2025 QUANTUM PROGRAM ❯ Day 11 ❯ Quantum Simulation and Lie Theory (Dr. Korbinian Kottmann)

0:00 Welcome back to Wiser Quantum 2025 program. Today we have a very exciting session planned for you. In fact, two very exciting sessions planned for you. First, we'll be having Dr. Corbinian Kotman from Zanadu. He is interested in quantum simulation. There's a lot of open-source quantum software and Zadanu is involved in softwaredriven research that feeds directly into research driving penny lane. But yeah, Dr. Caribbean is here today to talk about a very exciting topic. Another fun fact about him which maybe you all should also be aware of, he's an excellent karaoke singer. So if you ever get a chance to meet him at some conferences or at some site visits, definitely catch up with him. Over to you Dr. Caribbean.

0:45 Thank you for that kind introduction and thank you for having me. Today's topic is going to be quantum simulation and le theory. Lee theory has been a topic that is very closely connected to quantum mechanics ever since its origins. If you if you read old high energy physics papers or any high energy physics paper theory plays a big role. Um but it's not been so super common or so widespread in quantum computing or quantum simulation. And that recently has changed again. There there's been some papers that use le theory. And yeah, I I want to give a basically a brief introduction to to le theory and how how it's connected to quantum

1:31 computing, quantum simulation and all that buzz. Um before getting into that, a little advertisement. So I most of the stuff that I'm going to talk about is kind of independent of Penny Lane. Penny Lane is the open- source software that we're developing like Zadu. It's quantum computing software built by researchers for research and a lot of the things that I'm going to talk about you can you can do in penny lane and there's eventually going to be some small code examples or even some bigger what we call demonstrations and those are also the the I mean I think this this is one of the biggest driving factors for people to use penny lane because these are great resources for a to learn about mathematical physical concepts and on

2:16 the other hand learn about coding and directly implement the mathematical and physical contents uh concepts in code. And the nice thing about these demonstrations is that they're continuously updated. So you will you should you should never have the problem that you're trying to run a demo and it's not working because you have the wrong version. As long as you're on the latest version, they should run there. Maybe there are few exceptions, but for I would say like 99% of the demos, this this should actually be the case. Um yeah, so the the outline for today's talk is I'm going to basically introduce what what is leaf theory. And you don't need to be scared if you're if you're somewhat familiar with quantum computing. Most of the objects that I'm going to talk about you

3:01 already know. I'm just going to give them a slightly different name and we're going to do some tricks with them. And then we're going to see some examples in how le theory plays into quantum computing or quantum simulation. So our first kind of yeah we can call it an application is classical simulation also quantum simulation and quantum compilation and yeah what these entail in detail we're going to see in a brief minute also I I talked about this with her if you have any questions if you feel lost at some point during the talk actually feel free to just speak up or ask the question um I'd rather interrupt for a brief second and have everyone on board then I don't know go over something too quickly and and lose half the audience.

3:49 All right, so Le theory 101, all of the contents that I'm going to talk about today, they're they're essentially an accumulation of various demos that I wrote. So I'm going to put the the demo title in front of the sections. Um, so you can if if you want to know more about these things or just kind of revisit what I talked about afterwards, this is where you can find them. pennyai/demos or penny.ai/demonstrations. I/Q/ demonstrations. So le theory 101 in quantum computing we are all familiar with the with gates and with unitary circuits. So these are the the fundamental building blocks of quantum computing unitary circuits and a unitary operator is defined by the fact that the inverse of a unitary operator is just it's a joint.

4:36 This is also why sometimes quantum computing is referred to as reversible computing because all of the quantum circuits that you're that you're doing when they're unitary you can reverse them by just taking their their adjint. On the other hand, typically at the end of a quantum circuit, you want to measure some observable typically encoded in some Hamiltonian. And those Hamiltonians or observables, they are made up of hermission operators. And hermission operators, they're defined by the fact that the adjint of an operator is equal to the operator itself. This has some mathematical implications like the igen values being real and that yeah that is one of the reasons why they're describing observables because all of

5:21 the observable outcomes measurement outcomes are real numbers. Now, if you if you know these mathematical concepts and even mathematical objects, you're already pretty familiar with Lee theory because the unitary matrices and the hermission matrices, they make up the corresponding Lee group and Lee algebra that are relevant for quantum computing and they're both related with each other via the exponential map. You can view it anything that's in the exponent. This is describing the algebra and everything that's on the bottom so to say is making up the group. And you probably already at some point heard about the poly operators. They are in

6:08 some way a bit special because they are both hermission and unitary. So sometimes we we see poly operators appearing in the exponent and then we can we can view them as an as elements of an algebra or we can view them we can see them at the quote unquote bottom and view them as group elements. Now let me tell you what what do I actually what what is actually the group and the algebra that I'm talking about. So it's sometimes when you're in this picture sometimes you it's it's sometimes more easy to reason about a quantum circuit or some concept in quantum computing when talking about the algebra and sometimes it's more

6:54 convenient to talk about it on the group level. Um often it is yeah handy to talk about it on the algebra level. So let me tell you firstand what is the definition. So a lee algebra G. So this fractur G is a vector space equipped with a commutator. And importantly usually when we talk about vector spaces for example Hbert space we talk about wave functions. So really vectors in this case the vector space refers to objects that are made up of operators. So operators themselves can also form vector spaces. Yeah. And the vector space that we're talking about here is the vector space of of linear operators. And sometimes they're unitary and sometimes they're hermission. H sorry no they're

7:41 hermission. And yeah and to make this thing this vector space a le algebra we need the commutator. In principle there this this concept of lee algebra is more general than just with commutators. But for all practical purposes in quantum computing we only care about the case where this is the commutator. And an important property is that when you when you have a Lee algebra, it needs to be closed under the commutator. That means any element from your Lee algebra that you're taking and you take the commutator with another element, you should get a new element that is again in the Lee algebra. So you can never escape the Lee algebra via the commutator. So for example, let's take the poly operators X and Y together.

8:27 they don't make up a Lee algebra because if you take the commutator between X and Y, you get a new operator Z poly operator Z and that one is not in your original algebra. So this these these two two operators alone they do not form a le algebra. But what you can do is you can perform the lee closure and sometimes the lee closure is just an abstract mathematical concept where you say oh I have these operators that I care about and in order to reason about them on le algebraic level I'm just doing the leak closure without ever computing it or you can actually go and compute it. In this case it's it's pretty simple. we can just add the Z operator and and we're done. But in some cases, it's more complicated and we do have software tools to help you do that

9:15 automatically so you don't have to do it by hand. I mean, this case, this example is is pretty straightforward. But imagine you have some larger operators that are a bit more complicated. Um, then you probably don't want to do it by hand. And it's it's neat to have software tools helping us helping us do it. All right. Yeah. So this is this is essentially a lee algebra. It's a bunch of operators typic like it the concepts are more general than that. But for all practical purposes just think of it as poly operators. So poly operators like tensor product with with each other and sums of them. This is these are the le algebbras that we typically care about. They're composed of these elements and in order for them to be actual le

10:00 algebraas we need to close them via the le closure. And yeah, this is pretty much it already. So I think uh poly operators is pretty common in quantum computing. Commut commutators is pretty common in quantum computing. And we're just using the same objects and we're giving it new names and say this is now a lee algebra with this additional property that we're giving it that it's closed under the commutation. All right. So imagine you have some quantum circuits. So each of these boxes is a unitary gate and each of these unitary gates they're generated by some poly operator. So in that case we have like poly operator xx

10:45 uh these numbers. Oh I messed up messed up the indices. Well there should be just xxx on the appropriate wires and the z operators. So yeah, this is a circuit and we can look at the the gates that are are are generating that circuit and we can look at operators in the exponents that generate these gates and we can compute their Lee algebra. Now again as I told you before the Lee group and the Lee algebra they're connected via the exponential map. So the corresponding league group that this circuit is targeting or part of depends on the depth um how complete your mapping is but it's a subset of all of the unitaries in this group G this so

11:32 what I what I want to say is by looking at these generators and computing the leak closure we can tell we can say something about how expressive is this quantum circuit and in that case If we look at those generators, they're forming the so-called special orthogonal group. And this is one of three important semi-imple algebbras that we typically encounter in quantum computing. So you just need to know these three. This is a unitary group. These are just general unitary matrices. There's a orthogonal group or the orthogonal algebra which is in the on the group level it's the it's the matrices that are orthogonal. So kind of a bit more special than unitary with just real matrices. And we have the simplectic which is a a bit more niche I

12:19 would say. So maybe you've maybe you've seen it maybe maybe not. I think don't care about these things too much. And this is just to say by looking at the generators and by looking at the leak closure of these generators we can say something about what group this unitary circuit is mapping to. And yeah, this is a at this point very general statement and um doesn't have too many practical implications yet, but we're going to see some applications for le theory and some a bit more theory while we go there. So first of all, I want to talk about a classical simulation method. Let me give the spoiler already at this point that this simulation method is not too practical in the sense that there's very

13:06 rare very rarely an application where you can actually use this and the scales but it has some important theoretical implications in particular for baron plateaus but this is just a spoiler let let us let us walk there step by step again this is the contents that I'm going to show you now they're basically taken from this demo that I wrote. So if you want to know more details, feel free to check it out. Okay. So again, we're in the setting where we have some operators that are forming a Lee algebra and a very important concept in these Lee algebbras are the so-called structure constants. You can view structure constants as the coefficients

13:51 of decompositions of any possible commutator that you can take. And with that knowledge, the com the structure constants themselves, they're a full and faithful description of the Lee algebbras that we're looking at. Think of it like this. this uh on the left hand side we have some operator and on the right hand side we have some coefficients times again some bases of those operators. So this is just a linear decomposition of your left hand side and you do that for any possible combination of commutators and then you get the so-called structure constants. Now these structure constants they you can compute them by hand by just I don't know yeah basically by doing the linear

14:38 decomposition you just compute all possible commutators and then linearly decompose them and collect the coefficients. Um you can do that by hand of course but you can also do it in code. There's a function qml.structure constants on the le algebra that you're taking. Now these structure constants they inform what this this identity actually doesn't have a name. This is just what I or or we call it. We call it the adjoin identity. And the adjoin identity tells you how any of your operators from your lee algebra evolves if you sandwich it from left to right with u and udagger. And u and ud dagger is again generated by some element of your lee algebra. And when that is the case, you can rewrite

15:25 that in terms of the exponential of of some matrix. And that matrix is again the structure constants. Now this left hand side here, this is what is sometimes called a capital adjint action on the group level. And on the right hand side, we have exponent minus I the adjint action with the lower case on the algebra level. So this is why this is called the adjint action. And this is one way on the right hand side this is how you see it written on Wikipedia and this is sometimes not so handy to use and on the left hand side this is how I don't know you would encounter it in maybe in physics papers like the GSM paper and then you can use it for the technique that I'm about to show you

16:10 now. So importantly kind of take away from this page is there is the there are these structure constants. These structure constants completely describe our lee algebra and we can use that knowledge to inform how operators transform under group elements because that is essentially what is happening in a quantum circuit. Imagine you have some quantum circuit with again gates are generated by these generators that some some operators for all practical purpose think of them again as poly operators. So xx or just a z or xy z could be whatever some some poly operators you take the le closure of them and they make up your le algebra and now what you care about is

16:56 you want to you want to in the end measure again these elements their hmission operators so they're observables you can you can measure them and you can compute the expectation value and this is what you care about now and we keep track of all possible expectation values for all of our lee algebra elements ments and we have some initial state row zero that we know a priority. Typically in quantum computing our initial state is just a zero state. It could be something else. Then we can compute all of these expectation values and we get some number and we put them in a vector and that vector we call E for expectation value vector. Now how does that expectation value vector transform under evolution of

17:42 group elements from the corresponding league group and that is exactly what the adjoin sorry the adjoin identity tells us namely that all of these gates on the left hand side they correspond to some adjoin action. I put the joint identity on the bottom right so we can see them again. And so we see that any of the any of these vectors in in this trace they just bas you essentially just multiply that vector with the adjint ident to the i structure constants which is the adjoin representation on the algebra level. I think that sounded way more complicated than it should have. Uh the

18:28 takeaway here is you have some vector of your expectations and now the way you that vector transforms is simply by some matrix multiplication and that matrix multiplication is e to the i your structure constants. That is that is a takeaway. This is how this expectation vector of your initial values transforms. And now you might want to compute some expectation value of some observable that is a superposition with some coefficients here in the end. And then again this is just a multiplication of yet another vector from the left. This technique is called GSIM or Lee algebra simulation or maybe you find other names for it. And the kind of cool

19:16 thing about it is when we're doing classical simulations of quantum circuits or expectation values in quantum circuits, we typically work in a 2 to the n dimensional Hbert space and we need to keep track of that vector. And this is the reason why quantum computing is is there because we we we can't keep track of a 2 to the n vector for for n lar larger than I don't know 30 40 depending on how big your computational resources are. But this it scales exponentially. So you you're kind of stuck here. Um now GSIM tells us that we can perform all of this in in with matrices and vectors that are of the size of the dimension of your Lee algebra. So you might have a very large Hbert

20:04 space but a small le algebra and then you can perform GSM on it. Again, the spoiler, this is rarely the case, but it does have some interesting theoretical implications because while all of this was developed, at the same time, people figured out that the Baron Plateau problem, so this is kind of I'm I'm shifting gears here a bit. I'm going to to a different topic, but they're related. The baron plateau topic problem is in variational quantum algorithms where it happen it can happen that your loss function landscape becomes very flat and so on average when you initialize randomly you will be somewhere where there's no

20:50 gradient. So gradient descent doesn't work anymore and your variational algorithm doesn't train and then you run into a problem and this is the so-called barren plateau problem and this is yeah characterized by the expectation value. So like expectation here refers to averaging over different initializations of the gradient of your loss function and whether that scales polomially or exponentially. And when it's exponential we we call it a baron plateau. And now people have found out that this this value this expectation value of your gradient is inverse proportional to the dimension of your le algebra from your circuit. And this was in a paper in 2023 entitled

21:36 a joint is all you need. I mean the title is a bit longer but that's a catchy phrase that you can you can Google later. And so what this tells us is that yeah that when the dimension of your le algebra is polomial you don't get baron plateaus and if it's exponential you do get baron plateaus. But this on the other hand earlier I told you that we have the simulation method that scales as the dimension of your lee algebra. So that means when your dimension of your lee algebra is polomial you can perform gimm when it's exponential you get a baron plateau it doesn't work anymore now there's been some debate about this whether this means or implies that everything is doomed I personally have the opinion that that is not the case on one hand gimm almost never happens like

22:24 these lee algebbras you have to construct a very special lee algebra to actually get something that is pol polomial scaling the vast majority of them are exponentially scaling. That means on the on one hand it means we almost always have baron plateaus but yeah baron plateaus might not be the end of it. If you're initializing completely randomly then yeah baron plateaus really are a problem but if you have some more clever strategies than where to look and typically in physics problems you do have some initial bias on where to start your computation. um you're not completely lost and then you might not run into this problem. Another thing worth mentioning is that the GSIM is really only for expectation values. So

23:11 it does not tell you anything about sampling and a lot of quantum algorithms are about sampling. So you might have a polomial scaling some some algorithm that has a polomial scaling lee algebra and you do you are able to perform gimm but that only tells you something about expectation values and not necessarily about sampling. So you might still need a quantum computer to run that same algorithm doing sampling. Okay, this was a bit of a detour, but you can see how Lee algebraas really is are are the central players in all of these debates that are kind of going on the last few years. Another so this has been kind I would I would say more concerned with classical simulation. Another interesting

23:57 application of this although this is actually not phrased like that but if you look at the math you can actually see it. You can perform essentially GSIM but on a quantum computer. So your this has been discussed in this paper here archive 2407 I mean you see the number or you can just look at this at this demo. So the problem is quantum simulation as in time evolution and what you're doing is you define some shadow state and that shadow state is is essentially the same as the vector we've seen earlier with the expectation values. So you encode all of the expectation values of potential elements and you're encoding yeah you're yeah you find what they call a shadow

24:43 Hamiltonian and the shadow Hamiltonian you get from your original Hamiltonian projected onto the basis of your Lee algebra and then you can run a quantum circuit starting from this initial state using this shadow Hamiltonian as an evolution. and perform what they call shadow Hamiltonian simulation which essentially is GSIM on a quantum computer. It's a bit funny because this is not your typical setup where in the end you have some expectation values. Your expectation values are now the amplitudes of your state. So this is fundamentally different setup to how you typically do quantum computing and quantum simulation. It's an interesting use case. I'm actually not sure how

25:28 practical this is. Um, but to be honest, I I I I genuinely don't know. So, I don't have an opinion on this, but it exists and it is related to Lee theory and this is why it's worth mentioning here. Cool. So, for the last bit, last 10 minutes, we're going to do lee theory in quantum compilation. I'm not sure if quantum compilation is actually the correct term here. I'm just using it referring to the fact of translating something that is a classical description of some physical problem into something that we can run on a quantum computer. That is what I consider quantum compulation and this is what these two demos or these two topics are about. So this going to be split in two parts. In the first part we're going to talk about the fixed depth Hamiltonian simulation via carton

26:14 decomposition. And another very important concept in le theory. So earlier in in gimm we we we saw the structure constants and we saw the a joint representation. A new very important concept here is the so-called kak decomposition or sometimes also called carton decomposition. Technically they're two different things but sometimes there the terminology is a bit used interchangeably. So first of all a carton decomposition is you have some lee algebra and you split that lee algebra into a sub algebra. This letter here is very ugly and it's very hard to read. It's called this is fractur k. So this is small K and this small K is a sub algebra meaning that when you take the commutator of elements from K with

26:59 elements of K you will get something that is again in K and that is the definition of a sub algebra subly algebra and it has it has these additional curious properties reductive property and symmetric property meaning how the relationship between K and M. So K and M they're orthogonal complements of each other and they they're not arbitrary orthogonal complements they're done in a special way. If if they're done in a special way that it fulfills these two properties then we call this decomposition carton decomposition on the algebra level. Now we can further decompose the M part. M is not a sub algebra and we can take a sub algebra of M and any sub algebra of M has to be a billion. A billion means that all

27:46 elements they commute with each other. This is a special sub algebra like there's actually infinitely many of them and this is what we call like one of them we call a carton sub algebra. It's important because it informs the so-called kak decomposition on a Hamiltonian level. It means that you can write some Hamiltonian as some operator K which is unitary times another operator which is again hermission times again another operator which is unitary again um these K elements they're in the E to the I K this is a subgroup because K is a sub algebra and A is an element from this Aelion subgroup and we're going to in just one slide see how we can utilize this for quantum compilation just want

28:32 to meion mention that the kind of the carton decomposition or the kk decomposition. This is now on the algebra level equivalent. We can we can also do this on the on the group level. Now consider the problem where we want to do time evolution of some Hamiltonian. I showed you earlier how we can decompose such a Hamiltonian as a k ak decomposition on the algebra level. We can do that K and K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K1 and K2 are unitary and we can split them apart and we get something that is called fast forwarding. This is called a fast forwarding because K the case they're not they're not dependent on T anymore. So any T you put here this circuit

29:18 always has the same depth because the only thing that you're changing is this parameter here. And this is why this whole thing is called a fast forwarding. Now this of course it's it's almost like a cheat code cheat code. This is too good to be true in the sense that like if we can always do that we we solve all of the problems in quantum computing that are related to time evolution which is typically a problem that is not solved. It's something that is still ongoing research and people look for the best ways how to write this as a circuit. So there is some caveat and that caveat obviously is the dimension of all of these things because when these decompositions reach too large dimensions and that dimension is again

30:04 the dimension of your le algebra you you can't do anything anymore. So this whole setup, this fast forwarding is restricted to settings where your operators have a small lee algebra. Um this is still curious I would say because you can so far I think time evolution has been restricted to cases where you have few cubits but you could now in principle have a system where you have a lot of cubits but they're encoding a small le algebra and now you can do this fast forwarding another way to think about it when I talk with colleagues that are not familiar with these things and they talk about fast forwarding what they typically mean is this is something where we have an anal analytic solution. But now you can kind

30:49 of extend this notion of fast forwarding to this is something where we have an analytic solution or the lee algebra is small and then we can we can also do it. Again this is all explained in more detail in this demo. The second part of the quantum compulation bit that I want to show you is the question like of unitary synthesis. So think of the the scenario where you have a quantum quantum circuit given as a unitary matrix. So you're really given you're given a dense unitary matrix in terms of numbers of 2 to the n * 2 to the n. Now the question is how do I translate that into a circuit and this is the problem of unitary synthesis and there are three kind of standard ways of doing that. of

31:35 course up to some up to some number of cubits you can't do that arbitrarily large but for some subcomponents there's the quantum Shannon decomposition the Kesha Glazer decomposition and the block set xxx decomposition um the first two they're rather old they're like from I think the 2000 or 2010s and the block setx is is is recent is like a small improvement and it turns out that all of these can be uniformly described as carton decompositions or rather Recursive cardon de compositions. There is a set of allowed cardon decompositions that you can do and the most typical way is when you start with some unitary. You can do a U A3 decomposition followed by so in this A3 decomposition

32:21 you get a multiplexer of some again unitary gate with a smaller cubit count some multiplexed cubit rotation and again the other K on the other side and then you can further decompose this bit with an A decomposition. And now if you look closely, you're you're kind of you're counting starting on the on the top left. You're starting with a u 2 to the n and you're finishing with a u to the n minus one. So you can recursively repeat this procedure until you're left with just two. So you just have a single cubit unitary. So you can give me a matrix that is a unitary of some 2 to the n * 2 to the n and I can recursively decompose it using these two decompos these two carton decompositions. Going

33:07 further now the quantum shannon decomposition kesa glass blocks zxx they at some point they use slightly different choices to do some optimizations but they're all in this framework of just recursive carton decompositions which is new insights that some colleagues of me of mine found. We do have a nice, of course, there's a paper for it, but there's also a demo for it that you can check out. I think I'm good on time. Plan was something like 40 minutes. And with that, I want to conclude. Thank you for your attention. Honestly, this is the main takeaway that you should take away from this. namely that the objects that you're already familiar with circuits being unitary matrices and observables being hermission matrices they form the group the Lee group and the Lee algebra

33:55 respectively and they're connected with the exponential map and then there's all this stuff that I've briefly showed you that you can check out if you're more curious in the demos or in the respective papers yeah and with that thank you for your attention and I'm happy to answer any questions that you I'm sure you have. Thank you so much. That was awesome. We've already had a bunch of chat on Discord mainly around how wonderful this session is already is turning out to be. So, thank you very much. I'll start fielding the questions as they come through. I can see people typing. Does anybody have any questions? Please post them on Discord.

34:40 So, there was one earlier we just missed in the flow from James in New York City and he wanted to know if Let me see. Actually, if I copy this and put it in the Zoom chat, it may be slightly easier. Yeah. Okay. Did that come through to you? Yeah, I'm just looking for the chat, but I think I found it or not. So on the bottom bar, you should Oh, yeah. Oh, yeah. Yeah, got it. So the expressivity of a circuit and whether it

35:26 can be efficiently simulated by classical computer is dependent on whether the gates in a quantum circuit come from either s o n or simple n. Yes, I would say yeah. Yes. Though it's it's more like the the the gates they form some league group and you typically care about the dimension of that league group and how close it is. So when you when you when you have full universality meaning that you can you can go from any point in Hbert space to any other point in Hbert space that is when you have the full unitary group. So that is when you're in sum. Um now when you have a smaller dimension you know that you are restricted in some way. So there are some states that you

36:12 can own that you cannot that that you cannot reach. So you don't have full expressivity. And so I would say yeah in the end your expressivity of your circuit is determined by this the le group that your gates are forming. Yep. I hope that answers the question. So we've had another one come in. What do you believe what do you believe in when open when you open source your quantum code? What do we believe in? We I'm not sure. This sounds like a philosophical question almost but I think I think I get the gist. we we don't see any advantage from our point of view to close source our code and quite the opposite I think we actually benefit from people using our our tools and our

36:59 that we use internally on one hand just to improve the software itself because you might may have use cases that we haven't thought about and you can file a bug report if this if it's not working and on the other hand because you might use it to do some cool research showing that showing other people that quantum computing is is useful in some way and that is something that we benefit from as a quantum computing company I would say. Thank you. Ban is asking what is the significance of lie theory in quantum circuit design and compil and compilation. Yeah, I hope I hope my my talk gave gave a a brief insight into that question. um for quantum computing it has not been all too relevant. Some people in optimal

37:47 control have been talking about le theory in the early 2000 early 2010s I think. Um and yeah it's it's popped up again in the context of baron plateaus. It's popped up again in the context of quantum simulation of quantum compilation. So I I wouldn't necessarily say it's like it it's a thing you have to know and it it it's uh if you don't know it, you're not going to understand quantum computing, but I'm understanding more and more that it's super relevant and can be super like a really fresh new perspective on things that you already seen and you might be able to use tricks from Lee theory that you might otherwise not use if you didn't know about these concepts. So yeah, I would say it's it's a useful emerging technique that I'm I'm

38:34 happy to spread the word about with these demos and this kind of talk. Thank you. Kumar is asking, can you explain the curtain once again? Sure. Part maybe that was one was a bit quick to be fair. Like I'm not expecting everyone to see these slides and be like, "Yeah, I understand everything." um that would probably be unrealistic. The the the the biggest hope is that you get the big picture idea and then you go read the paper or the demo and really get the details. Um but just to be sure what you what you in the end want to do is you you want to have a decomposition of some hermission operator or some corresponding unitary matrix and you can

39:19 think of this as a generalization of the singular value decomposition or a generalization of the igen value decomposition. So your let's let's just for for the sake of brevity say value decomposition. So you have some hermission operator h and this one is equal to some u * some diagonal operator times another u dagger. And now this is a general generalization in the sense that you again have on the left hand side you have some u and on the right hand side you have some u but in the middle instead of having having a diagonal operator you have an element from an aelion subgroup and a billion means that it is commuting and the easiest way to represent a commuting or an aelion group is via

40:05 diagonal operators. So in a way yeah but it doesn't have to be right. So in in in in could be the case that they're diagonal, could be the case that they're not, but they're a billion. So they're they're switching around. So in that sense, it's a generalization of SVD or value decomposition. And that's probably like the the the main point there's a full like if if you really want to know like I'm I'm doing super I'm doing physics math here. So I'm I'm I'm being a bit loose with all the definitions and the terms that I'm using. If you really care about it a bit a bit slightly bit more mathematically sound, there's also a great demo on KAKD compositions where I think if you just Google KK de composition, this is the first you're going to find by my colleague David Vivik. And this is a bit more mathematically profound. And if you

40:51 want to really get the details straight, that's the that's the place to look. And it's honestly it's it's not just I'm biased, of course, but it's not just the the best resource for it. It's the only resource. Like there's no Wikipedia page for this. So that's the place to look. Thank you from Arthur in New York City. Can you explain the connection of Baker Campbell Housef to lie algebra? Yeah, in the end um they're all connected I would say in a way that how should I phrase that? If if you one way let's let's go here to the a joint a joint

41:39 identity in the end the camp ke baker cample house is telling you what is basically e to the i or e to the e to the a plus b how can I how can I stretch these apart not sure this is direct like I can directly explain it with the adjint identity But the identity identity is telling you something very similar in how is this trans how is an element from this group transforming under the under the under a unitary element sorry under an element from the from the lee group. In the end in both cases you get a big series of commutators. And if your Lee algebra is finite

42:27 dimensional, which is the case here for what we're talking about for what we're talking about in quantum computing, then this series becomes somewhat simple uh in in the case for when you're looking at how this is transforming under leak group element and it's essentially characterized by the structure constants alone because these are all you need to know. The structure constants tell you the basically the relationships between elements in terms of their commutator and what you have in the Baker Campbell House formula is also some a bunch of nested commutators. Um I don't want to say something wrong right now because I'm not sure if this is actually related to the adjint identity but it looks very similar but I I think I might be oversimplifying things. For sure these

43:13 things in both cases you care about nested commutators because also I mean what you do in the leak closure is essentially you look at all possible nested commutators all possible nesting all possible combinations of of commutators and that is how you get the leak closure. So there's probably some relation but I'm not sure actually with I don't want to say something wrong on record. Thank you. And Rodrigo has asked why is it useful to use the Kak decomposition in this setting? Yeah. So I guess they are referring to this setting. Why is this useful? Um typically when you want to do a time evolution of an operator, the most straightforward thing or actually not not just the most

43:58 straightforward thing but the most common thing people do is trotter expansion. And in Troter expansion, you're rewriting this kind of abstract operator as a series of gates that you can actually implement. So something e to the i some coefficient times a party word. And the thing is the longer or the yeah the longer your time is in your evolution the longer your sequence of gates. And this is a kind of fundamental thing of trotter expansion. Now what the KK demposition tells us is if we know how to do a KK K de composition of this operator, we can actually write it as something that has fixed depth. So the the K and the K1 and the K2

44:44 actually I think it should be only K and K dagger. Uh I'm not sure if this is working for K1 K2 but but anyway they they they're not dependent on T. So they're they're fixed. So the only thing that changes when you're changing T is this middle part. And also that one is fixed so that this is like a bunch of e to the i t sum operator and a bunch of them. So they're not going to grow. There's not the sequence is not going to get any longer. So your what you what you really can do is this is why this is called fast forwarding. You can get to arbitrary times for free without having to spend more more gates. Of course there's the caveat that this doesn't work always and you need to either have a small lee algebra dimension or you need to be with a few cubits.

45:30 Thank you. One other question. Vicki is asking how useful quantum simulation is if she you're working as a researcher in quantum field theory or any part of theoretical physics. I'm not working in quantum field theory. So I don't know how useful quantum computing is there. Yeah, I don't know. But I think most people that are doing quantum field theory are doing quantum simulation to some degree. And I guess your your classical your means of performing classical simulation are somewhat limited. You either have to do you either have to scale like to very large like high performance computing settings or you have to introduce approximation and both have their limits and if you if

46:17 you want to do the real thing people people typically do quantum simulation. I think we've yeah from a this is more like I guess like a this would be like a typical science application of quantum computing. So I don't think it's I think it's quite quite near but I'm I'm saying that under the cave that I'm not a quantum field theory researcher so not sure if I can speak on their behalf. Thank you. I think we have time maybe for for one more. So let's see we've got let's see if anything else is coming in here is one. Do we also use lie co-algebra and lie by algebra to model quantum systems? Wait, can you repeat the first part? Lie

47:03 algebra and lie by algebra. By algebra to model quantum systems. I'm not sure what by algebra is. Yeah, I like I hope I could show you how we can for example the GSIM is an application of that where you model part of your quantum computation in terms of Lee algebra elements that are transformed under the adjint action or adjint representation of your Lee algebra. And yeah, that's that's how one way to model quantum computation in terms of it's it's somewhat limited because you're you can only do expectation values but that's the case for for most classical simulation techniques. I would say they you like for example tensor networks also I mean you can in

47:50 principle compute also samples but it's quite expensive. So yeah this is one way to to model it with le algebbras. Thank you. If you could give the students that are online today three takeaways from today's session that you would really want them to go away with or even to kind of look into further, what would they be? Um, so I think the main takeaway for me is this picture here that the objects that you're probably already familiar with, they form the Lee group and the Lee algebra and they're connected with the exponential map. Then this is the first takeaway I would say. The second takeaway is that there is something called the

48:36 adjoined identity or that we call the adjoined identity I should rather say with which you can do classical simulation or quantum simulation in an interesting new way in an interesting new perspective that might not have too many practical implications in terms of like you might not perform crazy simulations that were otherwise not possible but I think there's some really interesting mathematical insights if you understand these techniques. The same goes for the carton decomposition. That would be my third takeaway. There exists something that is called carton decomposition or k de composition. These terms can be used interchangeably usually which is a generalization of singular value decomposition or value decomposition which is which we typically as physicists know and uh

49:23 these are generalizations on the group level and yeah there's a there's some great reading material. I would recommend you to check it out in more detail. That would be my those would be my three recommendations. Thank you very much and thank you so much for your time today. That was an amazing session. Thank you, Katie. Bye. Bye.