 Well, so I'd like to introduce David Mayer, Professor at Mathematics at the University of California, San Diego. So unfortunately, we lost a few weeks of David's visit due to the COVID, but still it's a period of time still to have a discuss with him at OIST. So today we have a seminar, very intriguing title, Quantum Data Science. So maybe you can start and we can ask questions afterwards. Thank you. So let me start by thanking Kai Sensei and the TSVP program for having me here. Everyone has been tremendously kind and helpful and I really appreciate this. This is not working. I'm using the one here. So as Kai says, I'm gonna talk about Quantum Data Science. Now it's a truism in academic talks that if the title has a question mark in it, the answer is no, but part of this talk is gonna be about how the answer to questions depends on the context. And so maybe the answer won't be no at the end. We'll see. My goal in the talk is to make it accessible to people who work neither on quantum mechanics or quantum computing nor data science, but at the same time have some things in it that will be new to people who do work on any of those topics. We'll see how well that works. I don't know. The part at the end will be the part that the experts may not have seen before and it's been done in collaboration with some people in my group. So David Rideout is a research scientist in my group at UCSD. Daniel and Grant were PhD students in my group and Maggie was an undergraduate who was working in our group for a little while. Okay, so with that, let me start. So quantum computing is in the news a lot. This is a story in Quantum Magazine a few years ago about Google and IBM clashing over this quantum computing experiment that Google did where they claimed to have accomplished or demonstrated a quantum supremacy. And IBM said, no, we could simulate that on a classical computer. They didn't actually do it. They just said they could do it. But I start with this because it illustrates the main point about quantum computing, which is this is Google's machine here. This is IBM's classical summit machine. And the point about quantum computing is maybe quantum computers can do things that classical computers can't do or at least they can do them more efficiently than classical computers can do them. And that's what the excitement about around quantum computing is about. So since then, both Google and IBM have continued on their paths. You might think that part of the reason IBM was critical was that they were also working on their own quantum computers. Here's a paper from earlier this year. This is Google's quantum group. And in this paper, they demonstrate that they can correct for noise that happens in their device. So part of the problem with the not yet usefulness of quantum computers is that they're very noisy and it's hard to get the answers that you want to get out of them, out of them. And one of the things that we would like to be able to do is have error correcting codes. And that's what they demonstrate in this paper. They demonstrate experimentally that they have error correcting codes and they scale in a certain sense. Even more recently than that, so less than a month ago, IBM had this paper where they said, well, we have our 127 quantum bit processor that has noise in it, but we understand the noise quite well. And so we don't need to have an error correcting code to correct the noise. We can just sort of figure out what the signal was before the noise because we know what the noise is. We can sort of subtract it out. I'm speaking very roughly here, okay? And they say, we argue that this represents evidence for the utility of quantum computing in a pre-fault tolerant error. So fault tolerance is the words that people use in the subject to say, we're going to have a fully quantum error correcting machine where we can correct the operations and deal with all the noise. And they're saying, well, if we understand the noise well enough, we can make some progress anyway. Okay, so these machines are the product of roughly three decades of science and technology development motivated by this result of Peter Schuart, completely theoretical result, which showed that if you had a quantum computer, you could efficiently factor large numbers. And this was important because our current crypto systems are based on the difficulty of factoring large numbers. So that got everybody excited and people started spending money on this and scientists started working on it. And now we talk about budgets and the billions of dollars for building quantum computers. But we haven't made that much progress in 30 years if they're still talking about 127 quantum bits. That's not very many bytes, right? That's pretty small. So what's the historical comparison here? People often make a historical comparison to classical computing, the development of classical computing, but I think there's a more apt historical comparison and it comes from this. So for those of you who don't read Latin, this is the 12th year of Queen Anne's reign. So this is Queen Anne, Queen of Britain, who became Queen in 1702. So this is 1714. And this is an act of parliament in the UK. So it's an act of providing a public reward for such person or persons that she'll discover the longitude at sea. So this was a real problem and they explain why in the preamble to the act. So whereas it is well known by all that are acquainted with the art of navigation that nothing is so much wanted and desired at sea as the discovery of the longitude for the safety and quickness of voyages. I'm reading this because the letters are so hard to read but I've read it before so I can do it. The preservation of ships and the lives of men. And whereas in the judgment of able mathematicians and navigators, several methods have already been discovered. True in theory, though very difficult in practice some of which there is reason to expect may be capable of improvement. Some already discovered may be proposed to the public and others may be implemented thereafter, hereafter. So this was an act which offered some large reward for the success of being able to measure the longitude at sea. Okay, so the methods have been discovered true in theory. That kind of sounds like quantum computing, right? So already 200 years earlier, Johann Werner had figured out how to determine longitude using the position of the moon. And Petrus Apianus had figured out how to measure the position of the moon. Here's a picture of how they do it by triangulating against the fixed stars to figure out exactly what the position of the moon is. Okay? Gemaphysius had described how to determine the longitude if you had an accurate clock in 1533. But these methods were still very difficult in practice. Again, sounds like quantum computing. When the longitude act was passed by the British parliament in 1714. So not until accurate clocks were constructed. So here's a page from the website of the Seiko Museum in Tokyo, which so they have a whole display on the evolution of clock technology. And here they had talked about the invention of marine chronometer by Pierre Rois. So it's actually funny to read this because of all the technical jargon in it. So he contrived an innovative détente escapement with a mechanism to reduce the impact imposed on the balance by the escape wheel during dramatic shifts in temperature, which is a problem when you're on a boat. Okay? As well as a fusée to keep the torque of the mainspring fixed. Unforeseen difficulties have been in his attempts to build an actual precision chronometer for many years. And then in 1764 equipped with an automated temperature compensated balance using mercury and then later he invented an innovative bimetal detached balance taking hints from the bimetal compensated mechanism developed earlier by Harrison, an Englishman. And then this really sounds like current science. Regrettably for France, Lois was forced to abandon 20 years of research for a want of funds. Unlike contemporaries across the channel, the French government offered no support for Lois' work. Okay, so it's a great museum, we should go there. Okay, so not until the accurate clocks were constructed and then the nautical almanac and astronomical ephemeris for the year 1767 was published by the Royal Greenwich Observatory, could these methods be implemented? So here's the nautical almanac from 1767, 250 years after the whole thing started. Okay, now I'm not saying that quantum computing is gonna take 250 years, but it's taken more than 30 so far, so okay. All right, so meanwhile, while the engineers and experimental physicists have been working on building quantum computers, mathematicians, computer scientists and theoretical physicists have been trying to devise good quantum algorithms for which you need a quantum computer to implement them. And hopefully solve some problems that people are interested in, maybe other problems then factor it. So to get sort of to the topic of this talk, here's an article from a few years ago, 2017, in Nature where they talk about prospects for quantum machine learning. And they say, machine learning techniques have become powerful tools for finding patterns of data. Quantum systems produce atypical patterns that classical systems are thought not to produce efficiently. So it is reasonable to postulate that quantum computers may outperform classical computers on machine learning tasks. Had I written this paragraph, I would have not have said to postulate, I would have said to hope. I think that might be slightly more accurate, but. And then they conclude the review, this is prospectus on future work, they conclude, however, the execution of quantum algorithms requires quantum hardware that is not yet available, okay. But that's not the only problem. The problem is also the algorithms, okay. So here's a standard data mining problem or machine learning problem, so recommender systems. So this is something that you interact with every time you watch Netflix and they recommend a movie to you or a show to you. So the problem is to take information on ratings by end users about end items to suggest new items to users, okay. And that's what happens on Netflix. So 2017, Karenidis and Prakash wrote a paper where they said, so here we have M users and N items, you can arrange that in a table or a matrix that's M by N. And if you assume that there's a, what's called a low rank approximation to that matrix, which basically means you can compress it into a smaller number of pieces of information than M times N, so good rank K approximation. Then there's a quantum algorithm that provides a recommendation in time, which is polynomial in K, like K squared or K cubed and polynomial in the log of M times N. So the log of a number is much smaller than the number and there's gonna be polynomial in that log, okay. So it's, you know, fast, okay. And they say, this is the first algorithm for recommendation systems that runs in time poly logarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for real world application. Okay, so this paper came out and the story is that a computer scientist at UT Austin, Scott Aronson asked one of his undergraduates, can you please try to show that this is really better than the best classical algorithm for solving this problem? So she worked on it for about a year. She couldn't show that it was better. She worked on it for a little longer and Ewin Tong, that's her name, still an undergraduate at that time, now she's a grad student, and she said, ah, there's a classical algorithm that also runs in the same time. So it was unknown before she did this, but then she did this. And so this was the first example of what we now call de-quantizing an algorithm. There's a quantum algorithm and it got de-quantized, it became classical, okay. All right, so, but all is not doom and gloom for quantum computing here. So here's a more recent paper from 2021 by Huang, Quang, and John Preskula, Caltech, where they find some bounds on quantum advantages in machine learning. And I'm not gonna read all this to you, I'll just tell you what it says. So what they looked at was if they did an experiment on a quantum system where they had some quantum state and they did things to it and then they observed the results incrementally, then they compared how well from those intermediate results they could predict what was gonna happen next, okay. And they showed that on average, a classical machine learning algorithm could do as well as the best quantum algorithm that they found. But if you wanted to get it right for every possible input, if you wanted to get a good approximation of every possible input, then the quantum algorithm was provably better than the best classical algorithm you could do. And this is based on information theory, that's sort of how a lot of these proofs work, okay. This is, it's a bit complicated and it required me to introduce too many ideas to explain this result. But this is a positive result for the ideas of quantum machine learning. Okay, so what I'm gonna talk about instead is sort of motivated by this paper which is posted in the archive in March of this year. So these two authors that are at IONQ, which is another quantum computing company. It's a, I don't know if it's considered a startup anyway. It uses IONTRAPS, hence the name. And the third author is in the Department of Psychology at the City University of London. And just checking the time here, that's what that 20 means, okay, I'm good. In this paper, they say, well, Patoz tells us that there are some models of human cognition which are quantum mechanical. I think we can justify our investment in quantum hardware by saying that we can implement these models quantum mechanically on our device, okay. So that's basically what this paper is. So what I'm gonna explain for the rest of the talk is what is this model for quantum cognition that they're basing here, basing this upon and should we take it seriously, okay. All right, so this has to do with this kind of problem. So this is a page from the Pew Research Center. So the Pew Research Center is a survey and polling company in the US that does polls constantly about various political and religious things. And they several times a year or annually, I don't really remember, they have this national information social security survey and they ask questions like this. So in 2005, the relevant question was, do you approve or disapprove of the way George W. Bush is handling his job as president? And then they ask, all in all are you satisfied or dissatisfied with the way things are going in the country today, okay. But then this part that I've highlighted in yellow, it says on form one, question one precedes question two and on form two, question two precedes question one. And that's instructions to the polling person. And the person is answering the questions, doesn't see that, okay. Okay, so why do they have this weird different versions of the poll? Well, it turns out that people who study polling scientifically recognized long ago that question order can be important in a survey. So these guys, Rugg and Cantrell, I feel like I left out an L there, I'm not sure. In constructing a ballot, care must be taken to place the questions in such a way that the answers to one question do not unduly influence the answers to succeeding questions. Okay, well, why would that be? Why would that happen? Well, you can go back way earlier than that. You can go back to Francis Galton in 1879. So Galton was a half cousin of Darwin and he's responsible for a lot of ideas in a lot of early ideas and statistics like correlation. But he was also someone who did psychometrics. He measured things, some not so appropriately, but anyway, avoid that. And here he's talking about an experiment he was doing on himself, okay? So he says, he was trying to understand his own thought processes. And he says, the plan I adopted was to suddenly display a printed word to allow about a couple of ideas to successively present themselves. And then by a violent mental revulsion and sudden awakening of attention to seize upon those ideas before they had faded and to record them exactly as they were at the moment when they were surprised and grappled with. It was an attempt like that of men allows in the Odyssey to constrain the elusive form of Proteus. Nice writing. He says, additional exertion and much resolution are required in carrying on the experiment to maintain the form of the ideas strictly unaltered while they are vivified as they have a strong tendency to a rapid growth both in definition and completeness. So the point is, as you answer a question or in this case, think you have the thoughts that were presented to him by this word, it can change his mind. It changes what you're thinking, okay? Now, 100 years later post the existence of quantum mechanics, these guys who are psychologists, they have sort of physics entity and they talk about, well, we understand since Galton at least, if not before that trying to pay attention to your thoughts, one thing can, or in other settings, one thing can influence another thing if you absorb it. And oh yeah, in physics, for instance, Heisenberg's Uncertainty Principle links basic physical processes to the consequences of measurement operations. Their goal was to develop a similar language in psychology relating theories of the generation of judgment and behavior to the effects of measurement, okay? So how well did they do that? Well, this is a good source of data to try to make such a theory, okay? So let's look at some data before we try to make a theory. So in that particular survey that I showed you, when people were asked question one and then question two, so Bush approve or disapprove and then country direction satisfied or dissatisfied, this is the numbers of the different kinds of answers that were returned, okay? And then when the, no, it's not me, okay. I don't know how to get rid of that. No, it's definitely not me, we'll just deal with it. And then if you ask the questions in the other order, this is the results that they got. So it's a little hard to compare these two because the numbers aren't, the totals aren't exactly the same. So let's look at the percentages. So I'm gonna call this arrangement of numbers turned into percent or fractions, I'll call that P and these numbers turned into fractions, I'll call that Q. And we can see that these numbers are different, right? This number here, which represents a lot of the answers is smaller than this number here. And similarly, this number here, which is the biggest one of these is bigger than that one. Okay, so the numbers definitely change under the two conditions. Now, of course, it was a different set of people. You can't ask the same person the questions in opposite orders, right? Because you would have already screwed them up, but statistically you might hope that this represents something about what happens to an individual person. So how are we gonna compare these two? Well, one way to compare them is to just look at the difference. So we're gonna take these numbers and subtract them from those numbers and we'll call that Z. And so that's this arrangement of numbers here. Now, if you think about this for a second, there had to be some negatives here because these numbers add up to one since they're fractions, these numbers add up to one since they're fractions. And if you take the difference, either all of these are gonna be zero or if some of them are positive then some of them have to be negative because the sum of these numbers has to be zero. But there's something more interesting than the fact that some of these numbers are negative. And that is that these two numbers are very close to the same, but with opposite signs, okay? So people who think about these things, notice this, okay? And these people, Wong, Soloway, Schifrin, and Boosmeyer, who are psychologists, they looked at a bunch of examples of these kinds of paired questions in survey data, all from Pew, included that one that I just showed you. And then they plotted these two numbers. So this is N1, this is N2. So N1, N2, they plotted those two numbers and they noticed that they tend to lie along this line which says that one of them is the negative of the other or that they add up to zero, okay? And when you compute the correlation of these, you get negative 0.82. And if you're skeptical about this one and this one because they're kinda outliers and throw those away, you still get a correlation of negative 0.73. Now, for any of you who've ever worked with real social science data, this is astonishing. You never get correlations that are that big. I mean, you're lucky if you get 0.3, okay? So getting something like that should make you suspicious, okay? And it made these people suspicious. They said, well, maybe there's some structural reason, some constraint here, which is making this happen. Let's see if we can understand what that might be, okay? To do that, they went back to this idea about physics that these guys had expressed. So I'm gonna tell you a little bit about quantum physics now. So this is the five minute lesson on quantum physics but without any math. So well, there's gonna be a little trigonometry at one point, but that's all. So this is a famous experiment in physics. I took this picture from Wikimedia. This is the Stern-Gerlach experiment. So what this experiment is, is you have a source of silver atoms here. The silver atoms get pumped out in a beam and then they go into this magnet where this is supposed to represent the fact that the magnetic field is not constant but it has a gradient. And then you have a screen here and you look to see where on the screen the silver atoms end up, okay? And the reason they did this is because silver has a magnetic moment. So it acts like a little magnet. The reason it acts like a little magnet for those of you who care is that it has 47 electrons and 46 of them are paired up. And so their magnetic moment is zero but the 47th electron isn't paired. And so the spin of that electron which is plus or minus a half if you like is a little magnetic moment. Now, classically, if you pass little magnets through a inhomogeneous magnetic field like this, you would expect them to sort of make a blurry blob in the middle. They'd just be spread out depending on how the magnet was oriented when it went in. But in fact, that's not what Stern and Gerlach discovered. They found that half of the silver atoms went up here and the other half went down here, okay? So this is a, I kind of think this should be on the first page of quantum mechanics textbooks instead of talking about the hydrogen atom. It's much simpler, but historically this came long after people thought about the hydrogen atom. So most physics textbooks don't talk about this until later on. But it's a beautiful example of quantum mechanics and the explanation is quite simple. So here's the explanation. So this is quantum mechanics on one page. So the state of the system is represented by a vector which in this picture is red and has length one. So its end point is on the unit circle, okay? So that means with respect to a particular coordinate system, it has an X coordinate, which I'm gonna call the down coordinate for the down beam and a Y coordinate, which I'm gonna call the up coordinate for the up beam. And the components of this vector in that coordinate system are of course cosine of theta and sine of theta if this angle is theta, okay? And then what quantum mechanics says is when you do an experiment on a system which is represented by such a vector and the experiment is to make a measurement with respect to this choice of directions, you're going to see down with probability the length here squared and you're gonna see up with probability the length here squared. And of course because cosine squared of theta plus sine squared of theta is one, that tells you that the two probabilities add up to one and so you get an answer, okay? That's quantum mechanics right there. Okay, now we're not asking one question, we're asking two. So maybe we should chain together two Stern-Gerlach apparatuses where the second magnet is rotated by some angle relative to the first one and make a second measurement on what comes out of the first one, okay? So this picture comes from Feynman's Lectures. So now we have to complicate our picture a little bit. The second measurement is represented by these blue, this blue coordinate system which is rotated by some angle in this picture it's rotated by angle omega relative to the first coordinate system. And then the result of that second measurement is that if the first measurement was down then you take that point and you project it up to the new coordinate system and you'll get down on the second one with probability the product of these cosines, right? So this length here was, sorry, not probability that point has length product of the cosine. So this length was cosine of theta and then this length here, the green length is gonna be that number times cosine of omega. And similarly over here if you wanna see what the probability of getting up on the second one if you got up on the first one is you take this length here which was sine of theta and then this little angle here is also omega because it's a coordinate system so it has the same relative angles to the axes times that cosine of omega. So the probability of blackup followed by blue up is sine theta, cosine omega squared and probability of black down followed by blue down is cosine theta, cosine omega squared, okay? Now of course there's also, you could get up on the first one and down on the second one down on the first one and up on the second one I didn't make the picture too complicated so I didn't include that, okay? Okay, so this is the idea that Wong and Busmeyer decided to use to model quantum cognition, okay? So they said a binary question is gonna be represented by a choice of coordinate axes for those of you who know linear algebra a basis in a two-dimensional vector space the axes or the elements of the basis correspond to the answers, okay? And the probability of each answer is determined by the length squared of that component and when a question is answered the state vector becomes the corresponding axis vector, okay? So it's quantum mechanics, okay? So let's go back to the data, okay? This was the data I showed you before. Now notice that we are interested in this phenomenon which I said must come from some constraint or it's just too unlikely to get such a big correlation. So let's compute it in this model, okay? So this is the up-up if you ask the black question then the blue question and the down-down if you ask the black question then the blue question but we can also ask the questions in the opposite order which is what they did in the survey, okay? So okay, so the picture is a little more complicated but if we ask the blue question first we project the state vector red onto the blue axis and onto the blue axis, so this line here and then we project onto the black axis or project back onto the black axis. So what are those lengths gonna be? Well, this projection has length cosine of theta minus omega because this angle here is theta minus omega and then it gets another factor of cosine omega when it gets projected down to here and similarly over here we're gonna get sine theta minus omega times cosine omega, okay? So this tells us what P11 is, what Q11 is, what P22 is and what Q22 is. If the initial state is theta and the relative angle between the questions the coordinate systems is omega, okay? So now we can compute what the difference is, okay? So, or we're gonna compute the sum. So Z11 remember was the difference between P and Q so it's P11 minus Q11, Z22 is P22 minus Q22. We can just look up here and plug them in. Notice that there's a cosine omega squared in every one of them, so I'm gonna factor that out. And then P11 has this sine squared theta, Q11 has this sine squared theta minus omega, P22 has cosine squared theta and Q22 has cosine squared theta minus omega, okay? I've kept them in the same colors so we can notice that we have sine squared theta plus cosine squared theta, which is one and minus sine squared theta minus omega minus cosine squared theta minus omega which is also one. So we get one minus one times cosine squared omega which is zero. So that says that this quantum question model explains that the two diagonal elements in the difference probability matrix should add up to zero. In other words, things should lie on that diagonal line. So these axioms imply what the authors call the QQ identity, the quantum question identity, namely that n two is equal to minus n one. Let me check the time. Okay, I'm good still. All right, so a little tiny digression for those of you who've done quantum mechanics before and you don't recognize this identity and you're wondering if maybe it's not a real thing and it's just because I set everything up in real system instead of in a complex vector space. No, this is a real thing. We just didn't bother to ever learn it in quantum mechanics. So here's the argument. Again, those of you who haven't done quantum mechanics before ignore the next two slides or something. So if psi is a qubit and we have two bases A and B, then P11 is gonna be the projection on A1 followed by the projection on B1 norm squared. Q11 will be projection on B1 followed by A1 norm squared, which we can write out like this. So that means the difference Z11 is gonna be this expression. And then this composition of projections, we can rewrite by noticing that A2, the other vector of A1 of the A basis is the identity minus A2, et cetera, write this out, multiply it out, lots of things cancel and you just end up with the negative of the same thing in the opposite order. So this implies that Z11 is equal to negative of Z22. I wasn't cheating by doing things in real. This is something from quantum mechanics that we just never bothered to learn. And this quantum question identity is approximately satisfied empirically. Wow. Okay, so clearly people answer questions quantum mechanically, right? I mean, we've explained this very surprising result by this quantum mechanical model. Well, not so fast. Yeah. Differently ordered questions were given with differences of people. Yes, yes. Differently ordered experiment were given with the same set of target responders and prospective teams that seem to be very similar each other more than two groups of people. Well, I mean, each individual atom goes through, right? You think? Yeah, I agree with you, of course. Yes, I mean, this is a good reason to be skeptical, right? I mean, it's really hard to imagine that there are any true quantum mechanical processes going on in an individual person which cause this to happen, right? There are a few people like Penrose who believe that, but most of us are skeptical about that. So, and to give the psychologist credit, they don't say that. They say, look, this is a model which generates the data as we see it and we don't see a classical way of getting this result. So despite not believing that people are like atoms, we're gonna go with it, okay? And what I'm gonna say next is basically, well, not so fast, okay? We should think a little harder about what we mean, what's possible classically, okay? Okay, so a little reminder about probability distributions for those of you who aren't economists. Economists think about probability distributions this way all the time. If I have a probability distribution over two things, so one outcome or the other outcome, the two numbers have to add up to one, which means that they have to lie on the line, P one plus P two equals one and they have to be in the first quadrant, okay? Similarly, if I have a probability distribution over three things, they have to lie on the plane in three dimensions where P one plus P two plus P three equals one and they have to lie in the first octant, okay? So this is the point a half, one third, one sixth, which lies on that plane, okay? Of course, we have four probabilities in P and in Q. So I need to draw a four-dimensional picture, which I can't, but I can draw a four-dimensional triangle because that's only three-dimensional, right? It's got three vertices and it's flat, it's two-dimensional. So it's a tetrahedron, okay? So a probability distribution over four things is a point in a tetrahedron, okay? And the data that I showed you, P is that point, okay? Q is that point, which is close but not the same, okay? And then the difference between them is that little red line that I showed you. And so it's P minus Q, so it's the second one minus the first one, okay? All right, so my picture of a tetrahedron shows us what all the possible probability distributions are. What are all the possible difference vectors between probability distributions? So that should also be some shape. It's a great problem to work out for yourself what that shape is. It's just sort of visual geometry to figure it out, but I'm gonna show you. It's this cube octahedron. So it's a cube with the corners cut off at the halfway point, okay? So all the little red vectors, which represent the differences between one probability distribution and another probability distribution over four things, correspond to single points in this cube octahedron, okay? So our difference arrangement Z is that point right there, okay? Now, we're interested in these two components of Z. So I wanna take this picture and project it into two dimensions, okay? So here it is, here are N1 and N2, which are these two components of this picture. And I've drawn this green hexagon to represent the range of possible values of that red point when it gets projected. And if you look at this, you can see going across here, that's this piece, then going down and then down, that's this corner, then going across the bottom is this, and then going up and then up, that's this and this, okay? So the projection of this thing is this hexagon and when we project a point, we get a dot there, okay? Now this is already suspicious, right? Because this thing is, you know, these points here are negatively correlated with each other, right? It's not just a blob, if they're high on this coordinate, they tend to be low on this coordinate and conversely. Of course, I've cheated slightly because I picked the two on the bigger diagonal. That's really only half of this picture, it's the half that I've shaded in here. The part that I haven't shaded in, those are the difference matrices where the off diagonal ones are the bigger ones, okay? So really I'm just projecting half of this thing onto here. And if it turned out that these two are the bigger ones, then I would sort of rotate it and do the projection of the uncolored parts, okay? Okay, so what happens when we do this projection? Well, if it turned out that we got sort of a uniform distribution on that hexagon, we could compute the correlation and we would find that the correlation is minus a half. That's a little calculation, it's not difficult, okay? Now that's not as big in magnitude as the observed correlation. But why would we think it would be uniform on here? There's no reason really to think it should be uniform. What if it's uniform over the whole cubic dehedron? So it's uniform in the red picture and we project it. Well, then we get these sort of dense areas here where the thick parts of that half get projected. And we compute the correlation of this as a little harder calculation, but still not that hard. And we get something which is approximately minus 0.54. Still not what we're aiming for, but an improvement, okay? Of course, why should it be uniform over Z, right? Maybe it should be uniform over P and Q, right? Okay, so let's do that. I did that by sampling. We get this arrangement of dots. Turns out that the correlation in that case is actually the same, that doesn't change anything, okay? All right, so it looks like maybe we're stuck, but we're not, okay? Let's go back to the data. Why did we bother to ask these questions in two different orders in the first place? Or we, Pew Research. They did it because they suspected that the answers were gonna be related, right? It wasn't like, do you prefer hot dogs or hamburgers? And do you think George Bush is doing a good job or not? It was the country going in the right direction and is George Bush doing a good job or not, okay? So we expect there to be some relationship between these answers, okay? How do we measure that? Well, we can measure the, if these were not categorical variables, but numerical variables, we would look at the correlation between the answer to the one question, the answer to the other question, we would expect it to be positive, right? But these are categorical variables, so that doesn't really make much sense. But because it's a two by two setup, a good proxy for that is the product of these numbers minus the product of these numbers. That's the determinant of the matrix. And if these were real valued variables, the determinant here is proportional to the correlation, okay? The determinant of this one is 0.149. The determinant of this one is 0.153. They're really close, okay? In fact, when you go through the Pew data set, the signs of the determinants of P and Q are always the same. They're either both positive or both negative, okay? So now let's sample uniformly in P and Q, but only keep the cases where they both have the same sign determinant, okay? You get this picture, and the correlation is, hey, look at that, minus 0.769, which is very close to what they saw empirically, okay? Actually, we can do better. So this is the hardest part of the talk, which I'm gonna ignore. We can look at what the probability distribution for the determinant is over random choices of points for the probability distribution in the tetrahedron, okay? And we get this black curve here. The histogram shows the histogram of the empirical data, what the determinants are in the actual questions. And we can see that even though the probability is biggest close to zero, if we pick a point at random, the probability is biggest at pretty substantial values, bigger than 0.1 when we look at the actual data, okay? In particular, the determinants in the Pew data are bounded away from zero, okay? So let's look at the cases where the product is actually not just bigger than zero, but bigger than 1.100, just a tiny bit bigger than zero, okay? Now we get these, and hey, look, the correlation is minus 0.879, okay? Which is even bigger than what they got when they threw out the outliers, okay? So we have reproduced the empirical negative correlation between the diagonal elements of Z using entirely classical probabilities. We didn't use any quantum stuff at all, okay? So your question is, well, maybe no, maybe we don't need to think about it quantum mechanically because look, we can do it classically, okay? All right, so I talked about a bunch of different things and it's time to wrap up, so let me give you a quick summary. So quantum computing is exciting because it promises to solve some problems more effectively than is possible classically. This has been demonstrated experimentally for some not so natural problems like the Google experiment, what that experiment was, I didn't tell you, is sampling the output from a random circuit, which is pretty useless. But it turns out that that sampling problem can't be reproduced classically, so okay? It would be nice if some data science problems could be solved quantum mechanically, that'd be great, okay? But no classical data science problem has yet been shown to have a more effective quantum solution. For example, the one I talked about was recommender systems do not, okay? But there are quantum algorithms that have been shown to solve some quantum data science problems more effectively, okay? So that's good. And that motivated this investigation of classical data that's maybe generated by a quantum model, but it's still classical data, okay? Namely this quantum cognition model. But if we take reasonable priors on classical probability distribution pairs for these two binary questions, this QQ identity is as close to being satisfied completely classically as it is in the empirical data. So we don't need the quantum model, it doesn't help us. So we need some stronger evidence if we really wanna judge that the quantum question model is what's going on rather than something classical, okay? However, the fact that this quantum question model was proposed led us to a better classical understanding of these survey data because my group and I did these calculations which had never been done before, okay? And the proposed quantum algorithm for recommender systems led to Ewin Tong writing down a classical complexity analysis which again doesn't seem to have been done before. So thinking about these quantum data science problems maybe eventually something useful will happen, quantum mechanically, but at the very least we're learning stuff classically that we didn't know before, okay? All right, so I will stop there and answer questions that people have. I thank you very much. So, just questions, any questions? I think this one works, right? Thank you very much for the nice talk, for example. In addition to the kind of confusion or skepticism that I shared earlier after the talk, there was somewhere else that there was an assertion along the lines that we still do not have the technology to do quantum computation. And the way that I understand it, maybe I misunderstand it is we know what we're expecting, we don't have the tool. And so is that what really the experts believe that we know what you're expecting but we don't have the tool and you're searching for. So the situation is that there are by now many noisy intermediate scale quantum devices where noisy means what you think it does, they have lots of errors. Intermediate scale means they're not very big, they don't have a whole lot of quantum bits in them. But they do something which is quantum mechanical but affected by noise. So, Google has one that's roughly 100 quantum bits, IBM has one that's maybe roughly 1,000 bits, I'm not sure right now. D-Wave does a slightly different model and they probably claimed several thousand bits at this point, I'm not really sure. All of these devices, they're physics experiments, right? They do some sort of quantum thing, it's a quantum system and you can set it up and you can control it and you can read things out. But none of it reaches the desired level of fault tolerance, meaning that it supports recursive quantum error correction to keep the results close to what they would be if there were no noise, okay? So all these companies which are spending lots of money on this are, well, they're trying to improve their machines to the point where they can get to that fault tolerant level. But they're also trying to figure out things that they can do usefully even with these noisy intermediate scale quantum devices. And that's kind of where the subject is at this point. People are sort of searching for useful applications of these. And if you read what the individual companies say, some of them say we can do useful things now. I'll only speak for myself, I'm skeptical that any of the things that can be done currently are useful in the sense that they can't be reproduced by a big classical computer. But it's improving, I mean, every year it's better, so. Very nice talk. So in data science, there's some problem called the feature section or variable section. This is like basically we want to say which feature is important from the data. We have the adventure vector, we want to say which feature is important. So this is basically the combinatorial program. So the best one is of course like most the easiest one is the brute force, but brute force is like we need to compute two to power the computation. This is very computationally expensive. So therefore in data science, so Michelle and we tends to use like, relaxation to the continuous and solve it. But for example, like I'm just wondering, I'm not super familiar with quantum computing, but I just wonder whether if you use the quantum computing whether we can solve these kinds of combinatorial problem or efficiently compute the current state of the art in classical of the machine learning approach. I just want. So I give sort of two answers to that. So one is there's certainly some hope of solving combinatorial kinds of problems with quantum algorithms. So there are examples of that. They're not directly data science examples, but they're sort of combinatorial question. The difficulty with taking any of the problems that people care about in classical data science and porting them over to a quantum machine is that most of the problems we care about in classical data science involve large amounts of data and all that data would have to be entered somehow into the quantum device. So really the only way you can expect to get a quantum improvement is if the classical algorithm that you want to run on the classical data is at least some high powered polynomial of the length of the data or maybe exponential as you suggest in the length of the data. If that calculation is large classically compared to the length of the data, then maybe it makes sense to input it into the quantum computer and run a quantum algorithm that might be more efficient. But in many cases, like if you think about a lot of image processing problems, for example, it's really inputting the data. It's really the size of the data that matters and the actual calculation you run, like a template matching, for example, to take a simple example. That runs linearly in roughly, I mean with a log factor I guess, but in the size of the data entry. So you shouldn't expect to get anything better quantum mechanically in that setting. So we're really looking for either data science problems which don't involve large amounts of data which seems like that's probably not going to be something we can't already do classically except factoring a large number or problems where the computation that you want to do classically is very expensive compared to the amount of data. So if you know examples like that, maybe that's a case that there's a hope for. Nobody has worked out such an algorithm at this point, but. Thank you very much. Thanks very much, very nice talk. Can I ask about the determinants in the empirical and the model? So it was interesting to see that in the empirical data the determinants were always positive where in the model you had to restrict it to be both positive. So in the empirical data, is that biased somehow because the questions they ask are so polarizing that you can't get a flip in the answers when you flip the order of the questions. No, it's just because of the order we write the columns in, right? So if we wrote the no answer and then the yes answer that would change the sign of the determinant relative to yes answer and then no answer. So it's just because they always write them in the order that makes it positive. But it might not have been. So that's why I was careful to say the product of the determinants is positive in case they happen to post me negative. Any other questions? Just ask one. So you talked about data, but still classical data on the sort of computation will maybe in quantum. But what is if really quantum data science? Yeah, so there's sort of quantum data, quantum science or quantum data, classical science. So I didn't talk too much about the Preskell and Huang and Quang result. So there's a really important detail there which is they have this quantum experiment where they make measurements intermittently. Now, when I say make measurements I might mean one of two different things. One thing might be I make a classical measurement and I pull classical information out of it like the Stern-Gerlach experiment. Another thing I could mean is the state of the quantum system, I have that interact unitarily with my quantum computer and then run some quantum analysis coherently with the experiment. And that's what they show gives a exponential improvement. So the data never becomes classical in that case. The future if we have the quantum internet, my computer my quantum computer generating data and then quantum data and it can send it to you. And maybe doing things like analyzing quantum Facebook at that point maybe is something you can do. That's right. As long as it never becomes classical. Any other questions? No, okay. So let's thank again.