very good estimate of exactly what the probability of annihilation is. So you’ve measured the Kauffman invariant of this knot.

Why is that interesting? Well the reason it’s interesting is because these Kauffman invariants are exponentially hard to calculate. The world’s biggest computer would not be able to calculate the Kauffman invariant of that knot even in 100 years. But these Anyons can do it; these Anyons can solve this exponentially hard problem. Now, that’s kind of an interesting thing to know just sort of fundamentally that these particles have a way of calculating something that our biggest computers still can’t do, but maybe we’re not so interested in calculating the Kauffman invariant of a knot. It turns out that this Anyon computer can do the same calculations as any so-called quantum computer can do.

Now, I’m not going to explain what a quantum computer is, but roughly a quantum computer is a type of computing device that uses the odd properties of quantum mechanics to do calculations that modern computers essentially cannot do at all. This particular type of quantum computer that uses Anyons and knots is known as a topological quantum computer because it uses the topology of the knots to do the computation.

Now, I will give you one short example of the other kind of things that quantum computers can do. Conventional computers, even your mobile phone, are very good at multiplication. If I give a computer or your phone these two very large numbers and I ask it multiply them together, in less than a millisecond it would come back with that super huge number as a result.

On the other hand, if I gave you this super huge number and asked you to find the two factors — find the two numbers which, when multiplied together, would give you this super huge number, it would take about 50 years of computer time. A quantum computer goes forward and backwards in roughly the same amount of time. This is kind of thing quantum computers can do: specific calculations that modern computers are unable to do efficiently, quantum computers can do very well.

There is a catch. The catch is that no one’s built a quantum computer. But this is what people are working on. People like myself and other computer scientists, physicists and mathematicians are quite interested in building these things in the next few years. This is why we’re interested in knots, world lines and quantum computation.

Thank you.

 

Related Posts

• Transcript of Angela Duckworth: Push Those Cell Phones Away
• Transcript of Abraham Verghese’s Harvard Commencement Speech 2025
• Transcript of JD Vance’s Commencement Speech at the U.S. Naval Academy – 5/23/25
• Transcript of This Is What the Future of Media Looks Like: Hamish Mckenzie
• Transcript of Elizabeth Banks’ Commencement Speech At the University of Pennsylvania