Braids & Anyonic Topological Computing

0:00 Got it? Okay. I'm just telling you, I think it's both. Alright, so, let me try to summarize and now, so you get a sense of what I'm going to do here. So, form processes are mediated by unitary transformations, and what I'm going to show you is one way to see how representations of the braid group can be used to make unitary transformations. So that's a mathematician's theorem, that unitary matrices can be kind of uniformly represented by braid. That is, that there are matrices which respect the properties of topological braids that are rich enough so that you can build lots and lots of unitary transformations from them. So that means that, so to a mathematician who imagines that braids are the real thing, then it means that quantum mechanics must have something to do with topology. But it doesn't really mean that. It's just a mathematical theorem about unitary transformations. And it might mean that if there were ways to have braiding kind of things happening in physics. And there are some speculations about that, which we'll see from my little slide show. And then the way you make this is kind of interesting from the point of view of thinking about discrimination and hierarchy. So we'll get to that. But that's sort of the outline, all right? So let's see. I remind you of this, that quantum mechanics in a nutshell is that the state of a physical system corresponds to some vector in a complex vector space, that physical processes in the absence of measurement are modeled by unitary transformations. and that if you if we're finite dimensional which I will be and I have a measurement basis of E1 through E1 those are the possible states that could be measured then a measurement means that one of them shows up and the others go away and it shows up with probability the absolute square of its complex coefficient

2:30 and if someone had invented such a probability theory And then you'll notice that unitary transformations would be exactly what would preserve the probability since unitary means that the sum of the squares will be constant. So if the sum of the squares is one, right. So that's quantum mechanics in a nutshell. And then if we were observing something, oh, this was just for fun. and it's actually a digression. What would a quantum knot be? Right? What's a quantum knot? Well, I guess a quantum knot, so I make a definition of a quantum knot that's sufficiently general so it might fill in to be useful someday. Namely, a quantum knot is some kind of blob which is a collection of possibilities of actual knots. And just like a quantum state is a collection of possibilities of actual states. And so when you observe it, I didn't draw the two different knots that this quantum knot is. I don't know whether this one is K or K prime, but some specific knot happened as a result. Just for the sake of reminiscing, this is where I got interested in this sort of thing. I was visiting an institute in France back in the early 80s and and I was looking for a place to eat and I ran into this physicist Mario Rosetti and we went off to to a crepe place and sat down and I said what do you do and he said I'm studying quantum knots and I said what and he said I'm studying I'm nodded quantum states in liquid helium. And I said, what? And he said, yes, you see, we have this theory in which we observe something like a vortex in liquid helium, and the observables give us enough information to see it as a nod. So something like that looked more complicated. And there is a theory due to Setti and Regi, about that subject. And their observables had to do with knot polynomials that would tell them about the knot and so on. Very interesting ideas, which I think have not been carried that much farther since the early 80s when they were working on it.

5:00 So there's something like the quantum knot idea that's in back of this. But what I'm going to tell you is not really about quantum knots, but rather about waiting and so on. But this is really the idea in the back of my mind about it that's still bothering me about that. Because I ended up visiting him and we talked about everything but that when I finally visited him. So we never got back to this. So, getting back to computing, I just reminded you that in very simple operations like this on two qubits, right, a qubit is a linear combination of a zero and a one, of course, and when you observe it, it's either zero or one, so that's why it's called a qubit. It's a quantum bit, and this is two quantum bits, and of course there are four possibilities, zero, zero, zero, one, one, zero, and one, one. You have a linear combination of them. And this operator operates on the individual ones by leaving it alone if the left bit is zero and changing it if the left bit is one. So it's a knot, but it's controlled by the first bit. It's a knot if the first bit is a one, and it's not a knot. the first bit is a zero. So this isn't a quantum knot, it's a disquantum knot. And it's unitary. Okay. And so that's the sort of typical ingredient of a quantum computer. You want to have some quantum processes, i.e. unitary transformations, that you could put together in flexible ways to make computations happen. Where computation means that you would have this concatenation and by some miracle, they could all happen without being disturbed and then you would measure at the end, having prepared a state at the beginning. And people are working hard at trying to make such things happen and you hear reports, maybe it's two years now, that they managed to factor the number 15 with a very small number of qubit, one MMR quantum computer. But I don't think things have progressed remarkably since then. The problem is, of course, that if you try to set up a quantum process, it's susceptible to being observed by its environment.

7:30 And so you're in trouble unless you configure something very good out. I guess I mentioned this the other day, but I'll mention it again just for fun. And everybody knows this experiment anyway, I suspect. If you think of it as a thought experiment, it's two different kinds of mirrors. a silvered mirror, which can transmit as well as reflect the light which has spin 0 and 1 for our labeling purposes, and an ordinary mirror. Now the ordinary mirror takes a 1 and turns it into a 0, or takes a 0 and turns it into a 1. So if you described it by a matrix, it would be this. The 0 state turns into the 1 state, and the 1 state turns into the 0 state. Right, so it's that. The other one that I've described here follows these rules. that zero coming in will reflect or transmit. When it transmits, it stays the same, but when it reflects, and when it reflects, it stays the same in the sense, I mean, it gets flipped, but there's no phase change. On the other hand, one undergoes a phase change when it transmits and flips when it reflects. So the matrix for this is that it's equally probable, so that means that for zero, what comes out is a linear combination of 0 and 1. And for 1, what comes out is a linear combination of 0 and minus 1. Or 1 in that case, net. And you'll notice this is a unitary matrix also. And the minus 1 was needed in order to make this matrix unitary, all other things being the same. So phase change is important. for the physics to work. There wouldn't be a quantum transformation if there wasn't a phase change in there. If you decided what your mirrors did was they transmitted or reflected, and on transmission, they basically gave you the same state back, and when they reflected, they flipped. Why then, you would need a phase change somewhere in order to make it unitary. So it's amusing to think about that. It's also interesting to think about other things, like the thought experiment that I've described here is a composition of these three matrices from the point of view of the matrices. It's also all the different pathways through the system if you think of it that way.

10:00 And those two ways of thinking about it are just equivalent, exactly equivalent here. And what happens in the end is that there's an interference here, so nothing gets through here. And on the other hand, there's a reinforcement here. So that means that if you started with particle in state zero here, then you get nothing coming out there the zero coming out there. And that's exactly what the product of the matrices is telling you. Okay, so that's an example. And it's also an example of what people are thinking about if they're trying to build a quantum computer, namely you might have various elements that you like and you try concatenating them and draw diagrams, discrete diagrams to concatenate them and so on. H is called the Hanover matrix. And Hadamard matrix turns out to be useful in lots of things. This slide is a little too detailed. I'm going to give you a diagram for it. But Hadamard, that we just had in the previous slide, does this. To zero, it gives you a linear combination of zero and one. And one gives you a linear combination of zero and one with a phase change, just what I had before. And what I'm interested in here is if I had a matrix, could I get at its trace? Which means, really, I want the diagonal elements of the matrix. I'd like their real parts, and I'd like their imaginary parts. So how would you get at that? By a quantum process and a measurement, right? And it turns out that you can, and I'll just show you the diagram process. You take an extra qubit, and the one you're interested in, which you want to apply U to. You apply Hadamard to the extra qubit. And then you use that one to do a controlled U. So if what's coming out is a 1, then U operates, and if what's coming out is a 0, then nothing operates. And then you measure the resulting state out here. So this is happening in the tensor product. And you compute the probability, and it turns out that you get that zero will be measured with the probability, which is given essentially by the real part of that matrix element, the psi-psi matrix element. So a little calculation shows you that. And the subject is full of things like that, where some cute calculation shows you that you can get some information one way or another.

12:30 And you can do the imaginary part as well. that if you had a unitary matrix, you could actually compute its trace by using a quantum computer, if you wanted to. It's interesting to me because it means that you compute certain things that I like to compute. There are many algorithms that are more complicated than that, as you know, like the famous Schor algorithm, which gets you factorizations of numbers. Now, here's an effect which promises to have something to do with those anions. And probably some of you know much more about this than I do. Quantum Halifat works like this. You have a magnetic field and a two-dimensional set up here. And you run a current this way, and then you end up getting current going perpendicularly into the current. And there's a first level of this where the current was measured and shown to go through various levels. They talk about the Hall resistance, but that's, you know, the apparent resistance for the amount of current that you see going in that perpendicular direction. And then there's a subtler effect called the fractional quantum Hall effect where there's finer structure in the levels of the thing. And then it was proposed that the fractional effect could be explained by anionides. That there would be collective excitations of the physics in a two-dimensional space that would have the property that when one of them was moved around another one, the wave function would change by maybe multiplication by a root of unity, something like that. And that theory would explain, can be used to explain the effect in terms of anions. So there's a theoretical level at which explanations of the effect come out of the postulation of these anions. But at the present time, there isn't clear experimental evidence that the anion, the collective excitations actually exist.

15:00 And on the other hand, if they did, then something like this could be described. You see, you would have the fact that if you exchange two of these quasi-particles, then it might multiply by something. And then you would have braiding in exactly the sense of thinking of exchanging particles as something like, as analogous to the evolution of the braid in time. at these in this case in this picture this one is moving back that way but just look at the picture up to this point these two particles got exchanged and some phase happened and then then these two got exchanged and these two got exchanged and the result was a cyclic exchange in this little picture so the braid that we're talking about is a time track of the way the particles moved in this two-dimensional space so that so the brains that I'm going to discuss are are really worldwide in that sense and and so it would be very nice if people actually were able to show that this could be done and even nicer for the people in quantum computing if if such things could be controlled now some Michael Friedman in particular got very interested in this possibility, and so there's some push behind this, and people I think are really trying to find out whether the anions exist in the sense that I'm talking about. Now, there's a kind of a mathematical point about how to think about this that's quite important here, and that is that two particles have to be sort of neighbors in order to think about exchanging them, because in order to understand what it means for them to be exchanged, you have to let them interact to find out what happened. At least that's the story I'm telling you. So if these two interacted and you found out that a certain thing happened, and then if they were exchanged and they interacted and the situation was multiplied by a phase, then you would know that something happened under the interchange. You certainly do not have continuous pathways of motion going around like that. You see, you don't really have that. They're quantum particles, they don't have trajectories in a classical sense.

17:30 So all you know is the end result of the braving if you knew anything at all. I've been thinking much about the basis of the notion of exchange. it's even very difficult not on the one hand to bring in the classical interpretation in which exchange is a process of exchange on the other hand that's not the effect you want it's a new concept it needs new forms but your discrete description here is not what they're invoking in order to produce their exchange forces not adequately I'm not sure what your question asks about it it's more of a statement it's more of a more of a statement I was querying, I mean, you're relating a remarkable exchange to the existence of exchange forces in particle theory. Yeah, it's not so much a matter of forces, but what would it mean to exchange the particles, right? Indeed, yes. So something new is imagined, which gives rise to, which is associated with the word exchange, that people imagine it. I see that, I've just hit a level of my ignorance, right? I'm not sure how I imagine that somebody is going to cause an exchange to happen. No, that is what I'm... If an exchange happened, it would be indicated by the change in faith. Yes, yes, that's fine. I think that is fine. I suppose I'm saying that they are having to imagine that something causes this to happen. I guess. That's the bit of the picture. that somehow somebody has to imagine isolating, isolating a quasi, isolating an electron, or isolating an app, something like isolating it. There's a lot of thought experiment that goes along those lines in the subject. if you imagine you have a cubit in a box and you carry it to New York or

20:00 something you can do that as a thought experiment so I mean I certainly hit my ignorance there in the sense of not understanding what it would really mean to exchange so we'll just have to leave it but there's not but in any case if So if you imagine that exchanges are detected by an interaction, then the interaction would be, well, we don't know about these particles, right? They're maybe producing other kinds of particles when they interact. And then that might interact with this guy and finally get this. So I have some processes that are possible. And I may set up a specific kind of process for the three particles here, right? Just to be specific. So this is always the test that I make. If it's always the test that I make in that order, then if I try to exchange these two and detect, I need to do a slightly different calculation. I really want to do this calculation, or this detection. So if I knew how to calculate what this one is in terms of this one, that would be good. So if I knew what the change of basis was between this process space and that process space. That would be very good. Now, let me go to the blackboard now and show you the simplest way of thinking about this. Because I'm going to be thinking about a single particle which can interact with itself to either produce itself or it can interact with itself to produce another particle. And that's going to be the basis out of which I build the simplest model here. And that's perhaps a bit amusing because the Spencer-Brown particle actually has that property. mechanical but it does have the property that's kind of a single entity which can interact with itself in two different ways it can interact with itself this way and produce nothing or it can interact with itself that way and produce something and this is a description of the properties of discrimination I remind

22:30 to. Namely, there is a distinction which is indicated by the very same thing. And there is an operation of crossing the boundary of the distinction from the inside to the outside or crossing the boundary of the distinction from the outside to the inside. And the inside is regarded as unmarked and the outside is regarded as marked. Alright? And in this altogether too clever notation, when you cross from the unmarked state, you arrive at the marked state. And when you cross from any state, it's written as cross from that state, you see. So you cross from the unmarked state and you end up at the marked state. then you end up at the cross from the marked state. So the cross from the marked state must be unmarked. So this is equal to, and I've written it as stuck within nothing. So that's the Spencer Brown logic about the discrimination. The discrimination in itself, the name of the discrimination from one side to the other are all written by the same thing just different ways of looking at the same thing and and then also the other equation is just that the name of the distinction is itself and so it doesn't need to be mentioned I carry my name tag it goes away so so those of it that's the that's the mark particle which has this property okay that it can interact with itself to either produce itself or nothing right and and we're going to unfold that so let's unfold that a little bit what are kind of process spaces couldn't have. So suppose that I want to look at the first one that we were looking at and let's suppose that we start with particles and we want to end up with particles at the end, right? Then what are the possibilities? Well, it could interact with itself to produce itself and then it can interact with itself to produce itself. That's good.

25:00 The other possibility is that it interacts with itself to produce nothing and then that can interact with that to produce the particle. In fact, that's the only thing it can do because nothing doesn't do anything. All right, so that's all. So this is a two-dimensional space. So I have a space V, particle, particle, particle in and particle out, and the dimension of this space is two. So, this could be the one qubit, and this could be the zero qubit. And then, V is the set of all linear combinations of V's, and that's my one qubit space. Now, the reason for expanding the picture of a qubit to being something like, saying is that the two processes that this can undergo become the two qubits, and then the qubit itself has, each qubit has some internal structure like this. And that internal structure could be subject to gradient, exactly because of the anions in there. So that means that it's possible in the case of the single qubit for the three-strand braid group, because there are three strands here, to be acting on SU2, right? I mean, you could have elements of SU2 that are matrices that represent three-stranded braids, okay? So in thinking about this situation, you could actually get braiding happening all the way down at the level of SU2. the point of this kind of construction. But in order to have it happen right, you would have to have this kind of recoupling transformation telling you what was going on. So we'll come back to that thing in a moment, but let's see. I've illustrated it here. See, if I want to do the braiding between these two guys on the right, then I will change basis over into here. Then I'll do the R,

27:30 which is just, it's just going to multiply by a phase. And then I'll change bases back. And I end up with a mapping from this process space back to itself. And that's going to be a mapping of a two-dimensional complex vector space for itself if it was this model. So it's going to be an element in SU2 or in U2. And that one, the gradient on these two, is F inverse RF, where R is that simpler one which just multiplies by phases. So, you see, it's not just going to be multiplication by phases. One will be just multiplication by phases, and the other one will be something a little more complicated. And that's what the structure will look like. You don't need to see this stuff. I want to now tell you a little bit more about the knot theory that can be used to make these models. So, this slide is just reminding you about knot theory and braids. The knot theory can be done diagrammatically and these are the basic topological moves. I think I bore you if I can hold on to this for very long. Braiding on that, it's very interesting to think about braids because they form a group and the generators of the group are little individual twists like this between adjacent strands. And there are relations in the braid group and they correspond to these done in a more vertical fashion. So they commute when they're far apart and so on. Then there's a non-invariant that I'm going to be using that's in back of this. And it goes like this. Again, I think it's something I've mentioned before. That you calculate it by expanding, by thinking of crossing as a kind of superposition of being smooth one way and smooth the other way. And then you do recursive expansion, turning, slowly turning the diagram into just loops. And then you add up the contributions of the loops, and you get the non-invariant. So it looks like a little partition function on the knock diagram. And there are some nice intricacies about checking that this works right, and so on. It's very elementary, but kind of non-obvious that such a thing would be useful. And it isn't obvious that it's related to SU2 here from the way I'm talking about it.

30:00 And I guess I won't try to talk about that story here. But the main point is that it's a very simple expansion that gets you the non-invariant. You look at the crossing, and when it's oriented this way, then you smooth it this way and you pick up an A and you smooth it that way and you pick up an A inverse and you just keep on doing that to calculate it. It's a generalization of Penrose's spin network ideas from a certain point of view. That business about the smoothing and resmoothing fits together into something a bit more systematic which is quite amusing to look at. If you think of a Dirac ket as a little thing going that way and a bra is going this way, then if you form the bra ket, you get a loop, and that corresponds to a scalar. And if you form the ket bra, then you get something which behaves like a projector, just as the notation does here. Namely, you multiply the ket bra by the ket bra, you get a loop in the middle, and the loop is a scalar, so it multiplies by itself that way. And then you can put them into tensor lines like this. So here's a Ket-Bron, here's an identity tensor line. And here's a Ket-Bron, another identity tensor line like that. And then when you multiply them together, you get this little string in the middle which pulls out. And you get that PQP is equal to P. Now, so the algebraic structure of these things in tensor products like that is called the tempered relief algebra. But what's amusing to me about this, one of the things that's amusing to me about this is that that pattern, the PQP should be P, or multiple of it, is a property of projections anyway in ordinary space. Because suppose that you had, say, a state that you're interested in. Here's one projector, and here's another projector, and the operation that P is, is to, I'll make that short, the operation that is P is to project on P, right? P as a projector is to project on P, so we do P, and then we do Q, and then we do P, and of course you just came back to a multiple of what P did.

32:30 so PQP is a multiple P okay so so that's a fundamental pattern which which shows up as a topological pattern when you when you make it into multiple lines like that it's interesting and very important for this work that it ended up falling into a topological kind of pattern like that now the other part on this slide is maybe better skips. I'll skip it. I don't want to talk about teleportation, so I'll skip this, I'll skip that, I'll skip this. How many sides there is to the skew? What? How many sides there is to the skew? I'm thinking about it. How many sides are there on the queue? Oh, the queue happens to have infinitely many sides. That shows that the state space is actually very large. Although it has finitely many sides, it just has a large number of sides. As large as I like. There must be some way to make it roll in other directions as well. Not as much as you like because you are limited to the . Well, someone pointed out the other day how 1, 1, 1, 1, 1 is equal to negative 1. And it's a good thing to only find a number of ones in there. So now this is what Penrose spin network theory looks like when you make it a bit topological. You have the expansion that corresponds to the not polynomial. But you have some other apparatus like the scalars have set, But never mind exactly how they're evaluated, right? We don't care. But they're evaluated somehow, these loops. But then you have some apparatus that's familiar from either network theory or from group theory. Namely, you have the analog or the sum over all the different permutations of something, okay? Which comes up when you're writing down the structure of representations of a group. And Penrose had this very clever way, originally, of making a network theory that was reflecting the way SU-2 behaved by making symmetrizers.

35:00 And his symmetrizers are simpler than these. They don't have the extra parameter. But this extra parameter means that braiding exists in the theory, whereas it didn't exist in his theory. And then an interaction vertex looks like this. A particle label A and a particle label B come in, come out if the networks fit together right. You have to have A lines here, B lines here. So I plus J is A, J plus K is B, and I plus K has to be C, you see. So it has to fit combinatorially. You have a combinatorial model for particles A and B interacting to produce a C. And if you think about it a little bit, you see that this is a familiar rule that we know I'm going to amend and recoupling theory, namely A plus B plus C is even, right? And A plus B is greater than or equal to C cyclical. So the particle interactions are being modeled by these little boxes of sums over symmetrized lines. That's a common tutorial level of the thing. If you only have two strands, and you did this symmetrization process plus the topology, you add it with two parallel lines minus one over delta times one of those little tetbras done vertically. And then in general, there's some formula for it that's recursive. So then the topological spin network theory is what you make up out of this. two lines, then it turns out that it multiplies by some factor. I guess on this slide, I've indicated one of various exercises that are fun to do. You see, I said that this was the sum over the different symmetrizations, which in this case is parallel and braided, because you don't, you have a braided permutation. And then you expand this line and collect the terms, and it ends up being But once you have this, you can see that it does act like a projector when you're doing things additively. You multiply it by itself and it gives itself back. If you were to tie two lines together at the bottom, this is a good exercise to do with your head. Tie two bottom lines together and you get a loop. But the loop's value is delta. So this is just minus the little guy on the top.

37:30 And that's the same as this and so it's zero. You can tie two lines together at zero, which is like saying that it's anti-symmetric. It's a generalization of anti-symmetric. Where are sums that coming from? I'm sorry. Projectors are sums of permutations. Yeah, these sums are also projectors. You know, of course, this is a projector. But these sums, these guys are projectors. If you multiply two of them together, you get one, you get it back. And if you tie two lines together, you get zero. And that's the key to the calculus of these things when you try figuring out various formulas about things in the networks. So the original idea for this formalism, the fact of pendulous, as I said, And you could do all sorts of interesting things with this. It's also explained in that early book, Quantum Theory and Beyond. It's sort of a cross between Anta, or early Anta. Okay, so now going back over to our process spaces. For psychologists, the process spaces occur Once they've been, or usually have occurred, once they get some clothing, and this is just some cultural background about the relationships, you can put clothing on the grass and then you can get some surface, you see the little lines, a little pair of pants upside down. and then you have like surface here which has been divided into these pairs of tanks and hence it has a graph on it like that and then if you wanted to label the graph with possible particle interactions then it's all closed up but you still have only a certain number of labelings of that graph because they have to satisfy Like, this is A and this is B, then this C is a certain finite number of Cs and so on. So there are some labelings that label the whole graph. And each labeling is a vector, just like each labeling over there was a vector. So you've got a vector space associated with the surface in that way.

40:00 And those vector spaces associated with the surface can then be used to get some invariance of three marifolds done. So that's the way people have thought about it. And so you can go back and forth if you want to between these surface pictures and the graphical pictures. And that's quite useful, but I'm not going to, I'm going to stay with the graphs. And then, for various reasons, these are the good identities, okay, that one wants. One wants, of course, that by something. Now what does that actually mean? It might mean that you sum over different possibilities, but the simplest case is that you just multiply by something and you keep the labels the same. So I'll assume that's what happens there. Then if you have some mixture of nodding and interactions, whatever it means, although it could mean something if you're thinking about two dimensions, right? In two dimensions it makes perfect sense you have a two-dimensional space and this is the time track, right? And so then as you went up the time track you got a single particle split into two and then this one split into two and then these two braided around one another and then these two braided around one another, right? And then the topologist would like it to be the case that the diagrams would behave nicely like that. You can just imagine pulling that down, okay? So that's a good identity to have. Well motivated by those top logical considerations that I was telling you about. In any case, you need these changes of basis. And then there are some combinatorial things that are just natural. This one particularly has to do with non-associativity. What I've done here is I'm thinking of A times B times C and then times D, right? And this is A times B times C times D, if you like, right? And these are different ways of associating things. And this is one little recoupling transformation from one to the other. Now, if I could do another one to get over to here, I could then do another one to get over to here. I can go this way to get these, and I can do one more to come back. And it goes around in a cycle like that,

42:30 or if you like, three forward, one, two, three here, or two to there, okay? That pattern is always there four things. And it might not be the identity. It might not be the case that if you compose this way twice, it would be the same as composed in this way three times. But in the simplest of worlds, that would be the case. It would be equal. And for topological purposes, it turns out that's a good thing to have. And then there's a mixture of braiding and recoupling that goes around in a Now, if you have all those, then the state spaces associated with the surfaces are independent of the way you divide the surface up into pairs of hands. And that's why topologists got interested in this, because they wanted these spaces to be independent. But the best of anionic theories then, from our point of view, would satisfy all these identities. And then from the point of view of the topology, the identities are easy to get. That's the nice thing. If you use the generalized spin network theory that I just described for you, then all these identities are automatically true just because things are topologically invariant and because of certain identities that are not hardly satisfied about the recovery to begin with. This identity here, for example, is what's called the Biedenhardt-Elliott identity in angular momentum theory. And the fact that the bravings work well with it is just a nice property of what happens when you make it braided. So these things can work. And now I wanted to tell you about this specific model that I think is fun to look at. And we have about 15 more minutes. Is that right? Yeah. So, P is the mark over there, and this is what I said before. And the question is, how could you make a model of something other than at the Spencer-Brown level, of something which interacts with itself to produce nothing or to produce itself? Well, if you shift icons over to the thing that we're looking at, it's right there. Namely, a two-strand box can interact with itself to produce itself, the strands add up correctly, or it can interact with itself to produce nothing.

45:00 Or it can interact with itself to produce a four-strand box, but I want that to be forbidden. Okay. Now, forbidding it actually ends me back up in a special case of the non-invariant that underlies the calculations. It ends up at a root of the ending. and one can find that out by just deciding that this is forbidden and calculating forward actually I'll show you but one can also know a lot about the recoupling theory and see immediately that if you chose the right root of unity the four strand box would disappear but that turns out to be the model you see that if you choose the correct root of unity then this nice icon different from the Spencer Brown icon to produce itself or to produce nothing is the impact of what will make this model, okay? So let's see a little more. This is what it looks like. There you see the two possible interactions I've indicated by dark mind for something and thought of mind for nothing. and then the recoupling transformations I've written in a slightly different way but it's just the same thing, right? That is I guess we better go to the strict fact to see that it's the same thing. I was writing that there ought to be a transformation from here to here, right? But if you look at that but that's really the same as saying that there should be a transformation from here to here if we were to bend something in the right direction. I guess we bend that down. All right. I mean, what I mean is, I will bend the wrong thing if I'm not careful, So let me not bend in angle, just indicate the middle line, right? Here's the middle line. Here's the middle line.

47:30 So you see, if you... Could you imagine picking this up and putting it over here? It's the same as picking this up and putting it over here. The left are just more in the karate thing than the right people. The left people, they are just more in karate. That's the only difference. Yeah, so this is a slightly more algebraic way of putting it. So then if they interact and produce nothing, and then they interact again to produce something in that way, it's a linear combination of these. And this one is a linear combination of these. And there's some matrix ABCDs. And then in the underlying guts of that thing, you use these projectors and figure out some of the parameters involved. is kind of fun diagrammatics to do these calculations, and you end up finding that the matrix is this in terms of these parameters. You don't need the details, but what's amusing is that, oh, well, you see, there's a symmetry here because if you just turned all these by 90 degrees, just turn them all by 90 degrees, you get the same equation. So that means that if you apply F twice, you should get the identity. so the square of F should be the identity and then you multiply this matrix by itself and you see that that says that the upper left-hand corner is 1 over delta plus 1 over delta squared which would be 1 and so you're getting the golden ratio coming in here and putting it putting that in and then adjusting them off and now this then you end with this and it isn't quite unitary you see it's not quite symmetric it's real so it's unitarity is just the fact that it ought to be symmetric it isn't quite symmetric but if you multiply every vertex in the model by a certain factor and you end up with this nice matrix uh the um inverse of the golden ratio square root square root minus that uh that's a nice little unitary matrix and that's the matrix that does the job of doing the recoupling. And so it works and it's that

50:00 simple in this way of thinking about it. Now what do we have? What we end up having is something that looks like this, just to show you the matrices. You have a recoupling matrix like that. You're at this root of unity and the phase change of the particles is given by this. So then the other grading matrix is the conjugate of this R by that F. You can have some fun seeing if that represents three-strand braids for example. Now it's called the Fibonacci model because of the occurrence of that root but it's also called the Fibonacci model because of the dimensions of these And that's another part of the combinatorics, just the base combinatorics that I'm using. So if you think of, if you forget about just this whole business about the spin networks, and just think about the puzzle of counting how many processes there are, we saw that in this situation there were exactly two, right? So let's see what happens if I add one more. So now I'll start with particle, particle, and particle. And I want to know what, and I want to end up with particle, and I want to know how many different things there are here. Okay, so this could add, this could interact with the particle, and then these two interact with the particle. That's Fetlis. This could interact with the stuff. Let me just, I'll just wipe these sequences off to the side, so that's one possibility. Another possibility would be nothing. And then could we get another nothing? No, you can't because you have to get particle. You can't, it's going to interact with particles, so it can't produce nothing. So you can't have two stars in a row. um and um and in this case that's all there is folks because no no no i'm sorry there's another one um we could also have um yeah this was this was uh determined but the one i i should have discussed first was this one. So that's dimension three. So that means dimension three. But you see what the rule is. The rule is that you have sequences of a particle and star such that you do not have two repetitions of star.

52:30 all sequences of two symbols such that one of them can't be repeated twice. So we might just look at that abstractly to see what happens. So I start with star and particle, and then star can revise the particle. I'm doing it on the right, so that's the only thing that can happen there because I can't repeat the star twice, whereas here I get this and that. And then from here, I'll just do one more. I can do this, or that. This could rise only to this, and this could rise to two, and that's one, two, three, four, five. and the next one is 8 and the next one is 13 and the next one is 21 and the next one is 34 and so on we've got a Fibonacci number of things if you want to convince yourself of that you could say row 0 is equal to star plus mark and then row n plus 1 is equal to So mark rho zero, rho n, plus mark the star mark, rho n minus one. If you think about that a little bit, that generates, and these symbols distribute over the symbols, right? This will recursively generate the lists, okay? and that shows you that it's giving you Fibonacci numbers. However, you can't see. But that means that the dimensions of the process spaces for this little model are Fibonacci numbers. Now, I'm not actually quite sure I understand why the scalars in the theory are also Fibonacci. The basic scalar in the theory is the golden ratio. That has something to do with the fact that the dimensions of the spaces are Fibonacci numbers,

55:00 but I'm not sure why that works. Now, the other thing is that these generate a lot of unitary transformations. The R and the F inverse RF generate enough transformations in SU2, or in U2, rather, so that it's dense in U of 2. So any element in U of 2 can be closely approximated by these bravings. now exactly that leads to the other question of course that's a technical question how do you find out your favorite matrix approximated by these and that's not easy but the fact that they're dense is just can be understood by thinking about an element in U of 2 looks like a rotation around the vector in 3 space with some extra structure and you think about the dance. And the similar arguments apply to the other spaces. So you end up getting enough transformations to generate enough unitary transformations to do quantum computing out of this little data. So I could show you, but I won't because we are out of time, how to generalize this way of making unitary transformations out of the recoupling theory to other roots of unity and so on and other things. But I think it's quite remarkable how this model actually does work and turns out to be the simplest thing you could do with the Brady recoupling theory. And that it somehow, at the epistemological level, is coming out of some considerations about discrimination and about making iconics for the properties of discrimination. That the iconics for the recoupling theory end up coinciding almost exactly with the Spencer Brown iconics is something to think about. just for fun I don't know how to skip forward except by doing this. something but it's probably not a good one. Let's skip forward by using a different method. I wanted to show you what I think is the last one. You can think about how would I compute

57:30 an invariant of a knot using a quantum computer. And this gives you a picture of how you could do that. Namely, think of this little network here, which is a very simple one. It's actually got two prongs here, two prongs there, and then it's got a prong and a prong going down. And I've set the conditions down here to be just zero, so all the labelings are up here. And now I do a braiding on that network, right? See, that's some braid applied to it. That's a unitary transformation in the theory, applying the braid. are purposes. It would have been two, but I had it generalized to arbitrary numbers of strands later on. So this is some braiding applied to this, and it ends up being a linear combination of the basis elements in the process space. Now X's and Y's appear here. And so there's this braiding matrix, and I want to figure out what is the value of the non-invariant, which means that I close the top. And closing the top ends up closing these tops, but a little loop at the end of a line in the recoupling theory causes it to be zero if these are not equal. And so I end up with just a single matrix element in the diagonal, or some element of the diagonal of the matrix. So I don't have to actually even compute a trace. I just end up having to compute a BII element of the matrix, some element of its diagonal. But I showed you earlier that the So therefore, it's possible to compute the invariant of this form of closure of the knot intrinsically by a quantum computer that was computing topologically in this way. So that's interesting from the point of view of topology and from the point of view of quantum algorithms. So I'll stop. Do you think it's possible to simulate a quantum computer inside a digital computer?

1:00:00 To simulate a quantum computer inside a digital computer? Oh, yeah, you could simulate it, but it will run more slowly, presumably, unless somebody figures out something like that. I mean, quantum computer is doing everything in parallel until you measure, and then it's not as powerful as a totally parallel thing because you only get, you know, the answer probabilistically. so anytime I talk about getting an answer I really mean I kept running that thing again and again and again until I got the relative frequencies of things but in fact if you say you're interested in some quantum algorithm and you go out on the web now you probably find with no trouble that people have like Shor's algorithm you could find mathematical programs that simulate it so you can understand some things about it by playing with the digital simulations. Kibanesh numbers appear here, maybe due to some recursivity in temporal algebra, but maybe in many other recursive algebras, Kibanesh numbers, also somehow appear. Can you comment on that? Is it more or less unique that, in many different respects, Stibonacci numbers appear here? Or it could be seen in many other algebras? Well, you don't have to get Stibonacci numbers if you have a different number. It has to do with, somehow what happened here is it had to do with just the utter simplicity of the number of particle outputs, right? we have one particle and two possible outputs and and the counts are coming out of that right in the thing i didn't show you where i have more i have particles and more outputs then i get different i'll get different dimensions so i'll get some recursive formulas for the dimensions but it has to do with uh all the forms definition of for interaction of the particles yeah more or less Well, if you thought that you were going to start with laws of form and apologize, that this is what you might have arrived at, right? In as much as he had a single logical particle.

1:02:30 I'm trying to get a grasp on what maybe this grading how you know what correspondence it might have to a physical is kind of what I mean. Like, if these were quantum vortex states, say, in a superfluid system, I understand the broken connectivity is associated with circulation. So that, in other words, the fact that the space is multiply connected and forms a vortex line is associated with the quantification and all that. Remember that the antionic stuff won't happen if you're in three dimensions because you could have transformed the loop continuously back. But I still could have braided structures like that in a superfluid. They would be geometrically braided rather than temporally braided. Right, right, right. Well, yeah, could be even temporal. And that's exactly what I'm trying to get at. I mean, what do you think the braidedness would be associated with, analogous to... See, broken connectivity is associated with circulation. Braidedness is sort of means that you can't shrink it to a loop without crossing. So, you know, it's the same kind of thing. Like, broken connectivity, you can't shrink a loop to zero because the space is multiply connected. It's multiply braiding, so what do you think, I mean, have you thought, what would be a physical kind of... Well, maybe you have something linked, right? That would be an example of braiding where you have something which is... Well, I could make that vortex line, you know. Vortex line like this. like that so that it would it would require energy for the two lines to come apart because they're they're moving through one another around yeah it's going to be some type of charge some kind because circulation is the equivalent of charge so so so you get quantized circulation and it gives you

1:05:00 charge and superfluid what what basically it's two-dimensional no electrodynamics. So this braidedness would probably be some kind of, you know, new kind of chart, so you'd have these little knots of some kind, attracting and repelling each other in a higher dimensional, you know, maybe they'd be the color, you know, those kinds of things, more complicated types of chart. So we leave your ruminations for you and Lou right away. Okay. Thanks very much. Two brief things. John Brake, could you just raise your hand? Sure. John prepared some work, but unfortunately he wasn't at night to be here when he should have been here on Saturday. And as you know, when you prepare the work, it's very, very desirable that you actually get it out and always get bad inside. And we actually don't have time now, but if anybody wants to stay behind, then what he says is, but he wants to discuss it with a few people. However, there is a council meeting now, and we're already on a bit late. So, if people stay behind, you get a chance to speak about, I'm sorry. It's a pleasure to come. It's about five minutes. Right. Secondly, Keith, if President Day, do you want to say a few closing words to this whole meeting? Ah, yeah. I'm willing to formally close the meeting and before I do so, to thank all who have contributed particularly the speakers have been some excellent talks and um the chairman of the meeting so often people forget how much work there is one of the persons and peter roland Welcome to Keith in that, and I think this is one of the best meetings for a long time. But you're always saying that. No, no, no. I'm not going to leave the class here. I'm not going to leave the class here.

1:07:30 Thank you.