## Quantum Advantage Theory and Practice with Di Fang

Download MP3The New Quantum Era, a podcast by Sebastian Hassinger. And Kevin Rowney.

Sebastian Hassinger:Welcome back to another episode of the podcast. This is another episode I recorded during my week at Rensselaer Polytechnic Institute for the launch of the IBM System 1 that they deployed on campus. I hosted a panel discussion on the 1st night and then I used a podcast studio in the theater in the event venue at Rensselaer that they provided to me, which was very nice of them, to grab various speakers for these 1 on 1 interviews. A couple weeks ago we ran, an episode that was a discussion with Professor Lin from UC Berkeley. Today we actually have an interview with a colleague of Professor Lin's, Professor Di Feng, an assistant professor of mathematics at Duke University and a member of Duke Quantum Center.

Sebastian Hassinger:She was actually a postdoctoral fellow at the Simons Institute For the Theory of Computing hosted by Professor Lin and Professor Vazirani. So we talk a little bit about her work in the, theory of quantum information science, the theoretical advantages of quantum information science algorithms, what the provable sort of basis for those advantages are and may be, and also the need for balancing sort of that theory side of things with practical side of quantum information science. It's a great discussion. We also talk a little bit about her involvement in the Google XPRIZE for quantum applications as well, which sort of brings all of her interests in the theoretical and practical advantages of quantum computing together. So let's take a listen.

Sebastian Hassinger:I've got a very special guest today. We're here at RPI at the ribbon cutting of the quantum computer that they've installed on campus. And I'm joined by Professor Difang, who's an associate professor of mathematics at Duke. Is that correct? Yes.

Sebastian Hassinger:Excellent. Oh, assistant. Okay. I knew it started with an a. Assistant professor at Duke.

Sebastian Hassinger:And thank you for joining me. Could you start by just introducing yourself and and talk a little bit about how you got into quantum information science?

Di Fang:Of course. First of all, thank you so much for inviting me. It's my great pleasure to join the podcast. My name is Di Feng, and I'm a mathematicians at Duke University. I, I'm also a principal investigator at the Quantum Center.

Di Fang:My research mainly regarding the theoretical aspects of quantum computing as well as quantum simulation. In particular, I do numerical analysis to quantum algorithms.

Sebastian Hassinger:Excellent. Excellent. And yeah. So, I mean, that's a really interesting area to me because, you know, in part, there's this kind of tension between provable efficiency or provable advantage in quantum and then the sort of open ended exploration of what can we do with these machines. So how did you sort of start to get interested in in the sort of the algorithm side of things?

Di Fang:Yeah. That's a great question. So, actually, I was trained as a mathematician. My PhD is in mathematics. I do have some exposure to quantum problem.

Di Fang:I have been working on my background is applied and numerical analysis for differential equations. I've been working on Schrodinger equations, neither better dynamics of an approximation no longer holds, how to prove various bounds, things like that. It was not until I went to, Berkeley as a more revisiting assistant professor, a postdoc position per se.

Sebastian Hassinger:Right.

Di Fang:And, it's a independent postdoc position. At the time, it is I think a interesting starting point for me was that I was actually teaching a, undergraduate high level undergraduate, proof based audio differential equation class. So in my class, I motivate different levels of physics, all can be described in differential equations starting from, you know, the Schrodinger equations and classical dynamics. And then we have statistical dynamics, and then we have fluid dynamics. There's different regimes of the dynamics one can consider in all different levels of physics.

Di Fang:And then I said, I'm very much interested in quantum, and so also for so this was like the motivation for my class. And I had a very interesting and super smart undergraduate student, math major. He came to me and he was like, Dee, you mentioned that quantum is where you you really love. And I'm, attending this another quantum class, which is a computer science, class. And then he draw me.

Di Fang:He want to sort of, you know, undergraduate students sometimes want to impress their instructors. Right. So he draw on my board a quantum circuit. And he was like, this is a super cool thing that I I learned in my other class. Since you work on quantum, can you give me some more insights Right.

Sebastian Hassinger:Right.

Di Fang:From a mathematical perspective? I was like, super cool, but I have never seen a Right. A quantum circus before. And then I also heard about this class from various different, revenues like, professor Biddy at our math department and other, folks, graduate students, and then I decided to go and audit it class. Right.

Sebastian Hassinger:Right.

Di Fang:The class was taught by Umesh Vazirani at our CSC

Sebastian Hassinger:department. A pretty famous name.

Di Fang:Exactly. And that's precise precisely the place, I guess, people want to start. But at the time, I really have no idea

Sebastian Hassinger:what is

Di Fang:quantum computing. I think Umesh class really brought me into this.

Sebastian Hassinger:So when you said you were interested in quantum in your class, you meant from, like, a purely mathematical perspective.

Di Fang:Right? Right? I meant, like, maybe classically, say, why Bronheimer holds? What condition Bronheimer no longer holds? If it no longer holds, what kind of bonds should we do in order to crack, and what kind of algorithms.

Di Fang:Most of the things we do are, say, in, quantum chemistry, either you do electronic structure problems or you do the dynamics. Once you have the potential energy services and non idiotic dynamic, transitions, what can you do with the nuclear part? So I was trying to justify mathematical understanding of all these different quantum capture treatment, and that was what have been very much interested.

Sebastian Hassinger:Right.

Di Fang:As you see, this is a quantum perspective, but definitely has nothing to

Sebastian Hassinger:do with quantum computing. Right. Yeah. Right. And I mean in that in that sort of applied mathematics area, is there is there sort of like you know, like the the mathematical, modeling that is exact but very, very difficult to work out and then sort of the the class are much more approximation based.

Sebastian Hassinger:Is that which

Di Fang:is the Exactly. So the motivation this is actually the precise motivation I will give you my proof based audio and differential equation class is that we have different levels of physics. The fundamental level is really the the most first principle per se is our Schrodinger equation.

Sebastian Hassinger:Right.

Di Fang:And then we have our particles which we see and the ourself, like our phone and so on and so forth. We give it a force. It have a movement. This is Newtonian's second, second law. And, so and then if we look at, much larger scales, we have the fluid dynamics.

Di Fang:How do we send our plane on the, on the sky and so on

Sebastian Hassinger:and so forth.

Di Fang:And then in between, there's the statistical mechanics. That's what how we describe plasma and so on. Those were so all of these, all of these different levels of physics, there are very important mathematical equations, differential equations. And, the the quantum level is a short integer equation. And the the, Newtonian level is the, ordinary differential equations and the the the, the classical dynamics Right.

Di Fang:And then the dynamical systems, and then the statistical mechanics is both my equations. And the, last level, we have the Nabors Stokes or the Euler's equations.

Sebastian Hassinger:Okay.

Di Fang:And the interesting thing is that if you start from the first principle from the mathematical viewpoint, all of these equations can be linked together by some parameters, sending some parameter either equals to 0 or equal to infinity, and their limiting equation will be the next level.

Sebastian Hassinger:Okay.

Di Fang:For example, from the Schrodinger equation, if we send the Planck constant goes to 0, we have the classical dynamics. And then from classical dynamics, if we send the number of particles goes to infinity, we will have the Boltzmann equation. And when we have send the Nusselt number goes to 0, we will get the fluid equations. All of these are very much interesting mathematical problems, like which sense of the convergence is host to and what are the conditions you can justify. And these are, like, very much very interesting mathematical problem that people in differential equations have been working on.

Sebastian Hassinger:Right. Right. And then so so you go to a mesh's class and you start to see sort of the quantum information science approach to it. And that's I mean, is it because it's applying sort of the the the linear algebra and the the power of the Hilbert space in in quantum computation that you're able to do more precise calculate or or is it the linkages between those 2 those levels that

Di Fang:That's a great question. So I think what fascinates me a lot is, because I have this picture from a mathematical viewpoint. We know that these different theories are linked together. So if one were able to solve the Schrodinger equation, many body Schrodinger equation for a molecule, then pretty much you can solve a large class of these problems. What fascinates me the most about quantum computing is this potential power of us directly simulating a Hamiltonian simulation or the full molecule.

Sebastian Hassinger:Right.

Di Fang:We can deal with electron, nuclei fully coupled molecule, and that is precisely a Hamiltonian simulation problem. Right. And this is what fascinate me so much because classically, what we always do is to decompose the electrons and the nucleolides into 2 separate procedures, assuming both primary 2 and nucleolides moving fairly slowly, and then we try to solve these biggest problems on in chemistry and united.

Sebastian Hassinger:Computation chemistry, I mean, the larger the the molecule, the the problem space, the more you're approximating and the further off you may be from an accurate solution. So if you're trying to link that up to those other higher level models, mathematical models, your your error bars are getting bigger and bigger.

Di Fang:Yeah. Exactly. Exactly. Exactly. Exactly.

Sebastian Hassinger:Potentially, if you if you exactly solve the Heisenberg equation for a large molecule, you could you could scale up from there and get better and better, sort of results from the model at larger scales.

Di Fang:Yeah. That's right. That's right. That's right. That's right.

Sebastian Hassinger:That's super interesting. Yeah. And, I mean, when you this is all, like, sort of in the abstract, what do you ever think about sort of what the, the practical implications of of that kind of of leap forward might be? I mean, you know, you think about quantum advantage, There's sort of a theoretical quantum advantage, and then there's a practical quantum advantage. Do do you think about maybe what the practical implications will be for for that type of of breakthrough?

Di Fang:Yeah. Sure. I I think, of course, from the hardware viewpoint, we definitely need a fault tolerant quantum computer very unfortunately in order to really demonstrate these theoretical advantages. From a theorist viewpoint, I by theorist, I mainly mean the, from a a algorithm viewpoint.

Sebastian Hassinger:Right.

Di Fang:I would say that maybe the theoretical road path, my personal theoretical road path, towards a theoretical quantum of advantage might be the following. I would say maybe 4 different levels. It doesn't mean that people should work on the next level until they figure out the previous level. This I think all the 4 different levels should be worked on simultaneously. So the first level, I would say, is very much theoretical computer science.

Di Fang:The, flavor is that given any computation problems that people want to ask, it can be anything. It can be your, optimization, your machine learning, and your, matrix multiple locations, and differential equations, and all different kind of things. And then we ask ourself that can we prove the upper bounds and lower bound for both classical algorithms and the quantum algorithms? And, in particular, say, the classical algorithm is there is, well, complexity class is interested in discussion. Is the, is the problem in PQP hard?

Di Fang:Then we have reasonable belief that this should be classically hard and quantumly easy. And, and also one can prove that given a computational problem, what is the lower bound that the the the at least if a classical algorithm needs to fulfill this cost and then what is the complexity lower bound.

Sebastian Hassinger:Okay.

Di Fang:And then we look at the quantum side. We want to find a quantum algorithm which we can prove a quantum upper bound within certain parameter. And then we start to ask ourselves, is there any separation between these 2? Is the separation exponential, super polynomial? Is it polynomial, or is there no separation at all?

Sebastian Hassinger:Right.

Di Fang:And then we I I think this is a first level, and then we will have some candidates. We can then look at the second level, which is for these candidates where there exist a quantum advantage or quantum separation, can we design the best possible algorithm, a fault tolerant algorithm

Sebastian Hassinger:as

Di Fang:a first step to match all of the possible parameters and not just be the, the the system size parameter, which is typically used in the first layer. It can be other parameters, say, the precision, the failure probability, and in the electronic the phase estimation problems, there can be overlaps and some others say mixing time and so on and so forth. So there are very similar parameters in the problem. We want to design the algorithm that has the best asymptotic param scalings in the sense that when these are either large or either small Right. And we have this big old notations, this is the best skating.

Di Fang:And we try to prove a quantum lower bound saying that this is the best one can do. And once you can match this upper bound and lower bound, then you can sort of declare victory at this level, and we have the optimal skating for tolerant quantum algorithm. And then I think the next level is maybe, many practitioner would be very much interested in is we for for the second level, the problem is typically over abstractions of a reality problem. Maybe we just say, okay, it's a s sparse matrix. It's a k local Hamiltonian.

Di Fang:But, from a practitioner's, perspective, what is the concrete h, the Hamiltonian? What is the concrete matrix? You know, a, say, optimization problem or in a physics problem. You want to look at maybe a particular molecule people are talking about, for MOCO, I think, and the Right.

Sebastian Hassinger:Yeah.

Di Fang:Different kinds of molecule. So what is for that particular Hamiltonian? It's not for a general Hamiltonian that is

Sebastian Hassinger:k local. Right.

Di Fang:For that particular hamiltonian and try different algorithm, try to cast out resource estimation per se.

Sebastian Hassinger:That's Fumoco just like, that's the, nitrogenase Yeah. Fixation. So is nitrogen fixation that Microsoft in particular has done a bunch of work

Di Fang:around Yeah. Yeah.

Sebastian Hassinger:That's right. Resource estimation of Yeah.

Di Fang:That's right.

Sebastian Hassinger:What it would take in terms of fault tolerant qubits

Di Fang:to Yeah. Exactly.

Sebastian Hassinger:Do that to solve that problem.

Di Fang:Exactly. Exactly. And then people got a number, like, how many gates we actually need are fault, fault tolerant quantum computer. And then, they try to bridge the gap between hardware and the theory Right. Which is definitely bring up the hardware awareness.

Di Fang:Right. Like, do we can we design our algorithm, say, you know, more robust way to certain features of the hardware and how to feed them and

Sebastian Hassinger:so on

Di Fang:and so forth?

Sebastian Hassinger:Or or find specific error mitigation techniques that work for your particular problem on this particular hardware. Right. That codesign kind of concept that comes up a lot. Yeah.

Di Fang:Yeah. Definitely. Definitely.

Sebastian Hassinger:And is that I mean, that's more of a, in in classical IT terms, that's more of like a hacking approach. Right? You're you're just trying to find, within the parameters that you have to work with. You're trying to find, tricks, basically, to to shortcuts or tricks that are gonna

Di Fang:Yeah.

Sebastian Hassinger:Squeeze more performance out of Yeah. What you are working with in terms of equation and the hardware.

Di Fang:Yeah. Yeah. Yeah. That's right. Maybe just one thing to mention that I found, fairly interesting may not be counterintuitive at all, but, is when we go from these optimal asymptotic scaling algorithms to the resource estimation.

Di Fang:In many cases, you will find that for a given problem, for a given procedure that you want to find, these the optimal scaling theoretical on paper algorithm may not give you the best number as

Sebastian Hassinger:Oh, really?

Di Fang:Outcome. Yeah. It really depends because these are based on the assumption that the system size is super large Right. Right. And the procedure is super small.

Sebastian Hassinger:But Right.

Di Fang:Say for chemical procedures and for actual system size, you may not always find that the best theoretical the optimal Right. Asymptotic algorithm gave you the best numbers.

Sebastian Hassinger:And that I mean, is it right that am I right in thinking that that in classical computing, there's been a similar sort of back and forth between heuristics and and, and, and, like, comply theoretical advantage. Right? I mean, there's been sort of algorithms that have been hit upon that work practically that took a long time to actually prove theoretically that there was any kind of efficiency there. Right? I mean, there is there I guess, is that just sort of assume that there's always going to be this back and forth between heuristics and theory?

Di Fang:Yeah. Definite that's a great question. So I will share my personal perspective on this. I feel like, well, people say before we have a super supercomputers and where our computation power before we have GPU and so on, people cares a lot about proving theories, especially, yeah, in the last century and we just started. We want to know that these are our guarantees.

Di Fang:We are currently really in quantum that stage, and that's why we care about proving bounds and so on and so forth. But I would think if one day we really have super large fault tolerant They work. Yeah. Yeah. No one well, people still very much they work.

Sebastian Hassinger:They work. Yeah. Yeah.

Di Fang:And no one well, people still very much care about crew prove, but they not that that at that that level, say, this big company, if it works, then great.

Sebastian Hassinger:It's like when the resource is constrained, it matters a lot to do all the theoretical groundwork. But once the resource is abundant, it's like, who cares? Just use it. Exactly. But if it works Yeah.

Di Fang:That would be my perspective.

Sebastian Hassinger:That's really smart. Yeah.

Di Fang:That currently, we are trying to prove this were, provable theorems and so on and so forth so that we have enough of a motivation so that we can get that available. We can really get a fault column in quantum computing large enough due to all

Sebastian Hassinger:these justification then

Di Fang:Yeah.

Sebastian Hassinger:In part. Yeah.

Di Fang:But then I feel like once we really get there and these maybe I will be out of job. Right. I know.

Sebastian Hassinger:That's what I was gonna say. I will

Di Fang:be very happy about that if we have fault tolerant fault. We have a fault tolerant quantum computer,

Sebastian Hassinger:and that's

Di Fang:the reason I'm auto I'm super happy about that.

Sebastian Hassinger:What do you do next, though?

Di Fang:Yeah. I can still well, I'm a

Sebastian Hassinger:straight way in the future. You don't have to worry about it now.

Di Fang:Yeah. Yeah. And and also, I think well, I'm mainly working on different mathematical tools with applications. I can go find the next, exciting applications. Yeah.

Sebastian Hassinger:So Well, and I think, you know, I mean, in part, it feels like the motivation behind the level of investment in quantum computing is because we there's a general awareness that we're kind of at the end of of what we can squeeze out of classical computing in terms of physical limits of the the chips and the density of the transistors and etcetera. Right? So, I mean, I feel like we've been in this very long period of abundance in computing resources. And if if it's quantum computing, if it's, neuromorphic computing, if it's thermodynamic computing, there's going to be, if anything, more and more sort of diversity in the way that we compute, going forward. So you're always gonna have some, new resource constraint to try to lay the theoretical groundwork around and try to to figure out how to best use the all these exotic kind of new resources that are coming on the on the stage.

Sebastian Hassinger:Right?

Di Fang:Yeah. Yeah. That's true. That's true. That's true.

Sebastian Hassinger:Really interesting. And you you mentioned, sort of the the the path forward. You sort of laid out a couple of different you know, like, looking at the theoretical, the purely theoretical sort of upper and lower bounds and then looking at the, the the sort of codesign or the optimizers. Are there other, areas that need, you think, more focus, more investment of time and and resources in this stage of of of where we are, you know, sort of in the earliest stages of logical qubits and fault tolerance?

Di Fang:Yeah. I think, I we also have discussions with some other folks at, this event, RPI. So I think, many of us feel like resource estimations current at this current stage should, be put more efforts into.

Sebastian Hassinger:Mhmm.

Di Fang:Because many of these, you know, we have a handful of theoretical exponential separation task including Hamiltonian simulations, Shor's algorithm for factory, and quantum linear system algorithms, and some of the potential quantum learning problems. And these problems, they already have, pretty much the near optimal scaling algorithms. The next question would really be, like, for concrete application and how do we what is the best in reality for these application algorithms? Different application can have different algorithms as a number. And I guess maybe you could mention that that's why Google is, is, putting $5,000,000 for their x price is for quantum application.

Di Fang:I find well, I'm very fortunate to be one of the judge on the panel, but Oh,

Sebastian Hassinger:cool.

Di Fang:Yeah. Yeah. I find that, the public's apportion say my friend who do not work on quantum computing at all, some of them heard about the news. Okay. Google is pulling $5,000,000.

Di Fang:This is how news, purchase, like, picture it.

Sebastian Hassinger:Right. Right.

Di Fang:Google pick, put in $5,000,000 to find the quantum applications. So my friends were messaging me was like they were freaking out. They were like, it doesn't mean that Google like, does it must because we absolutely have no applications for quantum computer. Otherwise, why a $5,000,000 cash price? I was like, no, no, no.

Di Fang:We know these are applications, but they were over abstractions, and these were not, you know, from a public's viewpoint, if I do not work on anything scientific, why should I care about Hamiltonian simulation? Why should I care about quantum linear system or, solvers?

Sebastian Hassinger:Like Right. Right.

Di Fang:We need something that is

Sebastian Hassinger:Grounding.

Di Fang:Yeah. Grounding that public can see that this is maybe, I don't want to pinpoint any particular application, but applications that is not on the certificate level, we have to write an equation to justify what it is to me.

Sebastian Hassinger:And and doesn't I mean, to back to your point about sort of, like, do you know, trying to figure out how to get the best performance from the hardware that you have. I mean, it feels like a real world problem is part of the forcing function. Right? That's part of the constraints that you have to work within. It's like, you know, this problem that industries, is facing or society is facing needs a faster solution or a more accurate solution, and those are kind of the success criteria that you need to work for towards you know, from an algorithmic perspective.

Di Fang:Right? Definitely. Definitely. Yeah. So far so, I work on a lot of Hamilton Hamiltonian simulation, and there were different levels, I think, people consider, say, in your error metric, you want to so so the in the first layer of the, the the road map that I I discussed, the upper bounds and lower bounds and whether it's BQP hard and so on and so forth, these are typically looking at the worst case scenario

Sebastian Hassinger:Right.

Di Fang:Which means you have a Hamiltonian dynamics, and then you look at the worst case observable and maybe the worst case, the, the the initial condition that you deal with. And then you try to pick the the bad guy Mhmm. And you try to prove this is very hard.

Sebastian Hassinger:Mhmm.

Di Fang:But, like, in reality, then the the when we go to the second level, we try to deal with generic problems. So for generic hamiltonian, not a particular hamiltonian, not the the worst case scenario. Can you prove a bound? And the the third there, I I would think this is, well, many of my research is regarding the bounds, but, what I prove a lot is regarding, so if I have a specific problem given at hand, I have a given initial conditions, I'm given observables, those problem typically has structures. Mhmm.

Di Fang:And then you can you don't have to deal with the worst case scenario. It can can improve your balance dramatically in many situations.

Sebastian Hassinger:But, I mean so, like, the reason you you sort of go for worst case is because that's where you can actually create the theoretical framework.

Di Fang:Yeah. That's right. That's right. That's right.

Sebastian Hassinger:As you said, you sort of set to 0 or to infinity.

Di Fang:Yeah. That's right. That's right. That's right. That's right.

Di Fang:That's right. That's right.

Sebastian Hassinger:That's so interesting. That's really interesting. So, I mean, is that the is that that's where your road map sort of leads to is is that, you know, the, the theoretical grounding and then working out sort of the the real world world world example, and then finding those, you know, those here those shortcuts of those heuristics that that actually lead to an application, so to speak.

Di Fang:Yeah. Yeah. Yeah. I would say that's the case. Yeah.

Sebastian Hassinger:It's very cool.

Di Fang:Yeah.

Sebastian Hassinger:So, like, you're, like, you're you're on the judge judging panel for the the Google XPRIZE, the quantum XPRIZE. That's great. By the way, when I I I was at CERN for the Open Quantum Institute, launch, just a couple weeks ago when when Google announced that that XPRIZE as part of the the Open Quantum Institute launch. So it's I think it's really terrific. As you said, it's sort of a a motivating kind of, and also communicating kind of tool.

Sebastian Hassinger:It's making it real for people. So what what, you know, since you're judging, you can't actually participate in any of those. So what sorts of projects are you working on over the, you know, the that you're looking to bring out results on in the next year or 2?

Di Fang:Yeah. So, I'm currently working on, I'm, I do, linear differential equations. And, in particular, short linear equation is a linear differential equation. So it's, one one particular case of that. So I have been working on, as I said, is typically when, people look at the judgment of the cost of algorithm, and the error metric is typically the worst case scenario.

Di Fang:Mhmm. I've you want to have a quantum circuits and you can prove that this is a cause and then it will work for arbitrary observables, arbitrary initial conditions. You just do not take those things into account.

Sebastian Hassinger:Right.

Di Fang:And, what I was working on is for given problems, what are the observables? What are the initial conditions? Say, typically, I can prove, in particular, I look at these unbounded Hamiltonian simulation problems. It sounds not a very common word, unbounded. What does unbounded even mean?

Di Fang:It's actually just the Schrodinger pde. So it just means that, in Schrodinger equation, we have momentum operators. These operators are, derivative operators, differential operators from l2 to l2, that's where the wave function naturally lives in. It is unbounded because you need the derivative to be bonded in order for after you take the Laplacian, the second derivative, it is sealed within a bonded, l two. So if you take the derivative of arbitrary l two function, the end product will not send you back to L2.

Di Fang:So it that's why if we look at these for unbounded Hamiltonian simulations, if we look at for arbitrary intrusion and arbitrary observables, you will always get the observe the operator non bound, which is the bound of the Hamiltonian operator, is infinity. And why do we even care? And the the reason is because if we look at, most of the quantum algorithm literature and all these Hamiltonian quantum most of these Hamiltonian quantum, Hamiltonian quantum algorithms will have at least a linear dependence on the operator norm of h.

Sebastian Hassinger:Okay.

Di Fang:So once the operator norm of h is infinity, then what does it even mean? Right? This is not a problem if we have so I was talking about this first quantization framework. This is not a problem if we look at the spin system and the the the the the many body Hamiltonian in the in, as a many body polys, the linear combination of many body polys, those will always be bounded. However, if we honestly look at the differential equations, those operators so first of all, what is the best truncations in the spatial degrees of freedom so that we can prove that with this truncation?

Di Fang:So after truncation, we can fuse the size of the Hilbert space, and this is the best truncation. This truncation will remain holds for the rest of the evolution time. This is very important because in many cases, say, in chemistry application and so on and so forth, people will just, make this number truncation n as a parameter. And the the end cause of the problem will have n as a parameter enter in. However, n is really coming from the error and then I choose n, which is how I truncate it.

Sebastian Hassinger:Okay.

Di Fang:So first of all, what is the best truncation, and how can

Sebastian Hassinger:you do

Di Fang:a balance for that? This versus the boundaries prove this can equally hold for classical as well. Because, yeah, these are

Sebastian Hassinger:Is that is that gonna lead to I mean, are you trying to get a more, accurate framework for for that resource estimation? Is that Yeah.

Di Fang:That's right. That's right. That's right. That's

Sebastian Hassinger:right. Gotcha.

Di Fang:That's right. And then, then, so once we have these tran truncations so so one may think that unbounded operator is not really a thing because once you truncate it, it's always going to be a finite dimension, and they will be bonded. So what's the big deal? It turns out just look at one dimension, very simple example. It's a oversimplified example.

Di Fang:The

Sebastian Hassinger:Simple is good.

Di Fang:Yeah. Yeah. Because one dimension Laplacian operator is the second derivative. If we try to try and, do the spatial discretization, I will we'll we'll have a finite dimensional matrix. That's not a problem.

Di Fang:So if we use n degrees of freedom to deal with this special discretization, it can be any of our favorite, can be plain wave discretization, finite difference, and the different orbitals, whatever. The point is that the degrees of freedom is n and the operator norm of these operator is also n. However, for n Hilbert space, we know that we only need log of n number of cubits to represent the space. So we are everything is great. We are doing great on using this quantum circuit, quantum algorithm, and everything is on the log of n, all of a sudden the operator norm of n comes in.

Di Fang:It become polynomial of the size. And this is definitely a large overhead that we do not want. But So this was very much what I've been working on. Yeah. It turns out, again, because this is unbounded operators, not every initial condition you put in is making sense because the if the new condition do not have any sort of bonded derivative, it's this should not be the space you look at in

Sebastian Hassinger:the

Di Fang:first place. So what I was able to show is that if we look at the specific initial conditions and there's certain regularities of which these are mathematically speaking, Sobib norms estimate that we do. So if these which means that the derivative have certain structures, derivative remain bounded in time and how we can prove all of those bounds. And then we can show that with respect to this, sort of special class of initial conditions, and we can get rid of these operator norm dependence Okay. And we can recover the the the the the log.

Sebastian Hassinger:Okay. And that's so so you're essentially making a step from the purely theoretical to the the real world application because you're you're assuming any kind of real application is going to have specific parameters. It's not gonna be unbounded. It's not gonna be those theoretical limits.

Di Fang:Got it. Yeah. That's right.

Sebastian Hassinger:That's right. Really interesting. And do you ever first, like, do you work with, with, sort of practical implementations on hardware at this stage, or do you sort of foresee doing like, taking taking it to that level

Di Fang:at some point? Yeah. Currently, unfortunately, I do not work with the hardware implementation. But since I'm at Duke Quantum Center Yeah. Our folks do have asked me, like, is there any of your algorithm can be implemented?

Di Fang:And I'm thinking about that, but I think the the current on on bond and hamiltonian, the there are various different algorithm. Sometime I prove a bound for existing algorithm, maybe others developed or some improve the bound, or I develop my own algorithm. But all of those algorithms are very much complicated

Sebastian Hassinger:to Yeah. I mean, there it does feel like when I've gone to QIP in the past, there's, you know, so many interesting papers being presented, but they're all starting with, like, well, presuming we have, you know, quantum memory of a certain size, we'll be able to do that. Like, there's still this this, boundary of, like, what's practically possible that's that's pretty limited, I think.

Di Fang:Yeah. Yeah. That is right. That is right. I think many fault tolerant quantum algorithms nowadays use maybe either QSVT, quantum singular radio transformation, or QSP as a building block.

Di Fang:And I would just love to see, say, the QSP alone be very accurately implemented on hardware level. And I think this is very much doable. I can come come up right now a simple example and write the explicit circuit. And I think that would be a very exciting thing to work with our hardware team at Duke Quantum Center.

Sebastian Hassinger:Well, you certainly have a lot of skilled practitioners at Duke. So

Di Fang:I'm sure

Sebastian Hassinger:Ken Brown and and, Chris Monroe and

Di Fang:Yeah. They are amazing.

Sebastian Hassinger:Yeah. They really are.

Di Fang:We also have other experimentalists Yeah. Norbert Link kinda working on very interesting things. Yeah.

Sebastian Hassinger:That's great. Well, thank you so much for joining me today. This has been super interesting. I've learned a lot, and I have a lot of questions to follow-up on. I've got research to do on my own, so I'm very appreciative.

Sebastian Hassinger:Thank you.

Di Fang:Yeah. Thank you so much,

Speaker 4:Sebastian. Okay. That's it for this episode of The New Quantum Era, a Production work is done by our wonderful team over at Podfi, Production work is done by our wonderful team over at Podfi. If you are at all like us and enjoy this rich, deep, and interesting topic, please subscribe to our podcast on whichever platform you may stream from. And even consider, if you like what you've heard today, reviewing us on iTunes and or mentioning us on your preferred social media platforms.

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