Session 2: Advances in Computer Applications in Data Science
4:10 pm – 4:35 pm GMT
Sudip Patra (OP Jindal Global Uni, Jindal School of Government and Public Policy, CEASP) – Quantum Computation: A New Frontier for FinTech
Thank you very much for inviting me in this very special conference and workshop. Actually, I’m not very well today so I’ll be a bit brief. Actually, as far as my research is concerned, I’m mainly working in two areas. One is a very emerging paradigm, which is quantum like modelling in decision science, and then applications to finance and other areas, which is like my main work. And then along with this, I’m also working in quantum foundations, which is more into quantum physics and philosophy of science. So here I’ll be very briefly talking about quantum like modelling in decision science and applications in finance. There are various interesting things which are happening in this field, quite a versatile field, for example, quantum games and financial market modelling, which I’ve been working with my quarters earlier, mainly, though, in this particular presentation, IoT, maybe presenting a general framework, but if more concrete papers or more concrete models you’re interested in, then I’ve referred to some of my recent works with Khrennikova 2019. Incidentally, Polina is also from Leicester University. So, there we applied one particular quantum finance model in asset trading, that is these investors diversity of opinion in that kind of an area, then, along with Professor Partha Ghose, who is a very renowned quantum physicist, we are working on some of the game theoretic papers etc. So, I’m based at op Jindal global university currently, and we also have a centre at OP Jindal University, which is CEASP, mainly working in complexity economics, I found that some of the papers presented earlier also related to networks and complexity modelling. So, it would be of great interest to us also to kind of network more in these areas.
Now, very briefly saying what is quantum like modelling? Now, there are a couple things here the first thing is that the qualifier like is quite important, because this simply means that it is not suggested here that there is a genuine quantum physical model of human brains, though that that itself is a very, maybe a viable project, many brilliant scientists are working in that area, but we are rather much more pragmatic, what we would like to do is basically to use various interesting mathematical frameworks, which have been used quite widely in quantum physics and quantum field theory mainly, and to see that whether they can throw some new light into cognition, modelling and its applications in financial markets and so on. So, there’s a huge empirical evidence from cognitive science about limitations of Boolean logical framework to describe decision making in general. For example, we know that the standard decision theory is very much based on set theoretic setups and from probabilistic modelling. We can think about Karmakar of decision making theory or measure theory. However, it has also been shown since the late 60s and 70s itself that when real agents in say financial markets situations or in rather controlled atmosphere they to choices, mainly on the ambiguity and uncertainty, then there are a lot of so called paradoxes or so called anomalies which arise which cannot be satisfactorily described by within the framework of say neoclassical decision theory or utility maximisation theory to give a very over simplistic name. For example, question order effects and failure of shear think principle conjunction fallacy, some of which I will refer to again these kinds of anomalies to appear and one standard way to address these anomalies would be to take the behavioural finance or behavioural economics route, and then try to either suggest that human beings most of the times, may not be fully rational. There’s bounded rationality models also, or for example, to use various kind of heuristics and thumb rules, that is a huge literature, behavioural finance, we know, but we are not in a way against behavioural finance literature, certainly that has informed us a lot. But along with this, we’d also like to remind that there can be better perhaps a mathematical framework, which would be able to retain most of the standard results of decision theory, but along with this also try to describe various so called anomalies in a more normal way, so, that we don’t really need to think that these are some pathological cases, this is some like kind of irrational behaviour altogether. So, that is basically what quantum like modelling would like to do, barring the language of quantum computation and quantum information theory. Later, I will also try to argue that there are other ways of introducing quantum computation in finance too, but this is a decision theoretic perspective, there are various other perspectives also, one very important and central notion in all of our modelling and maybe decision theory is the contextuality thing that is, whatever decision making we can think about, for example, some of the papers I was also going through in this great workshop, maybe contextuality can have a very important role to play in the final choice making or final asset allocations for example, so on. So, the context in which the decisions are made has a direct impact in the ultimate results and whether that can be satisfactorily modelled with the help of probabilistic modelling. So, that is like one of the central tenets of quantum like modelling, but definitely here also we have to be careful that we are talking about quantization. So, quantization is the central feature of quantum physics where, which is like reflected in quantum of least action or Planck’s constant that we do not need. So, we have to be careful to build up that kind of an algebra or that kind of a mathematical model, which would have contextuality and all the other desired features, but not directly going into some more hardcore physics factors.
Quantum like modelling which I have abbreviated here as QLM, is a comprehensive decision making paradigm. There have been many pioneers in this field. Emmanuel Haven, Andrei Khrenikov came out with the seminal textbook in 2013. And then, Youkalov and Sornette, they have been very much active in something known as quantum decision theory where the dynamics of decision making model comes in. Busemeyer et al from Indiana University Bloomington and others, they are a very strong group of cognitive modellers and so on. Now, here in our specific models, our specific framework that we are adhering to, we see that this contextuality is actually the central tenet of our modelling. And here there are quite a few choices that how to model contextuality in a decision framework. 2 most important choices are one it goes back to Niels Bohr, one of the founding fathers of quantum mechanics, Copenhagen interpretation. So, Bohr adopted the stance that the state of measuring device and the state of the object cannot be separated from each other during a measurement and constitute a dynamic whole. And that is what is actually thought to be contextuality in that particular paradigm. So it’s a kind of like an on analyzable or unseparable hole between the decision maker for example, in our case, the mental state of the decision maker and say the information environment. Now, there have been some recent works in that particular area to give a more mathematical framework, we have given one of them which I will go through there are other definitions of contextuality also for example, joint measurement contextuality or JMC that is basically focused on kind of joint probability distributions of random variables. There are certain interesting conditions to be made to us to see that whether the data with which we have working choice data for example, they really reflect contextuality or not. One particular condition is often called as no signalling condition that these are some interesting kind of like summation forms which have to be made. And then after that, we can look For whether there are further violations of inequalities like Bill inequalities or ch sh inequalities are quite well known in this particular literature.
Sudip can I just jump in there, just from a personal point of view about quantum physics and so on. How would, just give us a very simple example of say, you know, we are all familiar with state preference models, let’s say there are two states a probability and the sum of probabilities add up to one, how would that simple framework be made into a quantum physics framework?
Yeah, I’m just I’m just having them just in the next slide.
Yeah. So, basically, what we have here is, we can straight away go to this particular feature or maybe even this particular slide. So, in our modelling what we do basically we start with a particular cognitive state or a belief state of an individual or as a decision maker. And this particular belief state is thought as to be a kind of like a ray in the complex linear vector space, which is the Hilbert space. So, Hilbert space is our like state space, if you want to say in that way, and then certainly that particular state is represented with the help of certain linear combinations of basic states, which I will get back to say for example, you have two bases zero and one. So zero may be like kind of like yes state for the mental state and one may be kind of like no state for the mental state, and then you have a linear combination of zero and one, which actually would describe the ray in the Hilbert space to start with that is like the basic mental representation of the state. Now, the coefficients of this particular linear combinations are often called as to be probability amplitudes. Now, this probability amplitudes are not exactly equal to the classical probabilities, but it is only when you make a measurement on that particular state, that which particular final choice, that individual would make that whether either go for yes state or no state like a patient of mental state, then only those probabilities would be kind of materialised. So, what you do is basically you square the amplitudes in that linear superposition description and the square of the amplitudes actually becomes a classical problem. So, this is basically a correspondence between the quantum like description of the mental state or cognitive state and the classical probabilities. Now, in that particular picture, geometric picture, the most important thing that we have is that while you calculate the probabilities of a particular event happening for example, or a particular choice to take place, for example, like in this particular formula probability of A happening, which is a condition to another particular event or B and B is taking either of these values B1 and B2, then in this formula for total probability as we say, you have some extra interference terms coming up. And these interference terms which I have just written in red here, these are the interesting terms which comes up in this quantum probabilistic framework. And the claim as far as it goes of different papers is that these deviations that we observe in reality from the classical predictions as I was talking about earlier, but the anomalies for example, they can be described by suitable choice of interference stocks. There is a huge amount of research which has gone into this particular area, that different types of interference terms hyperbolic as well as trigonometric interference terms, the main purpose is to show that how the deviations occur for example, one particular example which is quite well studied nowadays, is the prisoner’s dilemma example in real life. So we know that in a typical rational framework, we have only one non-cooperative equilibrium, defection equilibrium in the first round of the game, until unless the game is repeated in financially and so on. But in real life scenarios, what happened is that under uncertainty and ambiguous scenarios, people really do not always adhere to that that natural prediction and that different types of behaviour which is coming up so those different behaviours, the probability of like deviating from that stubborn equilibrium can actually be shown to follow this kind of interference terms patterns, which are capturing the deviations from the equilibrium behaviour.
Can I, can I just ask you a little more about the interference factors? What sort of what is bring that about? It’s not random, it’s not random noise is it?
Yeah. So say for example, interference pattern actually, the name comes from it has a physics background, but we don’t have to get too much into it. The main thing here is that the set deviations from the classical results of total probability, so, if you calculate the total probability with the help of say any kind of a simple Boolean logical framework, you do not get this interference terms as these interference terms are like some kind of perturbative terms that you gave in in your results. Now, so, going back to our own mathematical framework, then what we can do is basically we can start with this particular state of cognition and we call it a sigh and then we have like two indexes here or we can say two parameters, one is C and one is Q, C is the context in which the decision making is happening. And Q actually denotes the questions which are asked to the agents. Now, if this is the mental state and this particular state is then one can say model as the complex linear vector space, which is the Hilbert space now the answers to these particular questions or say the response or to this particular mental state, when it is brought up or when it is kind of asked and the question is given by random variable, so, these are CQ, these are like the responses to the questions and again to index SOC for contest and q4 the questions if CSC a C prime are incompatible, one has complementarity the complementary two means that that different mental states which are corresponding to different contexts, which may not be compatible with each other all the time, so, they cannot be observed at the same time. These incompatibilities also to be understood in the sense of mutual exclusive comes context like in the quantum double slit experiment, where one or both slits are open, they cannot coexist with each other. So, these are like different kinds of exclusive context. So, we can you know, we give this representation of mental states in exclusive context. And therefore, one can go on in this way and show that say that like this kind of context like if C is like Alice ate a banana and then C’ is like Alice ate an apple. So, these are like two incompatible context with each other, we cannot like have both at the same time and they can be represented by these two different mental states CQ, and C’Q, which are not equal to each other. And therefore, the responses to these two states would also be different RC/Q and RC’/Q which are not equal to each other. Now, to represent this thing, we do not really need to get back to non commutative operators and so on which some models might have done RDF, but right now, we can show that it can be done in a much more simpler way not really going to bond commutation algebra, because if we go into non commutation algebra to represent this then we get into the representation of you know Planck’s constant etc which is not required in decision science.
So, basically the representation of this whole mathematical structure comes from First of all, doing two things first of all to choose the basis in which the state has to be written down to one for one particular basis will be a computational basis, like zero and one and here is where the language of quantum computation also comes in. So, zero is again represented by one zero column matrix and one by 01 matrix. And then we have this size data which can be represented in this way. Say for example, cos theta is zero plus e to the i phi sine theta get one with some suitable range of theta and phi theta and phi being the polar and the azimuthal angle on a unit sphere, which, which is often known as the pocket a sphere. And then what you can do is that you can keep on changing the theta and phi. So, these are the two parameters which you can keep on changing and that will show that how the mental state that you are representing on the unit sphere that keeps on changing over so, and then also you have Q which means different questions. So, if different questions also keeps on changing the mental change mental state also changes. Here is one particular example of our sphere. Now, the other thing that you can do always is that you can also change the context. So, either you can chande C or Q so, changing of the context will definitely mean that you are choosing a different basis. So, rather than zero and one basis, you are choosing something like a really linear combination of zero and one and so on that will change the mental state from CQ to C’Q for example. So, these are different flexibility you have in this particular model. Now, apart from this, you can also keep on a kind of get more and more general representation of the state, which often called as density matrix representation, I won’t be going too much into that is like direct products of kind of this SICU. Presentation, this actually gives us much more flexibility in introducing noise in the data also, for example, we know that in decision making often that there is a noise and choice, the last part of the whole thing would be like to construct suitable probability measures how to calculate the probabilities in this whole paradigm. That two things here the first thing is like for the updation of the state. And the second thing is that actual calculation of the probabilities that is actually done with the help of something known as to be pure VMs, which are known as positive operative measures into brevity of time as I won’t be able to go into that, but this is basically how this density matrix state row is updated from one state to another state. Say, for example, when new questions come in, or new interaction comes in, in the atmosphere, the mental state keeps on updating and so on. And you have a nice formula for calculating the end choice that is given by this trace formula of the projector multiplied by the density matrix projector is the same, this is simply the positive operator, which takes you to that particular one single choice like yes or no.
Yeah, so basically, I’d like just like to in here, that this particular paradigm is actually going into a lot of areas like one is like finance very much of which definitely, we can talk a lot later on that there are different types of tools which can be used in finance, like operator approach, ladder operators, and path integrals, etc, which can be taken from this language. And it has been shown very recently, that a lot of interesting problems can be studied. I just like to end on one particular problem, which has been talked about quite a lot, that is common knowledge formation. We know that in neoclassical economics, also common knowledge plays a very important role in theoretical modelling. Now, if the agents do follow a different kind of logic, rather than the standard Boolean logic, then it can be shown that the carbon knowledge theorem of Robert Amen, for example, that can be relaxed. And there are like different solutions coming up, rather than the standard solution of not possible to agree to disagree. And we know that in financial markets, there’s a lot of situations where divergence of opinions are more important rather than convergence of opinion. So in these areas, also, this kind of mathematical modelling can really help us and then certainly there is also an empirical part to it, which I’d be very happy to discuss later on. But this is a very brief overview, I should say. Thank you.
Hi Sudip. So I have a question regarding this particular modelling paradigm, I’m of course, new to it, very little idea about it, but I was just wondering that, how close does it come to some of the other decision modelling paradigms like fuzzy logic and all because a number of terminologies that you were using. For instance, when you were talking about Boolean logic not following a Boolean logic, or maybe in preciseness of information and stuff like that, a lot of these kinds of things are spoken about by experts working in the area of fuzzy logic as well. And at the same time, there is a lot of mathematicians who criticise people working in the area of fuzzy logic by saying that, okay, whatever these people are talking about, can very well be done with just probabilities. So, what’s your take on that and how close this particular area comes to maybe some of these other paradigms that people are looking at. And of course, the other thing that I observed was that you’re taking some sort of a function of probability as well. Right? In one of the equations, I saw there was some function of probability with interference and those sorts of things. And there it comes very close to maybe other sort of techniques like prospect theory as well, where you take a function of probability and then you create the utilities and everything.
Yeah, I think that is a very, very interesting questions. And I will say that there are a lot of interfaces, there is no doubt about that, as some kind of commonalities also one can observe, but quantum like modelling as of now is rather quite acting, kind of working with the real data also. And also, I will say that the deeper questions for example, the question of contextuality, that can be, in a way, model better with the help of this quantum like modelling, because it also has a very solid theoretical perspective to how to form your approach towards contextuality, and others. And definitely, there are different ways of introducing. So I haven’t just given one particular presentation, there are other ways of representing quantum like modelling too. And also one important area, which definitely I didn’t get time to get into is basically the game theoretic part of it, there is a very good amount of research was happening in using this framework in understanding and designing real games. So this is perhaps one way in which it is bit different from the other paradigms, where we can really design real games with the help of say, quantum computation, if you like to say, and one of the aims of designing these games have been to generate superior strategies or say superior equilibrium outcomes, like one very important game, which has the study’s prisoner’s dilemma. And I think that from the late 90s, maybe 1999 or something, there have been very interesting protocols which have been designed by mathematicians, where using certain interesting features like entanglement in a particular system, you can generate superior equilibrium outcomes. So these are some of the advantages that you can have with this framework.