WEBVTT 00:00:00.000 --> 00:00:11.360 Thank you so much, Henriquet, for the invitation to Erlangen. 00:00:11.360 --> 00:00:16.160 This is actually my first time in Erlangen, and I was really delighted to be reminded 00:00:16.160 --> 00:00:19.720 actually that Emmy Noether was born here and studied here. 00:00:19.720 --> 00:00:22.000 And Noether's theorem is actually one of my favorite theorems. 00:00:22.000 --> 00:00:25.560 I'm a physicist, by the way, my background. 00:00:25.560 --> 00:00:30.760 And of course, Noether's theorem is a beautiful way of seeing how mathematics actually impacts 00:00:30.760 --> 00:00:35.200 something so deep in physics, like symmetry and conservation. 00:00:35.200 --> 00:00:37.640 What I'm doing now is quantum computation. 00:00:37.640 --> 00:00:42.040 Actually this is another example of where having a physicist's way of interpreting 00:00:42.040 --> 00:00:47.160 the world, it really helps us rethink what we mean by algorithms. 00:00:47.160 --> 00:00:51.080 So I mean, you guys, I'm sure most of you know much more about PDs than me. 00:00:51.560 --> 00:00:56.760 So I work with Reza Jeng, who's an expert on PDs, and some of our other collaborators. 00:00:56.760 --> 00:01:00.960 But what about the quantum simulation part of this talk? 00:01:00.960 --> 00:01:07.360 What we really mean by that is actually what happens when your computer itself actually 00:01:07.360 --> 00:01:13.080 obeys quantum dynamics and not classical dynamics, and the information now is embedded inside 00:01:13.080 --> 00:01:15.240 your quantum states. 00:01:15.240 --> 00:01:23.080 And now when we think about simulating PDs beyond Schrodinger's equation, then we also 00:01:23.080 --> 00:01:26.280 need to modify the way we think about algorithms. 00:01:26.280 --> 00:01:31.680 And I'll show you how we've been doing this in recent years. 00:01:31.680 --> 00:01:37.720 So I don't need to explain ODs or PDs to you guys, and their importance in scientific computing. 00:01:37.720 --> 00:01:43.360 But what we'll do say is that, as you very well know, in classical numerical algorithms, 00:01:43.400 --> 00:01:45.760 a big problem is the cos of dimensionality. 00:01:45.760 --> 00:01:50.240 So the high dimension, it becomes exponentially more difficult. 00:01:50.240 --> 00:01:56.400 But what we do know is that if, a particularly good example of that is say, n-body Schrodinger 00:01:56.400 --> 00:01:57.400 equations. 00:01:57.400 --> 00:02:02.680 When you have high order, high dimensional equations, it becomes exponentially more difficult. 00:02:02.680 --> 00:02:08.120 And this is when people like Paul Benioff and later Feynman suggested, well, what happens 00:02:08.200 --> 00:02:13.960 if you want to simulate Schrodinger's equations on actually a quantum device itself, that 00:02:13.960 --> 00:02:16.560 itself obeys Schrodinger's dynamics? 00:02:16.560 --> 00:02:21.200 And there you can actually alleviate the cos of dimensionality. 00:02:21.200 --> 00:02:24.640 And now we think, well, Schrodinger's PDs, that's just one particular PD. 00:02:24.640 --> 00:02:28.200 What about all these other PDs that everyone's interested in? 00:02:28.200 --> 00:02:34.160 So maybe if we can map these onto Schrodinger-like equations and put them on a real quantum device, 00:02:34.160 --> 00:02:38.720 we can also help alleviate the cos of dimensionality for those problems as well. 00:02:38.720 --> 00:02:42.920 So therefore, the very first step to do that is to actually do this mapping. 00:02:42.920 --> 00:02:47.360 How do you turn non-Schrodinger-like equations into Schrodinger-like equations? 00:02:47.360 --> 00:02:50.560 And this is something that we call Schrodingerization. 00:02:50.560 --> 00:02:52.800 The name, I think, is self-explanatory. 00:02:52.800 --> 00:02:56.440 And we'll introduce these a bit later on. 00:02:56.440 --> 00:03:02.040 And since I have an idea that you guys are mostly classical, like numerical methods and 00:03:02.080 --> 00:03:07.120 classical machine learning, I'll just say, give a little bit of background on quantum 00:03:07.120 --> 00:03:11.000 computing and what we mean by quantum simulation. 00:03:11.000 --> 00:03:16.120 So here, I talk about obviating the cos of dimensionality. 00:03:16.120 --> 00:03:20.240 This is really saving on time complexity, all the time steps. 00:03:20.240 --> 00:03:24.800 But actually, if you want to go beyond classical infrastructure and use quantum devices, it 00:03:24.800 --> 00:03:29.360 can also help with other things, like memory, energy costs, communication, and so on, which 00:03:29.360 --> 00:03:30.760 I won't talk about. 00:03:30.760 --> 00:03:36.600 But also the fact that it's inevitable, because as our chips get smaller and smaller and smaller 00:03:36.600 --> 00:03:39.000 and smaller, they will hit the quantum scale. 00:03:39.000 --> 00:03:41.880 They'll no longer obey purely classical laws. 00:03:41.880 --> 00:03:45.360 So you can't have these circuit equations that work in the same way. 00:03:45.360 --> 00:03:46.920 So it's really inevitable. 00:03:46.920 --> 00:03:52.040 So then you guys are mathematicians saying, well, this is an inevitable future reality, 00:03:52.040 --> 00:03:57.240 that we really have to confront quantum mechanics in one way or another. 00:03:57.240 --> 00:03:59.160 How do we think about algorithms? 00:03:59.160 --> 00:04:05.520 This is a key message that I want to present here, is that now the computational techniques 00:04:05.520 --> 00:04:11.640 and methods that people have been doing for the last 40, 50 years, we really have to rethink 00:04:11.640 --> 00:04:16.520 that, because the actual infrastructure in which the computation is based on, that changes. 00:04:16.520 --> 00:04:21.120 And therefore, even the mathematical language that we use, that also has to be modified. 00:04:21.120 --> 00:04:24.400 And I'll give you some examples of that today. 00:04:24.400 --> 00:04:27.920 And just for those of you who haven't seen Schrodinger's, oh, well, we actually saw this 00:04:28.160 --> 00:04:33.600 in the previous talk, but this is Schrodinger's equation, and it evolves by unitary evolution. 00:04:33.600 --> 00:04:38.920 And the chief behavior of that is dominated by the Hamiltonian H, which is a Hermitian 00:04:38.920 --> 00:04:41.760 operator. 00:04:41.760 --> 00:04:44.080 So what do we mean by quantum simulation? 00:04:44.080 --> 00:04:45.520 Why is it a hard problem? 00:04:45.520 --> 00:04:50.720 So what we mean by that is that if you start off with a quantum system at time t equals 00:04:50.720 --> 00:04:56.880 to zero, and then you want to ask the question, well, what is the state after a certain amount 00:04:56.880 --> 00:04:58.480 of time t? 00:04:58.480 --> 00:04:59.520 What happens? 00:04:59.520 --> 00:05:03.560 So if you have a system that's made up of quantum bits, and I'll introduce that a little 00:05:03.560 --> 00:05:06.080 bit more later, because think about these as two-level systems. 00:05:06.080 --> 00:05:10.520 If you have n two-level systems, how do you predict what happens? 00:05:10.520 --> 00:05:17.360 So if they obey Schrodinger's equation, now even to represent the information of these 00:05:17.360 --> 00:05:22.120 n two-level systems, you need classical entry of 2 to the power of n, so purely memory. 00:05:22.120 --> 00:05:25.720 But now dynamics, now a Hamiltonian dynamics. 00:05:25.760 --> 00:05:28.600 Now that's 2 to the power of n by 2 to the power of n. 00:05:28.600 --> 00:05:34.200 So even to actually to process that, that becomes extra-mentally more difficult. 00:05:34.200 --> 00:05:41.440 And actually today already we're reaching the limit of what classical devices can simulate. 00:05:41.440 --> 00:05:45.760 So on the order of hundreds, but now we already know on real quantum systems we can actually 00:05:45.760 --> 00:05:46.760 go beyond that. 00:05:46.760 --> 00:05:51.040 And I'll also introduce something today what we call continuous variable quantum modes, 00:05:51.040 --> 00:05:53.480 and for that you can go into millions. 00:05:53.480 --> 00:05:56.520 And we're actually doing some experiments related to these. 00:05:56.520 --> 00:05:59.640 So I'll introduce the language first. 00:05:59.640 --> 00:06:03.640 And there are two types of quantum information I like to talk about. 00:06:03.640 --> 00:06:07.040 One is something you probably would have heard of, which is quantum bits. 00:06:07.040 --> 00:06:10.920 Everyone is familiar with bits, zeros and ones, and all your numerical algorithms are 00:06:10.920 --> 00:06:14.720 really reduced to bits in the end. 00:06:14.720 --> 00:06:16.440 But why do we work with bits? 00:06:16.440 --> 00:06:20.920 That's actually deeply related to physics, because information doesn't exist by itself. 00:06:21.320 --> 00:06:25.240 It has to exist in a physical substrate, and the way that the physical substrate represents 00:06:25.240 --> 00:06:26.600 that information. 00:06:26.600 --> 00:06:31.440 So when Turing presented his model, so you have a tape and you have holes that are punched 00:06:31.440 --> 00:06:33.760 or not punched, so that's zero or one. 00:06:33.760 --> 00:06:37.600 But you can't have stuff that's in between, like a half-punched hole. 00:06:37.600 --> 00:06:39.760 So you wouldn't have that in Turing's model. 00:06:39.760 --> 00:06:46.240 But now instead of using tapes with transistors on and off, so instead of using these systems 00:06:46.240 --> 00:06:50.840 to represent information, what happens when we use, for example, atoms to represent information? 00:06:51.400 --> 00:06:56.120 So one way that we can use atoms to represent information is, for example, energy level. 00:06:56.120 --> 00:06:59.720 Your ground state, your lowest energy state, and your first excited state. 00:06:59.720 --> 00:07:04.920 You can say, okay, well, lowest ground state, that's zero, first excited state, that's one. 00:07:04.920 --> 00:07:11.400 But it turns out that in physics, you can have states that's not either zero or one. 00:07:11.400 --> 00:07:18.280 And in physics language, we call this a superposition, but it's really not equivalent to a mixture. 00:07:18.280 --> 00:07:23.480 It's not like a probabilistic coin, like half the time it's tails or half the time it's heads. 00:07:23.480 --> 00:07:30.280 And it's very difficult to actually explain, because evolution hasn't equipped our minds to deal with this. 00:07:30.280 --> 00:07:33.720 So it's much better to think about it abstractly, mathematically, 00:07:33.720 --> 00:07:38.360 that we actually represent these states in complex Hilbert space. 00:07:38.360 --> 00:07:43.240 So for a single qubit, you can think about the x-axis going as the ground state, 00:07:43.240 --> 00:07:46.840 the y-axis as the first excited state. 00:07:46.840 --> 00:07:50.520 And in most states, it's a vector in this space. 00:07:50.520 --> 00:07:53.880 And what do the components, alpha and beta, mean? 00:07:53.880 --> 00:07:59.880 Well, the absolute squared value of that represents a probability that you observe your atom, 00:07:59.880 --> 00:08:03.960 either in the ground state or the first excited state, with that probability. 00:08:03.960 --> 00:08:10.760 And then if you have multiple, say, n qubits, then it's in 2 to the power of n-dimensional Hilbert space. 00:08:10.760 --> 00:08:15.800 But the mathematical language that we use here are vectors and matrices. 00:08:15.800 --> 00:08:19.000 So Schrodinger's equation is linear, so we can use linear algebra, 00:08:19.000 --> 00:08:20.840 and therefore, all we have are matrices. 00:08:20.840 --> 00:08:24.360 And the way that we represent dynamics, again, is using matrices. 00:08:24.360 --> 00:08:31.240 How you transform one finite dimensional vector into another finite dimensional vector with matrices. 00:08:31.240 --> 00:08:34.680 And we can represent these, again, by sums of Pauli operators. 00:08:35.480 --> 00:08:38.840 And we can represent them in lots of physical systems, 00:08:38.840 --> 00:08:41.400 ion spin systems, superconducting qubits, and so on. 00:08:42.600 --> 00:08:43.880 So that's information. 00:08:44.360 --> 00:08:47.320 And I said the dynamics you can represent by matrices. 00:08:48.440 --> 00:08:53.400 But if we want digital evolution, and this is analogous to what we do classically, 00:08:53.400 --> 00:08:57.880 is that we concatenate our evolution into a lot of steps. 00:08:58.440 --> 00:09:01.560 So if we have universal devices, this is what we can do. 00:09:02.520 --> 00:09:07.800 And if we have a fully digital, programmable, universal quantum device, 00:09:07.800 --> 00:09:11.560 then we can really use it to represent any unitary matrix. 00:09:12.280 --> 00:09:16.760 So long as we have smaller systems, so the smaller gates, 00:09:16.760 --> 00:09:20.600 and we can always compose them into any unitary gate. 00:09:20.600 --> 00:09:26.760 Now, however, to actually realize this in a fault-tolerant way, this is kind of not near-term. 00:09:27.560 --> 00:09:29.000 So we really have to wait a while. 00:09:29.000 --> 00:09:32.680 We actually don't know exactly how long we need to wait before we do something reasonable. 00:09:32.680 --> 00:09:38.040 And certainly, if we want to simulate something like ODIs and PDs, we're really not there yet. 00:09:38.520 --> 00:09:39.020 Okay. 00:09:40.760 --> 00:09:47.080 On the other hand, physics actually provides us with systems which are called continuous variable, 00:09:47.480 --> 00:09:48.280 quantum systems. 00:09:49.080 --> 00:09:52.760 And in fact, in quantum computation, people don't mention this very much, 00:09:52.760 --> 00:09:55.000 because a lot of them, they come from computer science. 00:09:55.000 --> 00:09:57.640 They're actually in computer science from the Turing model. 00:09:57.640 --> 00:10:00.280 So they're very biased towards discrete systems. 00:10:00.280 --> 00:10:05.160 But as a physicist, we know that most continuous systems, 00:10:06.120 --> 00:10:09.240 most physical systems are indeed described in a continuous way. 00:10:09.240 --> 00:10:12.600 And indeed, the PDs that you guys work with, they're inspired by nature. 00:10:12.600 --> 00:10:13.320 They're continuous. 00:10:14.680 --> 00:10:17.480 And indeed, there's continuous quantum information. 00:10:18.040 --> 00:10:20.840 And for those of you who might have heard of wave function, 00:10:20.840 --> 00:10:24.360 and actually, this is the representation in terms of wave functions. 00:10:24.360 --> 00:10:28.600 And here, you can think about this as really living in infinite dimensional, 00:10:28.600 --> 00:10:29.800 okay, Hilbert space. 00:10:29.800 --> 00:10:32.120 And this X here, you can think about as position. 00:10:32.120 --> 00:10:34.680 And we know that things like position and momentum, 00:10:35.400 --> 00:10:37.560 so these are all continuous quantities. 00:10:39.240 --> 00:10:41.080 So now, what are the operators? 00:10:41.080 --> 00:10:43.720 So before, we had matrices, so that language. 00:10:43.720 --> 00:10:45.880 But now, because we're living in infinite dimensional space, 00:10:45.880 --> 00:10:47.480 we need to use operator language. 00:10:47.480 --> 00:10:49.240 So again, this is different mathematics. 00:10:50.520 --> 00:10:53.640 And there are two key operators I want to introduce. 00:10:53.640 --> 00:10:56.360 So we have what we call X hat. 00:10:56.360 --> 00:10:58.280 So this is a position operator, 00:10:58.280 --> 00:11:02.680 and it acts on a position eigenstate that reproduces for you the position. 00:11:03.320 --> 00:11:04.920 And P represents momentum. 00:11:08.600 --> 00:11:11.400 And this is important for us later, that if you want to represent 00:11:13.160 --> 00:11:16.920 what happens when you multiply your wave function U of X by X, 00:11:16.920 --> 00:11:20.840 then you act upon that state by a position operator. 00:11:20.840 --> 00:11:27.240 But if you want its derivative, then you actually take a derivative of that function. 00:11:27.240 --> 00:11:31.640 So X goes to X hat, and d dx goes to I P hat, 00:11:31.640 --> 00:11:33.000 where I is your imaginary number. 00:11:33.640 --> 00:11:39.640 And another really important property to look at is that if we look at the inner product 00:11:39.640 --> 00:11:42.520 between the eigenstates of position momentum, what do we notice? 00:11:43.320 --> 00:11:44.840 It's e to the power of I X P. 00:11:46.040 --> 00:11:49.240 And this is really what we see when we do Fourier transforms. 00:11:49.720 --> 00:11:51.160 So this is a factor that's involved. 00:11:51.160 --> 00:11:53.720 And this will also be important for us later, 00:11:53.720 --> 00:11:57.480 because it turns out that actually in most powerful quantum algorithms, 00:11:57.480 --> 00:12:01.560 you actually need to do Fourier transforms and inverse Fourier transforms. 00:12:02.280 --> 00:12:06.040 And if you want to do these on qubit-based systems, it's actually very, very difficult to do. 00:12:07.240 --> 00:12:09.720 But here, we actually see it's actually really very natural, 00:12:09.720 --> 00:12:12.440 because what you're doing is that you're just transforming a basis 00:12:12.440 --> 00:12:15.640 between a position basis to a momentum basis. 00:12:15.640 --> 00:12:17.160 So it's like you're measuring instrument. 00:12:17.160 --> 00:12:19.880 If you won't measure momentum, you can be measuring position. 00:12:19.880 --> 00:12:23.560 And just by making this change, you're actually doing a Fourier transform. 00:12:24.120 --> 00:12:27.800 So this is very easy to do for things like on optical systems. 00:12:28.760 --> 00:12:31.080 And this will also be important for us later. 00:12:32.360 --> 00:12:37.640 And here, so these continuous variable systems are represented by things like lasers. 00:12:37.640 --> 00:12:40.840 I don't have a laser pointer today, but I usually do this with a laser pointer. 00:12:41.960 --> 00:12:46.360 So the state that comes out of a laser, that's directly continuous variable, quantum state. 00:12:46.360 --> 00:12:49.960 You have light inside cavities, so you have freely moving atoms, and so on. 00:12:51.480 --> 00:12:52.600 Now, what about evolution? 00:12:53.480 --> 00:12:56.920 So before we talk about discrete evolution, breaking into lots of gates, 00:12:56.920 --> 00:13:01.480 and I said we're really far away from realising that on a real quantum device. 00:13:01.800 --> 00:13:05.080 However, all quantum systems that we know, including you and me, 00:13:05.640 --> 00:13:07.080 we evolve through continuous time. 00:13:09.640 --> 00:13:11.880 So if we have time evolution through continuous time, 00:13:11.880 --> 00:13:13.880 this is very relevant for you guys who study control. 00:13:14.840 --> 00:13:20.520 Because it means that if we're able to use control theory to find the appropriate parameters 00:13:20.520 --> 00:13:24.520 of the Hamiltonian, to engineer the final state that we want, 00:13:24.520 --> 00:13:26.360 then we've actually solved our problem. 00:13:26.360 --> 00:13:30.920 So we don't need to go through doing these universal quantum devices. 00:13:31.400 --> 00:13:35.320 Specialised devices where you simply control certain parameters using control. 00:13:36.520 --> 00:13:39.640 And again, we have lots of quantum systems that do that too. 00:13:39.640 --> 00:13:44.040 Naturally, they're already there in the labs, and they're not special quantum computers 00:13:44.680 --> 00:13:47.080 that you hear about that Google has, and so on. 00:13:47.080 --> 00:13:48.520 They're just natural in the lab. 00:13:48.520 --> 00:13:49.640 Everybody already has them. 00:13:50.760 --> 00:13:59.640 Okay, so for all these kinds of computing, we have discrete and continuous ways of encoding 00:13:59.640 --> 00:14:03.480 information, either in quantum bits or in these quantum modes or Q modes. 00:14:04.040 --> 00:14:06.920 Or in the processing itself, we can either have discrete time, 00:14:07.480 --> 00:14:10.520 you chop it up into lots of gates, or continuous time. 00:14:10.520 --> 00:14:15.960 And mostly what I'll be talking about today is the right bottom corner, 00:14:16.520 --> 00:14:19.000 and this is what we call quantum analogue computing. 00:14:19.000 --> 00:14:21.400 And it turns out the method that we have, shredding organisation, 00:14:21.400 --> 00:14:24.600 is particularly suitable for quantum analogue computing. 00:14:24.600 --> 00:14:27.240 Of course, it can also be suitable for discrete computing, 00:14:27.240 --> 00:14:32.920 but in this technique, no other method can do this except ours right now. 00:14:34.760 --> 00:14:39.000 And just to give you a taste of how powerful quantum modes these Q modes are, 00:14:40.360 --> 00:14:41.960 because most people haven't heard of these. 00:14:44.120 --> 00:14:47.960 If we think about the poster child for quantum algorithms, 00:14:47.960 --> 00:14:50.600 Shor's algorithm, so this is about breaking RSA. 00:14:51.240 --> 00:14:56.680 So we know to factor a very large number n on a classical device, it's very difficult. 00:14:56.680 --> 00:15:01.080 But for quantum algorithms, you can have a poly log n type of algorithm. 00:15:01.800 --> 00:15:05.960 So classically, to break the current RSA encryption will take longer than the lifetime of the 00:15:05.960 --> 00:15:09.720 universe. But if you use a quantum device, even though it won't take so long, 00:15:09.720 --> 00:15:12.680 but you actually need a lot of qubits, a lot of quantum bits. 00:15:14.440 --> 00:15:20.040 And this is work I did many years ago, and we call this the Power One Q mode model. 00:15:20.040 --> 00:15:25.960 And this is where you can actually replace all these perfect non-noisy quantum bits, 00:15:26.680 --> 00:15:29.080 in the original Shor's algorithm, with a single Q mode. 00:15:30.200 --> 00:15:32.840 And that is already sufficient for factoring. 00:15:34.200 --> 00:15:36.840 But of course you say, well, why haven't people done this? 00:15:36.840 --> 00:15:39.000 Well, it turns out you have to actually exchange 00:15:39.000 --> 00:15:42.120 number of qubits with the energy required in a Q mode. 00:15:42.120 --> 00:15:43.880 But at least this gives you a basic idea. 00:15:43.880 --> 00:15:48.120 Because after all, a single quantum mode is really an infinite dimensional quantum system. 00:15:49.480 --> 00:15:54.760 All right, so now to our question of how to simulate classical ODIs and PDs 00:15:54.760 --> 00:15:57.240 with quantum devices, how do we do the transformation? 00:15:57.240 --> 00:16:00.760 Well, first of all, we have two types. 00:16:00.760 --> 00:16:03.880 One is we have a system of ordinary differential equations, 00:16:03.880 --> 00:16:07.560 or we have a D plus one dimensional partial differential equation. 00:16:08.120 --> 00:16:11.960 And quantum systems or Schrodinger's equation can also have one of these two types. 00:16:11.960 --> 00:16:16.280 So what we really want to do is we want to map these guys onto these 00:16:17.720 --> 00:16:19.160 Schrodinger type equations. 00:16:19.160 --> 00:16:22.280 So you have a system of D, ordinary differential equations, 00:16:22.280 --> 00:16:24.440 and you really want to map to the left-hand side, 00:16:25.240 --> 00:16:28.440 where now your Hamiltonian is now in this matrix language. 00:16:29.000 --> 00:16:37.480 It's a finite dimensional matrix, and then you have a log D size system. 00:16:38.280 --> 00:16:41.640 But if you want to simulate a D plus one dimensional PD, 00:16:41.640 --> 00:16:45.560 then you really want to map onto a Schrodinger equation something like the right-hand side. 00:16:47.720 --> 00:16:50.040 And this is how Schrodinger's equations are normally introduced. 00:16:50.040 --> 00:16:51.720 You have momentum operators, you remember? 00:16:52.360 --> 00:16:55.080 So you remember the rule, and you have the potential, 00:16:55.080 --> 00:16:58.120 and you remember the rule when you have a derivative, it goes to IP hat. 00:16:59.640 --> 00:17:03.640 And this is what happens when you have a Laplacian in your Schrodinger equation, 00:17:03.640 --> 00:17:05.960 it goes to momentum squared. 00:17:05.960 --> 00:17:09.480 And this you guys will probably remember from high school, like in physics, 00:17:09.480 --> 00:17:10.440 what do we mean by energy? 00:17:10.440 --> 00:17:13.960 Because Hamiltonian represents energy, you have kinetic energy plus potential energy. 00:17:13.960 --> 00:17:15.560 And what is kinetic energy? 00:17:15.560 --> 00:17:16.920 That's basically momentum squared. 00:17:17.560 --> 00:17:21.480 So all that should fit with the classical intuition. 00:17:22.120 --> 00:17:26.840 And so therefore what you see is that in a system of ordinary differential equations 00:17:26.840 --> 00:17:32.440 is more suitable to represent by discrete systems, meaning qubit systems. 00:17:32.440 --> 00:17:35.000 But if you want to simulate partial differential equations, 00:17:35.000 --> 00:17:37.000 then you really want to use these qubits, 00:17:37.000 --> 00:17:39.160 because now these act on infinite dimensional systems. 00:17:41.400 --> 00:17:41.800 All right. 00:17:42.600 --> 00:17:48.680 And when I talk about analog devices now with quantum simulation, 00:17:48.680 --> 00:17:52.040 we can have analog, we said before, and we can also have digital. 00:17:52.600 --> 00:17:54.840 And just as classical cases, we have digital, 00:17:54.840 --> 00:17:57.960 and this is where we have this cos of dimensionality problem. 00:17:57.960 --> 00:18:00.600 But we can also have classical analog devices, 00:18:00.600 --> 00:18:03.320 which all you guys are probably too young to remember. 00:18:04.440 --> 00:18:05.880 But they did have those. 00:18:07.400 --> 00:18:11.240 But with classical analog devices, they actually don't solve PDs very well, 00:18:11.240 --> 00:18:13.720 because you have to discretize your PDs. 00:18:14.440 --> 00:18:16.680 But it turns out for the quantum case, I'll show you later, 00:18:16.680 --> 00:18:19.000 we don't have to do any discretization at all. 00:18:19.640 --> 00:18:23.720 We actually preserve the continuity of the PD, 00:18:24.360 --> 00:18:27.240 and that's actually because we utilize quantum entanglement, 00:18:27.240 --> 00:18:28.920 and then that's another quantum property. 00:18:29.640 --> 00:18:33.080 And we actually showed that the quantum cos is linear in dimension, 00:18:33.080 --> 00:18:35.480 and no discretization is necessary. 00:18:35.480 --> 00:18:40.920 And we can also use this method to map it onto qubit dynamics as well. 00:18:40.920 --> 00:18:44.440 And again, that would be polynomial in dimension, 00:18:44.440 --> 00:18:48.600 and also logarithmic in the discretization size of the system. 00:18:52.360 --> 00:18:57.160 So the basic idea, I want to say, so this is Schrodingerization work. 00:18:57.160 --> 00:19:02.760 So we have the qubit-based algorithms as well as the continuous algorithms. 00:19:02.760 --> 00:19:07.240 And I think I'll present the analog form for you, 00:19:07.240 --> 00:19:09.320 because I think it's cleaner to see. 00:19:09.320 --> 00:19:12.360 And we can always discretize that later, if we want. 00:19:14.520 --> 00:19:18.200 So now let's have a look at the original Schrodinger type of equation. 00:19:18.200 --> 00:19:21.000 And now let's look at a more classically motivated equation, 00:19:21.000 --> 00:19:21.960 like the heat equation. 00:19:22.840 --> 00:19:25.720 So what do we notice about these two equations? 00:19:27.880 --> 00:19:29.960 They basically look the same, right? 00:19:29.960 --> 00:19:30.760 Except there's an I. 00:19:32.520 --> 00:19:35.080 Everything else is really the same, and you're like, oh, it's so close. 00:19:35.800 --> 00:19:37.960 So mathematically, what can we do? 00:19:37.960 --> 00:19:40.120 And of course, people say, well, oh, it's very obvious. 00:19:40.120 --> 00:19:41.320 It was just an I, right? 00:19:41.320 --> 00:19:43.320 Let's just get rid of that I. 00:19:43.320 --> 00:19:46.840 We take I to the left-hand side and make time imaginary. 00:19:46.840 --> 00:19:48.200 And this is what people did. 00:19:48.200 --> 00:19:50.040 So this is called an imaginary time evolution, 00:19:50.040 --> 00:19:52.520 sometimes called Wick rotation for physicists. 00:19:53.480 --> 00:19:56.440 But now the problem is that in imaginary time evolution, 00:19:56.440 --> 00:20:00.600 solution in time t does not actually correspond to the solution 00:20:00.600 --> 00:20:04.920 of the heat equation at that time t, because it's not a real time. 00:20:04.920 --> 00:20:07.320 And in fact, if you want to do quantum algorithms with that, 00:20:07.960 --> 00:20:10.920 you actually need to do partial tomography at every step 00:20:11.720 --> 00:20:15.080 before you know what evolution you want to perform. 00:20:15.880 --> 00:20:18.840 So this is actually, I think, not a good method, 00:20:18.840 --> 00:20:22.840 even though mathematically you just do t goes to I.T. 00:20:22.840 --> 00:20:26.520 So now our question is, is there a real time way 00:20:26.520 --> 00:20:29.480 of making a non-Trodinger equation trodinger-like? 00:20:32.440 --> 00:20:32.840 All right. 00:20:32.840 --> 00:20:36.840 And then, but before we go on, just by looking at this, 00:20:36.920 --> 00:20:39.400 we're saying if t goes to I, t can solve the problem. 00:20:39.400 --> 00:20:42.840 And when we think about I going to the imaginary argon plane, 00:20:42.840 --> 00:20:44.680 we also think about going to an extra dimension 00:20:44.680 --> 00:20:45.720 by introducing this I. 00:20:46.360 --> 00:20:48.280 So already in your head, you're like, actually, 00:20:48.920 --> 00:20:52.840 it suggests by adding a dimension, somehow this might help us. 00:20:54.120 --> 00:20:57.320 Because this is what's happening with this imaginary time. 00:20:58.120 --> 00:21:01.080 But here, instead of adding an imaginary extra dimension, 00:21:02.520 --> 00:21:06.280 coming from the argon plane, here we're introducing a real, 00:21:07.240 --> 00:21:08.040 extra dimension. 00:21:09.080 --> 00:21:14.040 And we introduce a transformation that we call the warped phase transformation. 00:21:15.000 --> 00:21:16.280 So now we have the heat equation. 00:21:17.480 --> 00:21:20.200 That's all u, T of x. 00:21:20.200 --> 00:21:22.840 But now let's introduce a new spatial dimension, psi. 00:21:23.480 --> 00:21:29.320 And when psi is above zero, we multiply this u by e to the power of minus psi. 00:21:29.320 --> 00:21:32.840 This is the warp factor along this extra dimension. 00:21:33.720 --> 00:21:35.160 What's the point of that? 00:21:35.960 --> 00:21:39.160 Well, let's see what equation w obeys. 00:21:39.960 --> 00:21:45.160 And it turns out this w obeys something, this heat equation in phase space. 00:21:45.800 --> 00:21:48.760 So it basically kind of looks like the heat equation, right? 00:21:48.760 --> 00:21:53.000 Except now on the right-hand side, instead of acting on w, 00:21:53.000 --> 00:21:57.240 it acts on the first derivative of w with respect to psi. 00:21:58.760 --> 00:22:01.080 And you're saying, well, now you're just making things more complicated. 00:22:01.800 --> 00:22:03.800 But let's think back to our original question, 00:22:03.880 --> 00:22:06.920 which is how do we get an extra i in a real way? 00:22:08.120 --> 00:22:14.760 Well, because now this is an extra derivative of w with respect to psi and not w itself, 00:22:15.800 --> 00:22:19.240 we can have a way of getting an i if we take a Fourier transform. 00:22:19.240 --> 00:22:21.400 Because if we ever take a Fourier transform, we take a derivative, 00:22:21.400 --> 00:22:22.760 this is how we get an extra i. 00:22:23.720 --> 00:22:26.120 And this is what we can do. 00:22:26.760 --> 00:22:30.520 So we have a Fourier transform in this extra dimension, psi, 00:22:30.520 --> 00:22:31.880 and then we take the derivative. 00:22:32.040 --> 00:22:37.560 Okay, then we get an extra factor of i eta, where eta is a Fourier mode. 00:22:37.560 --> 00:22:39.000 Okay, just psi at. 00:22:39.000 --> 00:22:43.560 So now what we see, what we get in blue, this is really a Schrodinger-like equation. 00:22:45.000 --> 00:22:47.560 So this is Schrodinger's equation for each Fourier mode. 00:22:47.560 --> 00:22:58.680 And now t, you can think about as being dilated by a real number, eta, instead of bi-imaginary number. 00:22:58.680 --> 00:23:02.840 So here you're saying, okay, well, this is like having an infinite number of Schrodinger's equations. 00:23:03.480 --> 00:23:05.720 So how do we actually represent this in a real system? 00:23:06.600 --> 00:23:09.400 Well, it turns out it's actually very easy to represent this in a real system, 00:23:09.400 --> 00:23:11.800 even though it's an infinite number of Schrodinger's equations. 00:23:13.000 --> 00:23:16.200 But remember, a single corner mode that can be represented by, 00:23:16.200 --> 00:23:19.560 you know, say a single laser beam, that itself is already infinite dimensional. 00:23:19.560 --> 00:23:24.360 So it really means that in a physical system, all we need to do is to add an extra mode. 00:23:25.160 --> 00:23:29.880 Okay, and that can actually capture, you know, having this infinite eta. 00:23:29.880 --> 00:23:34.920 Okay, so remember the rule, okay, whenever you have, you know, your x, it goes to x hat. 00:23:35.640 --> 00:23:40.360 Now eta is actually acting as a spatial mode, okay, so it can go to eta hat. 00:23:40.360 --> 00:23:43.080 And when you have a derivative, it goes to momentum. 00:23:43.080 --> 00:23:47.400 Okay, so then, you know, this heat equation system really transforms 00:23:47.400 --> 00:23:52.360 to a Schrodinger system with a Hamiltonian that actually the first part, okay, it looks 00:23:53.160 --> 00:23:59.000 just like, you know, kinetic energy plus potential energy, okay, p squared plus v of x. 00:23:59.000 --> 00:24:03.720 But now a tensor product, because now we added a mode, has a tensor product of eta hat, 00:24:03.720 --> 00:24:05.720 okay, acting on this extra mode. 00:24:06.600 --> 00:24:11.720 Okay, so now this is telling us if we have a quantum system that has this Hamiltonian, 00:24:11.720 --> 00:24:16.200 then we're actually able to simulate the heat equation, okay, using a quantum device. 00:24:17.160 --> 00:24:22.520 Okay, and this general methodology works for, you know, any linear, okay, PDs, but, 00:24:23.720 --> 00:24:27.800 yeah, so also inhomogeneous ones and so on, but it's a little bit, you know, complicated. 00:24:28.600 --> 00:24:32.120 So I'll just explain, you know, the general methodology for simplicity, 00:24:32.120 --> 00:24:36.040 okay, for a linear homogeneous PD that's first derivative, okay, in time. 00:24:36.040 --> 00:24:41.160 So remember the rule, okay, you can rewrite all these equations as, you know, 00:24:41.160 --> 00:24:48.200 dU dt equals to minus iA of U, where whenever you have x gets promoted to an operator, x hat, 00:24:48.200 --> 00:24:53.000 when you have a derivative, okay, then it gets promoted into a momentum operator. 00:24:53.000 --> 00:24:57.320 So you end up with an operator A, okay, that's made up of, you know, 00:24:58.680 --> 00:25:02.040 momentum operators and position operators, okay. 00:25:03.000 --> 00:25:10.200 Okay, so, but with any of these operators, it can be very easily separated into a completely 00:25:10.200 --> 00:25:15.320 homission form and a completely anti-homission form, right, so if A is completely homission, 00:25:15.320 --> 00:25:20.760 that's just your regular Schrodinger-like equation. If it's completely anti-homission, 00:25:20.760 --> 00:25:27.320 that's exactly like your heat-like equation, okay, so, but if it's anything between, okay, 00:25:27.320 --> 00:25:34.040 then any operator A can be written as a sum, okay, A1, which is homission, minus iA2, 00:25:34.040 --> 00:25:39.960 where A2 is also homission, okay, where A1 is just a sum of A plus A dagger and A2 is just 00:25:39.960 --> 00:25:45.960 A minus A dagger, okay, well, with an i. So then the Hamiltonian that we have is, 00:25:47.480 --> 00:25:52.280 yeah, I don't have a pointer, but if you can see, it's equal to A2, tensor product eta, 00:25:52.280 --> 00:25:56.360 and that part I already saw before the heat equation, okay, this completely anti-homission 00:25:56.360 --> 00:26:03.320 part, tensor product with our eta hat, okay, plus A1, which is a completely homission part 00:26:03.320 --> 00:26:07.960 of the equation, tensor product the identity, right, and why does it have this form? 00:26:07.960 --> 00:26:12.680 So if you remember, if you have something that is completely heat-like, okay, so the opposite of 00:26:12.680 --> 00:26:20.840 Schrodinger's equation, okay, in a sense, then you have to utilise your extra mode, okay, and in fact, 00:26:20.840 --> 00:26:26.600 having this tensor product actually implies there's entanglement, okay, between these modes. 00:26:27.320 --> 00:26:33.560 But if your operator has the completely homission part, which is A1, that means it's already 00:26:33.560 --> 00:26:37.560 Schrodinger-like, so you actually don't need the second mode, and that's why it's tensor product 00:26:37.560 --> 00:26:43.160 identity, okay, because you actually don't need to utilise this extra dimension. So in a physical 00:26:43.160 --> 00:26:47.960 system, if you have access to a Hamiltonian of this form, then you can actually simulate differential 00:26:47.960 --> 00:26:53.160 equations, okay, of this form, okay. So as an example, if we have, you know, a very high-dimensional 00:26:53.160 --> 00:26:58.360 Fokker-Planck equation, and if you have an example of linear drift and additive noise, then we have 00:26:58.360 --> 00:27:04.360 the Hamiltonian on the bottom right-hand side, and actually, if you're a physicist and you see 00:27:04.360 --> 00:27:09.480 this, you're actually saying, actually, this looks very familiar, okay. So for the second part of 00:27:09.480 --> 00:27:14.680 this term, this is something that people see all the time in linear optical systems, okay. Now the 00:27:14.680 --> 00:27:18.440 first part of this Hamiltonian is a little bit tricky here, that there are parts of it that can 00:27:18.440 --> 00:27:23.800 be seen in superconducting systems, okay, and that there are actually transformations we do to make 00:27:23.800 --> 00:27:28.360 it even easier. So we are actually currently conducting experiments to actually simulate, 00:27:28.360 --> 00:27:36.600 okay, equations of these forms. Okay, so sort of the summary, okay, it's actually, you know, 00:27:36.600 --> 00:27:41.960 incredibly easy to do. If you have any linear system of ODEs and PDEs, okay, you find out 00:27:41.960 --> 00:27:46.840 they're completely Hermitian and anti-Hermitian, okay, parts of your operator A, right, and if 00:27:46.840 --> 00:27:50.920 you want to simulate this on a quantum device, what do you do? You find a Hamiltonian of the form 00:27:50.920 --> 00:27:57.480 A2 turns a product eta hat or x hat, okay, same thing, plus A1 turns a product the identity, 00:27:57.480 --> 00:28:01.320 okay, and you can go back and forth, okay, and, you know, and because everything is kind of 00:28:01.320 --> 00:28:06.920 linearly related, it's very simple to do, okay. And if you want, you can also simulate this on 00:28:07.640 --> 00:28:12.280 qubit-based systems, so you can have a qubit representation of x hat, which is, you know, 00:28:12.280 --> 00:28:17.640 a diagonal matrix, okay, from minus n to n, where n is your discretization size, okay. 00:28:17.640 --> 00:28:23.000 Okay, yeah, and this is, again, like another flow chart, okay, of what you can do, 00:28:23.960 --> 00:28:29.080 and this is also applicable when you have, you know, time-dependent Hamiltonians, okay, as well. 00:28:30.040 --> 00:28:33.560 Okay, and here you can see it's, you know, very easy to go back and forth. 00:28:35.320 --> 00:28:38.680 And for those of you interested in saying, well, what does it actually look like, 00:28:38.680 --> 00:28:45.480 okay, on a device, well, what do you need to do? Well, what we need to do is that we need to prepare, 00:28:45.480 --> 00:28:49.880 okay, the initial condition of our system in a quantum state, and remember we have this extra 00:28:49.880 --> 00:28:55.000 mode, okay, so we actually need to prepare the initial condition of this extra mode, and it's 00:28:55.000 --> 00:28:59.640 actually related to applying the warped phase transformation, we need to, you know, multiply 00:28:59.640 --> 00:29:04.680 things by e to the power of minus psi, but it turns out, so that turns out that, you know, 00:29:04.680 --> 00:29:09.880 it's good to prepare a quantum state, okay, of the form above, so the integral of minus, 00:29:09.880 --> 00:29:15.320 e to the power of minus psi, but it turns out that this state is actually very, very close 00:29:15.320 --> 00:29:20.840 to a state that directly comes out of your laser, okay, what we call a coherent Gaussian state, 00:29:20.840 --> 00:29:25.240 okay, it's very, very close, so experimentally this is what, actually what we're using, 00:29:25.240 --> 00:29:32.600 and in the end to actually find the, you know, answer to our problem, okay, we need to make 00:29:32.600 --> 00:29:39.480 a measurement just on our extra mode, okay, so if we measure momentum, okay, so long as our detector 00:29:39.480 --> 00:29:44.680 says, momentum's above zero, okay, then we can be guaranteed we have the solution to our problem, 00:29:44.680 --> 00:29:51.400 and if we have a d-dimensional linear PDE, then here all we need are d plus one-dimensional quantum 00:29:51.400 --> 00:29:57.000 modes, okay, and as I said before, actually in, you know, linear optical systems, we can actually, 00:29:57.000 --> 00:30:01.960 d can actually get up to a million, okay, so this is like a million-dimensional RRPD, 00:30:04.200 --> 00:30:10.200 okay, so, so, so just a summary, okay, if we have, you know, d-dimensional like PDE, if we want to 00:30:10.200 --> 00:30:15.400 simulate this with analog quantum devices, then the quantum cause is linear, okay, in dimension, 00:30:15.960 --> 00:30:20.120 and we don't need to discretize anything, okay, we didn't, you know, do any numerical tricks, 00:30:20.120 --> 00:30:25.960 okay, this is exact mapping, if we have, I mean, I didn't show you this, but we could also map this 00:30:25.960 --> 00:30:32.760 onto qubit-based systems, okay, and in many cases these can also be efficient, but whereas classically 00:30:32.760 --> 00:30:37.400 we see they either have the course of dimensionality or you're really forced to do a discretization, 00:30:37.400 --> 00:30:45.000 okay, and, you know, where is the quantum, okay, part in all of this, well, so this is actually 00:30:45.000 --> 00:30:50.680 important also for foundational type questions, because we can actually see it's so easy to map, 00:30:50.680 --> 00:30:56.120 you know, any linear PDE into our quantum system, we can actually see which parts of the PDE, 00:30:56.120 --> 00:31:00.520 okay, visual PDE maps to which part of the Hamiltonian, and this is something that you cannot 00:31:00.520 --> 00:31:06.760 see if you, you know, discretize and scramble everything up into lots of gates, okay, so 00:31:07.560 --> 00:31:12.520 all the structure of the PDE is completely destroyed, okay, we can't, you know, see that, 00:31:12.520 --> 00:31:16.680 but here it's very, very clear, and what we do know is that in quantum systems, for example, 00:31:16.680 --> 00:31:20.360 in continuous variable quantum systems, there are certain types of gates which, you know, that, 00:31:20.360 --> 00:31:24.920 you know, may not be classically simulable, okay, and we know that some which are, so we can actually 00:31:24.920 --> 00:31:29.000 see which type of PDEs are in fact classically simulable, even though we're performing on a 00:31:29.000 --> 00:31:35.960 quantum device, and which ones are likely, maybe not so, okay, all right, so, like this little 00:31:35.960 --> 00:31:40.120 summary table, so we, you know, we have lots of examples of PDEs that we can do, the Louisville 00:31:40.120 --> 00:31:45.400 equation, heat equation, Fokker-Planck, Black-Scholes, a wave equation, a Maxwell's equation, okay, 00:31:45.400 --> 00:31:49.640 and so on, and, you know, we can also apply to lots of different, you know, boundary conditions, and so on, 00:31:50.440 --> 00:31:55.560 and I don't say here, but we can also use this for linear algebra problems, okay, preparation of ground 00:31:55.560 --> 00:32:00.040 states, which is important for optimization problems, and preparation of thermal states, which is 00:32:00.040 --> 00:32:03.400 actually important in machine learning, okay, if you, you know, think about quantum and Boltzmann 00:32:03.400 --> 00:32:07.960 machines, and so on, for generative learning, okay, so there are also lots of applications there, 00:32:09.000 --> 00:32:15.240 okay, and, you know, there are a lot of benefits, okay, to this, because here, as you see, we take 00:32:15.240 --> 00:32:19.880 advantage of this exact mapping, so that we don't need to perform any discretization, okay, and we 00:32:19.880 --> 00:32:25.320 have real quantum systems that naturally are continuous, okay, so we can have, you know, good 00:32:25.320 --> 00:32:30.120 specialized devices, right, that can simulate these PDEs without resorting to universal 00:32:30.840 --> 00:32:36.520 quantum devices before they're available, okay, we can also, we don't also, don't need to discretize 00:32:36.520 --> 00:32:41.880 in time, okay, but also the formalism is flexible enough that if you want to use qubits, you can, 00:32:41.880 --> 00:32:47.800 okay, if you want, and in fact, we're developing a software that allows you to, well, it's a 00:32:47.800 --> 00:32:53.880 dedicated for, you know, PDE, like using our method for simulating PDEs on classical devices, well, 00:32:53.880 --> 00:32:59.560 it's really simulating the quantum dynamics on classical devices, okay, so, and here, we can use 00:32:59.560 --> 00:33:04.920 discretization, and it avoids a lot of the complications, also, of, you know, techniques 00:33:04.920 --> 00:33:08.680 that you'll be familiar with, and, you know, classical numerical methods, like if you want to 00:33:08.680 --> 00:33:12.360 use matrix inversion methods, then you know, you need to know condition numbers and all of that, 00:33:12.360 --> 00:33:16.840 blah, blah, blah, but here, we don't need to worry about that if we stick with a continuous formalism, 00:33:16.840 --> 00:33:23.400 okay, yeah, and again, the cost is in D in terms of number of modes, not exponential, okay, in D, 00:33:24.680 --> 00:33:27.880 and again, you know, there are, you know, lots of opportunity for more, you know, 00:33:27.880 --> 00:33:35.000 foundational exploration on what is quantum and what is not, okay, all right, so, since I have some 00:33:35.000 --> 00:33:40.040 time, like as well, so, as I talked about here with short ingorization, okay, so, where we, you know, 00:33:40.040 --> 00:33:45.560 added one dimension into our classical system, and now, we can actually use, you know, quantum 00:33:45.560 --> 00:33:51.240 dynamics to simulate this classical dynamics, but we have other tricks, okay, as well, where, 00:33:51.240 --> 00:33:55.880 you know, adding a dimension really helps, okay, so that problems become simpler by lifting to a 00:33:55.880 --> 00:33:59.880 high dimension, okay, because remember that in quantum simulation, okay, because we don't have 00:33:59.880 --> 00:34:06.520 this cos of dimensionality issue in the same way, so, if you're adding, you know, k extra dimensions, 00:34:06.520 --> 00:34:11.640 you don't need to have an exponential cost in k, okay, that you would need to pay classically, 00:34:12.440 --> 00:34:16.680 so, we have, you know, other tricks that we use, okay, but adding this extra dimension include, 00:34:16.680 --> 00:34:22.680 if we have non-autonomous systems, we actually show how linear non-autonomous systems can become 00:34:22.680 --> 00:34:27.720 linear autonomous systems, okay, by adding an extra dimension, or maximum two, okay, 00:34:28.760 --> 00:34:35.560 before quantum systems one, and also, if we wanted to uncertainty, uncertain problems, uncertain PDEs, 00:34:36.120 --> 00:34:42.440 by adding some extra dimensions, we can actually make them deterministic problems, okay, 00:34:43.400 --> 00:34:48.360 and for certain types of non-linear problems, okay, for certain kinds of non-linear PDEs, 00:34:48.360 --> 00:34:53.560 by adding some dimensions, we can actually make them completely linear, and then solve for this, 00:34:53.560 --> 00:34:59.640 okay, as well, so, again, all these methods of increasing the dimension, now, they can make the 00:34:59.640 --> 00:35:04.120 problem simpler, but classically have to pay this exponential cost, but now, if we can map them onto 00:35:04.120 --> 00:35:10.280 the quantum system, we don't, okay, so, that allows us to use these tricks, okay, so, this is the idea, 00:35:10.280 --> 00:35:15.320 okay, so, usually, like classically, you want to turn a high dimensional problem into a low 00:35:15.320 --> 00:35:19.480 dimensional problem, okay, but for quantum, we have a little bit of a lead way, where we can have a 00:35:19.480 --> 00:35:23.800 few extra dimensional problems without too much extra cost, quantum mechanically, so, we can kind 00:35:23.800 --> 00:35:27.640 of go the other way, okay, so, how to turn low dimensional problems into one that's slightly 00:35:27.640 --> 00:35:34.200 high dimensional, okay, to make our lives a bit easier, so, one of these methods is, you know, 00:35:34.200 --> 00:35:39.720 for non-autonomous PDEs, how do we turn linear non-autonomous PDEs into linear autonomous PDEs, 00:35:40.280 --> 00:35:45.160 okay, and I think there's also an interesting trick, you know, classically, and this is really 00:35:45.160 --> 00:35:50.120 about adding an extra clock, okay, so, for example, we have a quantum dynamics on the left-hand side, 00:35:50.120 --> 00:35:55.080 which, you know, is non-autonomous, so, your Hamiltonian is dependent on time, what you can 00:35:55.080 --> 00:36:01.160 do is that you can add an extra dimension, okay, we can call S, which acts as like a clock, okay, 00:36:01.160 --> 00:36:07.480 so, and if, in fact, when you think about what time is, okay, there's no actual time dimension, 00:36:07.480 --> 00:36:12.680 it's like the way that we represent time is always through space, okay, if you're looking 00:36:12.680 --> 00:36:17.080 at an analogue clock right there that's using angular degrees, spatial degrees of freedom to 00:36:17.080 --> 00:36:21.240 represent time, okay, if you're looking at a digital clock, that's, you know, again, digital 00:36:21.240 --> 00:36:26.280 spatial degrees of freedom, okay, and this is really captured here, where you use S as a continuous 00:36:28.200 --> 00:36:33.720 clock, where, you know, the position, okay, along the, yeah, so, if you're reading the position, 00:36:33.720 --> 00:36:37.880 okay, along the clock, that's actually telling you the time, and you need to tell the system 00:36:37.880 --> 00:36:42.840 that you're actually using this, okay, time, that time actually moves, and this is why you have, 00:36:42.840 --> 00:36:49.960 you know, DWDS, remember, DDS is like, or derivative is like momentum, okay, and this is actually 00:36:49.960 --> 00:36:54.520 telling you as your time goes, it actually shifts, okay, so, by adding one dimension, 00:36:54.520 --> 00:37:01.320 you're turning the system on the left-hand side into a PDE on the right-hand side that looks just 00:37:01.320 --> 00:37:07.480 like your original PDE, except you're adding a momentum, okay, to this clock, okay, so this is 00:37:07.480 --> 00:37:14.280 the DWDS, and your time T has become S, because it's now transformed into this spatial degree of 00:37:14.280 --> 00:37:19.640 freedom, okay, so this is by adding one extra dimension, and note that this is very different 00:37:19.640 --> 00:37:25.400 to the way that people normally introduce clocks in here, okay, so, if you introduce it without 00:37:25.400 --> 00:37:29.320 adding this momentum operation, then you're not really telling the clock what to do, okay, 00:37:30.120 --> 00:37:34.840 so in that sense, you're actually solving a coupled system, okay, of your clock DS, 00:37:35.880 --> 00:37:43.320 your clock system S, as well as your system U, but there, this coupled system is non-linear, 00:37:43.320 --> 00:37:49.080 okay, but whereas if you tell the clock to move, okay, in your original PDE, then it doesn't become 00:37:49.080 --> 00:37:54.440 a coupled system, it becomes a single PDE, okay, that actually already has your time embedded in 00:37:54.440 --> 00:38:01.400 there, okay, and that's why it's linear, all right, so now, so this is by adding one dimension, you 00:38:01.400 --> 00:38:08.760 turn a time dependent, a non-autonomous Schrodinger equation, okay, into an autonomous linear equation, 00:38:08.760 --> 00:38:14.840 okay, in green on the right-hand side, so if you started off, then with any linear PDE, we know from 00:38:14.840 --> 00:38:21.000 Schrodingerization, right, that we can actually change into a Schrodinger-like system by adding 00:38:21.000 --> 00:38:26.440 one dimension, right, so which means that if you want to map any non-autonomous linear system into 00:38:26.440 --> 00:38:31.480 an autonomous linear system, maximum, you need to add in two dimensions, okay, and that's sufficient, 00:38:33.000 --> 00:38:38.680 you know, in terms of implementation, it's actually also really simple, okay, so we prepare, 00:38:39.240 --> 00:38:43.800 you know, the initial condition of our system, and again, we have an extra mode, and this extra mode 00:38:43.800 --> 00:38:48.600 is really our clock, okay, and we initialize this clock, and you know, and that notation just means 00:38:48.600 --> 00:38:56.040 we initialize at zero, okay, of our clock, and then you see the evolutions e to the minus i h bar t, 00:38:56.040 --> 00:39:02.600 where now this h bar itself does not depend on time anymore, okay, so whatever your time dependence 00:39:02.600 --> 00:39:08.600 was, you're actually changing it to an operator s hat, which is your clock, okay, but now you have 00:39:08.600 --> 00:39:14.680 to add in, okay, so this h bar is, you know, h of s hat plus this momentum operator, which now moves, 00:39:14.680 --> 00:39:19.320 okay, your time, and in the end, what we do is that this with this extra mode, we don't need to 00:39:19.320 --> 00:39:24.200 make any measurements, we just throw it away, okay, and after throwing away the state that we have in 00:39:24.200 --> 00:39:31.480 the end, we can guarantee that to be very close, okay, to the state that we want, okay, so this is 00:39:31.480 --> 00:39:37.160 a very simple protocol, okay, for doing this, yeah, and then we also have some, you know, other methods, 00:39:37.160 --> 00:39:42.440 again, you know, you know, seeing how, you know, physics really helps with this, so suppose we want to 00:39:42.440 --> 00:39:50.520 solve uncertain PDs, okay, so what do we do there? So in our PD problem, it means, what we mean by 00:39:50.520 --> 00:39:56.360 uncertain PDs, we mean the coefficients of our PDs are stochastic, okay, so they depend on stochastic 00:39:56.360 --> 00:40:01.160 variables, which are sampled from some stochastic distribution, and one way of solving for these is 00:40:01.160 --> 00:40:04.840 that, you know, you're doing the sampling, okay, so you're getting different coefficients each time, 00:40:04.840 --> 00:40:09.240 and each time you get a different coefficient, you solve this PD, but then you need to solve 00:40:10.120 --> 00:40:14.120 many different PDs, okay, like, because each time you have some different coefficients, 00:40:14.120 --> 00:40:18.120 and then you take some ensemble average, okay, maybe get some statistics from that, 00:40:18.120 --> 00:40:21.960 but that actually means that if you have to sample many times, or if you have, you know, 00:40:21.960 --> 00:40:26.920 a very large number of stochastic variables, you need to solve, you know, many different PDs, okay, 00:40:26.920 --> 00:40:33.240 and that's not very efficient, okay, so the question is, is there a way of directly capturing 00:40:33.240 --> 00:40:39.960 these ensemble average quantities without solving the PD so many times? Okay, and there's one really 00:40:39.960 --> 00:40:44.600 beautiful way, I think, where, you know, physics really comes into this, and this is what we call 00:40:44.600 --> 00:40:50.360 quantum stochastic Galerka method, okay, and this is how it works. So suppose we start off with a 00:40:50.360 --> 00:40:55.960 very simple convection equation, okay, of du dt, where, you know, c of j are, you know, you can 00:40:55.960 --> 00:41:02.200 think about that as your speed, okay, and your z's are your stochastic coefficients, whereas your x's, 00:41:02.200 --> 00:41:07.240 you know, that's just space, okay, and now let's suppose that your stochastic variables, 00:41:07.240 --> 00:41:13.160 they're sampled from some Gaussian distribution, okay, which is very natural, okay, so then what 00:41:13.160 --> 00:41:22.680 we can do is that we can expand our solution u, okay, into a discrete, okay, well, an infinite sum, 00:41:22.680 --> 00:41:28.760 where little n here are natural numbers, okay, and they're expanded in terms of polynomials, 00:41:28.760 --> 00:41:34.440 okay, where u of n are the coefficients to the polynomials p of n, and it turns out that if we 00:41:34.440 --> 00:41:41.480 associate n with particle number, okay, so say for, you know, performing an optical experiment to 00:41:41.480 --> 00:41:46.840 simulate this, and then we associate, you know, these natural numbers n with, you know, the 00:41:46.840 --> 00:41:50.200 particle number, so meaning that if we're making measurements on these systems, our particle 00:41:50.200 --> 00:41:56.680 detector, right, you know, is it, you know, turning on when n equals to 0, 1, 2, 3, okay, so if we 00:41:56.680 --> 00:42:02.280 associate n with that, and we associate, you know, x with, you know, the position quadratures, then it 00:42:02.280 --> 00:42:08.280 turns out that the inner product of these actually gives us Hermite polynomials, okay, so which means 00:42:08.280 --> 00:42:12.920 that mathematically, if you associate n with, you know, this real physical, okay, particle numbers, 00:42:13.480 --> 00:42:20.520 then just by making particle number measurements, we can actually naturally take into account 00:42:20.520 --> 00:42:29.320 this Hermite polynomial, okay, way of expanding, okay, our solutions, okay, so, and then it turns 00:42:29.320 --> 00:42:35.000 out that if we want ensemble average of our solutions, okay, we actually don't need the 00:42:35.000 --> 00:42:40.360 full set of solutions, so all we need are, you know, the zeroth coefficient, okay, u of 0, 00:42:40.920 --> 00:42:46.360 okay, of our solution, and to get the variance, we need the variance of, you know, u n squared, 00:42:46.360 --> 00:42:50.600 okay, of these coefficients, but the maximum n doesn't have to be very high, okay, because for, 00:42:50.600 --> 00:42:55.800 you know, smooth enough functions, you know, and, you know, can go, you know, be four or five, 00:42:56.360 --> 00:43:02.040 and which means that, you know, physically when you're making, doing an experiment, all you need 00:43:02.040 --> 00:43:06.040 to do is, you know, when doing these experiments, when you're making photon number counting 00:43:06.040 --> 00:43:10.680 measurements, you just need to calculate the probability to which you don't get any photons 00:43:10.680 --> 00:43:16.200 on one side, okay, and that will tell you the ensemble average, and then you only need to 00:43:16.200 --> 00:43:20.680 measure up to, you know, photon number up to four or five, and that's enough to actually tell you 00:43:20.680 --> 00:43:26.120 the variance of the ensemble solution, okay, without you sampling, okay, so many times, okay, 00:43:27.400 --> 00:43:33.800 so this is one method, okay, for uncertainty quantification, yeah, and if we have, you know, 00:43:33.800 --> 00:43:40.840 L stochastic variables, okay, we can also do this, and this is another method, an early one that we 00:43:40.840 --> 00:43:45.400 have, and this is related to the warped phase transformation, and again, if we have, you know, 00:43:45.400 --> 00:43:51.720 heat equation as an example, and assume, now we have A of z, okay, as a diffusion coefficient, 00:43:51.720 --> 00:43:57.080 but now z is the stochastic variable, and now we have a warped phase-like transformation, where 00:43:57.080 --> 00:44:02.760 instead of, you know, e to the power minus, you know, p, it's, you know, minus A of z, okay, p, 00:44:02.760 --> 00:44:11.160 and now if you ask what equation this, you know, your w, okay, function obeys, now the stochastic 00:44:11.160 --> 00:44:17.000 term actually disappears, okay, and now we can actually simulate this and compute the ensemble 00:44:17.000 --> 00:44:24.040 average, okay, directly, okay, and we also have quantum algorithms for those, and finally, I won't, 00:44:24.040 --> 00:44:30.600 you know, say any details about this, but you can, you know, come and ask us later, so how about 00:44:30.600 --> 00:44:35.000 non-linear ods and pd's, okay, because everything I mentioned here is linear, and that's because 00:44:35.000 --> 00:44:38.920 Schrodinger's equation is linear, okay, so if you have a non-linear equation, then you should have a way 00:44:38.920 --> 00:44:43.640 of turning, right, your non-linear equations into linear equations before we can apply our method. 00:44:45.240 --> 00:44:51.400 So two examples of these, so for those of you who work in control theory, you're very 00:44:51.400 --> 00:44:56.600 interested in Hamilton-Jakobi equations, and, you know, for Hamilton-Jakobi-like equations, 00:44:56.600 --> 00:45:04.360 for hyperbolic non-linear equations, then there is a method of, you know, adding some dimensions to 00:45:04.360 --> 00:45:09.160 make them completely linear, okay, and in fact, if you have non-linear Hamilton-Jakobi equations, 00:45:09.160 --> 00:45:14.120 you need to, you know, add in, you know, twice, okay, the dimension, okay, roughly, until it becomes 00:45:14.120 --> 00:45:20.520 a linear Louisville-like pd, okay, and if you had, you know, m initial conditions, you can actually 00:45:20.520 --> 00:45:25.400 make it into a Louisville equation with, you know, a single initial condition, and once we have that, 00:45:25.400 --> 00:45:29.640 you know, pd, then we can actually put this into a quantum algorithm and actually output observables. 00:45:31.160 --> 00:45:35.800 And, you know, what happens when you have more general, okay, pd's? Well, we can discretize these 00:45:35.800 --> 00:45:40.600 into systems of ODE's, and we know that, you know, for you guys who work on, you know, particle 00:45:40.600 --> 00:45:46.520 methods, okay, you know, it's easy to see how if you have, you know, a dynamical system, okay, with, 00:45:46.520 --> 00:45:50.840 you know, lots of particles, then if you look at the probability distribution, then that obeys, 00:45:50.840 --> 00:45:55.880 you know, a linear pd, okay, even though your particles, they actually obey non-linear ODE's, 00:45:55.880 --> 00:46:02.280 okay, so we can do this, but actually this is not so efficient, okay, so we don't get much quantum 00:46:02.280 --> 00:46:11.960 advantage actually out of this, okay. Yeah, so the basic message, okay, in all this program of using 00:46:12.840 --> 00:46:18.440 quantum systems to simulate our pd's is that, you know, whenever we have some, you know, problems 00:46:18.440 --> 00:46:22.760 that we want to deal with, okay, we realize we actually need to find methods where we increase 00:46:22.760 --> 00:46:29.960 the dimension, not by very much, by a little bit, and we can actually solve lots of problems, okay, 00:46:29.960 --> 00:46:35.560 with, you know, on these quantum devices, okay, that don't have this cursor dimensionality. 00:46:35.560 --> 00:46:39.960 And this is like a small summary, okay, of the work we've been doing the last couple of years, 00:46:40.760 --> 00:46:45.400 so on linear pd's, you know, non-autonomous systems, on certain pd's, non-linear pd's, 00:46:46.200 --> 00:46:50.760 systems on non-linear ODE's, certain boundary conditions, and we've recently also been 00:46:50.760 --> 00:46:56.280 doing some explicit circuits for these, so that later we hope to share with you an open source 00:46:56.920 --> 00:47:01.640 kind of program where, you know, you can actually, you know, use it yourself and try it out and 00:47:01.640 --> 00:47:05.800 actually be able to simulate these. We can also use these for linear algebra, ground state, and 00:47:05.800 --> 00:47:09.400 thermal state preparation, okay, it wants as well, so if you want to ask me about that later, 00:47:09.400 --> 00:47:13.160 you can. Yeah, and thank you, and welcome to visit us in Jatong. 00:47:36.200 --> 00:47:42.200 So, 00:47:42.760 --> 00:47:46.840 this is a generic problem for all quantum algorithms that you can't really interrupt, 00:47:46.840 --> 00:47:52.120 okay, like in between to see. So, for, like, usually what we do, like, at least in the first 00:47:52.120 --> 00:47:56.280 stages when performing experiments and these things, you know, we can actually try, you know, 00:47:56.280 --> 00:48:02.120 simulate, like, classically, like, what happens, okay, and then, you know, so using, you know, 00:48:02.120 --> 00:48:06.840 classical control for quantum problems, right, so it's in that, because this is what, you know, 00:48:06.840 --> 00:48:10.600 when you're doing any kind of experiment, that's what you need to do anyway, so we're really 00:48:10.600 --> 00:48:15.080 treating this on, you know, at the experimental stage, right, rather than the computational stage 00:48:15.080 --> 00:48:19.720 as to, you know, how we would error correct, like, on its own, but people, so for, you know, 00:48:19.720 --> 00:48:25.640 discrete systems, people do work on error correction protocols, like, for these, where, you know, you 00:48:25.640 --> 00:48:29.560 make partial kind of measurements, or they call, you know, mid-circuit measurements, and then you 00:48:29.560 --> 00:48:33.080 can, you know, there are things you can do, but at this stage, we're not thinking about that. 00:48:33.720 --> 00:48:34.220 Yeah. 00:48:47.720 --> 00:48:55.240 Oh, yeah, so, of course you can, yeah, so, but the idea is that we want to start, you know, 00:48:55.240 --> 00:48:59.320 as simply as possible, because actually, the more systems you have, the more complicated, 00:48:59.320 --> 00:49:05.720 actually, the controls become, right, so, and the idea here is that, you know, before, you know, 00:49:05.720 --> 00:49:10.440 these qubit, you know, type of systems, or, you know, these inhomogeneous systems, where you have, 00:49:10.440 --> 00:49:15.800 you know, or heterogeneous systems coming in, you know, we want to demonstrate they're already, 00:49:15.800 --> 00:49:19.560 they're interesting PDEs you can simulate even on the simpler systems, but of course, 00:49:19.560 --> 00:49:23.960 you know, the simpler systems don't work, then, you know, later on, we want to go to these hybrids. 00:49:23.960 --> 00:49:24.460 Yeah. 00:49:29.320 --> 00:49:43.400 Oh, actually, every quantum system, every quantum system are actually, in fact, qubits, 00:49:43.400 --> 00:49:49.720 right, there are no qubits, actually, in nature, right, we make them qubits, right, and this is, 00:49:49.720 --> 00:49:55.720 you know, similar to, you know, in classical information, zero and one doesn't really exist, 00:49:55.720 --> 00:49:59.480 right, even in our brains, when we think about zeroes and ones, these are analog signals, 00:49:59.480 --> 00:50:03.080 right, on neural networks, you guys who work with machine learning, right, also, there's nothing 00:50:03.080 --> 00:50:08.120 digital, I mean, in the end, you know, you have an analog signal, and then you have to select, 00:50:08.120 --> 00:50:14.120 okay, you know, if it goes to zero or one, okay, so, and this is how, you know, so people try very 00:50:14.120 --> 00:50:19.000 hard to take, you know, naturally continuous systems and make them discrete, because, you know, 00:50:19.000 --> 00:50:24.120 as we do arithmetic and things like that, we try, we like to think in a discrete way, right, but 00:50:24.120 --> 00:50:29.640 we're arguing that for problems like PDEs, which are naturally continuous, it makes more sense, 00:50:29.640 --> 00:50:33.720 at least at the beginning, right, to use, you know, natural continuous systems, because actually 00:50:33.720 --> 00:50:37.800 controlling them, you know, controlling quantum systems is very difficult, and to make the 00:50:37.800 --> 00:50:42.840 discrete versions, as well as making them all coordinate in such a, you know, very complicated 00:50:42.840 --> 00:50:47.560 way, and you have to use millions of gates, actually, even with the simplest PDEs, so with 00:50:47.560 --> 00:50:52.280 a heat equation, for example, you need millions of these gates, but whereas, you know, with, 00:50:52.280 --> 00:50:56.440 you know, these type of methods, we just need one, basically, but, you know, controlled in a nice 00:50:56.440 --> 00:51:10.440 way, right, and that's the comparison. Okay, thank you again.