 Okay, let's start. Well, good morning to everybody. Thank you to all of you who have been coming here to join us for this special event. Thank you for everybody to join us. Today is a very special, it's the starting point we think of a major initiative in Trieste. We call it the TQT, is the Trieste Institute for the Theory of Quantum Technologies. And it's a collaboration that we announced a very, essentially one year ago, the event of the Rack Medal Ceremony last year. And today we bring it to the conclusion by the official signature of the agreement collaboration between the three institutions. CISA, the University of Trieste, and ICTP. So, it's a real pleasure to have this kind of events and we can show that the three institutions being very, very close and working together. Actually, thank the scientists from each of the institutions, in particular also to Stefano Maurizio for joining us in this collaboration. We have great expectations, not only for the region, but also for the mission of ICTP to promote also this big field, which is getting more and more important with time to have it available to as many possible scientists as possible worldwide. And so I think that starting with this initiative could be a beginning of something big as you can, you know that there's this flagship project from the European Union that is promoting this field of quantum technologies and we would like to be a very important component of that project. And within the region, there is a lot of interest and there is already some expertise in the different institutions that we, all of them together, could make it much bigger as the sum of its parts. So in that sense, we are looking forward for this special event. And so before proceeding to the signing ceremony, I would like to ask Stefano Maurizio to say a few words. Thank you very much for having organized this event. I think it's a great thing for Triesta, for the Triesta system, which is not made only by ICTP in CISA and the University of Triesta, but it's an entire system that is moving towards new frontiers of science and technology. Here the point is perhaps not the quantum technology, but rather the data. Data everywhere. Data is going to be the very big change that we were gonna face in the next, I would say 10 years, perhaps 20 years. We will be submerged by data. I know I was a teacher of database at my university and I always told my students that data and information is different. You don't have information if you are not able to extract information from the data. And that's the point. Already now, data science and the scientific computing are covering a paramount importance in the development of the, let's say, infrastructures of developed countries. And when I say data everywhere, this is the situation because we have the internet of things which is creating an enormous amount of data. But tomorrow, and we'll be tomorrow morning, we will have the internet of nano things with terrible and incredible implications in what is called the virtual twins, the digital twins. Internet of nano things will generate and collect and transport and transport an enormous amount of biological data. It's not science fiction thinking that each one of us will have a digital twin stored somewhere in a big data environment. And all this, again, I said is not science fiction. This is actually what will happen. And in order to deal with all this data, we need at least three things, but at least, I'm saying at least three things. First, computer power. High performance computing. Even though the new, let's say, development of the petaflop technology will suffer from data transmission. There will be no problem in coupling, in getting an enormous amount of information done, but the transmission among CPUs will be one of the problem. The storage will be the second problem. The algorithm will be a problem, but probably the most important one will be data transmission and cybersecurity. You can imagine how challenging would be a possible, a potential hacker, perhaps at a service of some political party huckering data of regular people. This is very scaring. This has happened. This has happened in US. Not in terms of internet of nano things, of course, but that was a signal, very important signal. So in this respect, I think what we are starting today now in Trieste, and it's not a case that we started in Trieste, it's adventure, is let's study quantum computing, but not only computing, quantum communication in general, because we are facing in the future an evolution which will be non-linear. Today, up to now in computer science, we experience a sort of linear development, exponential perhaps, but expectable results were achieved. In the future, this will not be the case. I'm convinced that quantum computing will not solve all the problems we have so far. But with quantum computing and quantum technology, especially for data transfer, we will be able to tackle problems. And most probably knowing the quality and the quantity of work done here in Trieste also give some, let's say, suggestion for technology transfer to industries. Big industries, IBM and Google are very much involved in this, but also Eurotec, thank you, Siagri for being here. They are also very active in this. So it's not a quantum technology in the cloud made only for scientists for publishing papers in nature. This is something that I'm expecting to have very important, let's say, effect on the territory. In the area, in the, let's say, spirit of technology transfer for the benefit of our industries. As I said already, it's not a case that all this has happened and will happen in Trieste. Thanks a lot. Instead of going on along the comments, I would like to profit of the opportunity of opening the theoretical quantum technology institutes in Trieste with a personal recollection that might be useful also for the young people here in the hall. I did my master thesis on belts inequality and at the time the field was considered as sort of philosophical. And I still remember that the head of the commission that declared me a master in physics was, he liked to joke, but at the same time he was the president of the Italian Physical Society and my thesis advisor. And he said to me, I declare you unemployed in physics. So I always thought of this sentence and I never said in public about this story and I never wrote this story in any of my papers. So this is a perfect opportunity to remember this old recollection was 1977. I think it can be also a message for the young people to follow curiosity, to which most of the research that then as an outbreak is motivated by curiosity of the scientists. And there are certainly many topics in this area of research that are still puzzling and interesting to go deeper and also in other areas of physics. I hope that you will have a nice talk with Nyancio Sirac and have a nice day. Thank you very much, Stefano Fondizio. It's a great inspiring words from both of you. Thank you very much. I understand the deputy mayor is here, Paolo Paridoria. I would like to say some words in coming from the mayor of the piazza. Yes. Buongiorno a tutti, scusatemi. Io potrei affrontare anche un saluto in inglese, ma preferisco rinunciare perché è un po' ruggine, quindi mi capiterete lo stesso e mi perdonerete. Porto il saluto ovviamente della sindaco di piazza che ha avuto un impegno imprevisto e improvviso, quindi porto anche il saluto della città di Trieste. In queste occasioni si nota e lo noto di persona come Trieste abbia le sue eccellenze anche in cose che magari non vengono alla luce ogni giorno, magari vedendole passando per la città. Ma queste sono quelle eccellenze che danno lustro a questa città, che danno il segno di quanto questo instituto, di questa joint venture anche tra i vari istituti possa portare degli obiettivi e risultati incredibili da quel poco che ho visto, che sono ricito a leggere, perché una questione di fondamentale importanza e sarà, vedevo la criptografia, quindi la gestione anche di nuove formule di scrittura in sicurezza in modo tale che i dati possano essere protetti nel miglior modo possibile. Io dico, c'è anche un problema di controllo di chi controlla i dati. Su questo ovviamente la parte scientifica è meno interessata la cosa, perché l'importante è sviluppare le nuove tecnologie, ma dall'altra parte, altrettanto, è importante sviluppare tutte quelle forme di controllo per dare comunque la possibilità di gestire in sicurezza l'enorme progresso che la tecnologia s'offre per quanto riguarda la gestione dei dati. Io vi ringrazio, vi auguro un buon lavoro e un buon di raggiungere gli obiettivi che vi siete prefissati, perché sicuramente sono importanti, non so, per questo ovviamente, ma per l'umanità intera. Grazie, buona giornata. Grazie mille. Okay, so before we go to the highlight of this event, we now proceed to the important part, which is the signing of the agreement within the three institutions. So we have here the documents and you will witness how the three of us sign the three documents. I think it's done. I have a second option. So I think this is your last one. So this is finished. You finish everything. No, this is finished. I'm not here. This is finished. Okay. Photograph. Okay. I'll leave it there for now. Very good. So well, this is a historical moment for all of us, so I think it was very good. Okay, so now is the highlight of this event, is that of course all of these activities are because of science and we always claim that it has to be not only an initiative in science, but you have to have excellence in science and as a way to demonstrate it, we have to attract the top scientific ideas and we were extremely lucky to have Ignacio Sirac, who is one of the world leaders in this field of quantum information, that he agreed to come and give us a special colloquium. So let me say a few words about Ignacio. So he was educated in Spain in Complotencia, but then he was a professor for some time in Innsbruck, in Austria, where you notice there has been a source of a lot of discoveries in this field, and right now he is the director of Max Planck Institute in Munich. He has been there for several years and of course, given his big contributions to this field of quantum information, he has several awards and I can list them, the Prince of Asturias Prize for Science and Technology, the National Prize for Research, the Franklin Medal, the Nilsborg Medal, the Wolf Prize. That's important for us unfortunately because I have to confess he has been always mentioned as a candidate for the Iraq Medal, but we cannot give it because if they have a Wolf Prize, we cannot give the Iraq Medal, but at least that tells you the level that Ignacio has. So recently he got the Max Planck Medal from the German Physical Society, which is probably the most prestigious prize given within the country. In general it says that for his contributions from fundamental mathematical calculations in quantum information theory, modeling of quantum antibody systems to concepts for the implementation of quantum optical systems. So it's a great pleasure to have here Ignacio. He agreed to give a talk and the talk will be the title is Quantum Simulations and the difficulty of solving quantum antibody systems. Join me to welcome Ignacio Serac. Okay, good morning. So first of all, let me congratulate you for this great initiative that you have about this Trieste Institute for theoretical quantum technologies. I think that putting together very strong institutions here in Trieste will make a unique place in the world to develop this field of quantum technologies. So it's a great honor for me and the privilege to be here in this inauguration and to give a talk about quantum technologies. So they asked me when they asked me to give a talk that should be, I mean, not too complicated, so a little bit more like for, let's say, general students of physics and things like that, so the first thing that I have to do is to apologize to the experts that I see that there are several here in the audience because some of the things that I'm going to say will be very superficial for them, but nevertheless I hope that this is good for some of the people. Okay, so you have heard that we are able to tame and control the microscopic world, the world of atoms, molecules, electrons, photons, the world of quantum physics, then this promise certain applications and in particular it provides new ways of processing information and building computers that promise to make possible some solution of some problems that so far otherwise would be difficult. It also promises us new ways of communications based on quantum physics that would make communications more efficient and secure and also new ways of sensing. And as I was mentioned in the presentation so this will for sure have impact in industry and technology in data processing and all this everything that comes with the processing of data and artificial intelligence and machine learning and so on. But here I want to focus in some other applications that maybe they are not so much highlighted when one presents the world of quantum science and technologies and this is how this field may also have contributions to change science and in particular in fields like chemistry, material science, maybe higher in the physics. So this is what I want to concentrate here so how this quantum technologies different aspects of this quantum technologies may affect our development of science itself. And the message that is behind my talk is that some of these science in particular the one that study chemical systems or materials or even higher in the physics actually is typically confronted with a huge challenge when you want to make predictions in these areas and the reason is that in all these systems when you want to describe them then so typically what you do is that you put them on a lattice so you make a model out of them and at the end this lattice corresponds to some particles some objects which interact with each other that in physics we describe there are some Hamiltonian and to make predictions about the behavior of these systems is typically connected to studies on quantum mechanical problem, many body problem on these lattice systems and even if these lattice systems is composed of the simplest objects so you would think that in these lattice we would have like a two level system a qubit in each of these lattice sites then you want to make any prediction we'll have to solve a quantum mechanical problem so we'll have to write at the end the solution will be some state some quantity on the system which even for the case of qubit will be according to the laws of quantum mechanics of a linear superposition of all possible configurations so this means that whenever we want to make predictions I don't know if this is make predictions so typically we have to solve a problem and at the end we have to give what is the wave function that we are studying and the information is in this coefficient this coupler is coefficient in each of these configurations so the general state of our system would be a linear superposition of configuration where all the qubits are state zero or all the qubits in state zero and one in one or the other one in one and since there are two to the end configurations then it means that we will have to compute and store these two to the power end coefficients and this is an exponential law so that we try to do it even with the most advanced computers we run very soon out of resources and so in particular we need an exponential memory so we need a memory that grows exponentially with the number of lattice sites that we have in the system and we will require also because we have to store these two to the end coefficients and it also requires the computation of this coefficient an exponential time because we will have to do at least one computation for each of the coefficients so this is the reason behind so this means that in order to solve many body problems in physics chemistry or high energy physics or condensed metaphysis, atomic physics so the resources the computational resources both time and memory have to grow exponentially with the size of the system with the number of particles that we have with the volume that we have and it makes it impossible to solve for whenever we have lattice systems of the order of 30 40 qubits we cannot go beyond that apart from that and this will be important but as I said before so maybe we can trade a little bit of space and for time so imagine that our computer we want to solve this problem is finite it has 10 to the 80 particles or the size of the universe and that we cannot go beyond that then can you still solve problems so we can trade this space for time so maybe we don't need an exponential space but then it will take longer to solve these problems we can do this computation and it turns out that then the computational time so we have only polynomial resources in space so polynomial small one small memory then the computational time not only grows exponentially with the size of the system but grows super exponentially it grows like 2 to the power and to the power n of the system and so that's a huge increase and that's why many problems in science whenever we have to deal with many other systems become very difficult whenever we don't have all the methods that approximate the solution so so what I want to say or talk about here is ways around this problem that are offered by quantum technologies and I will concentrate on three different methods so one of them would be a quantum computer so how a quantum computer could help us to solve these kind of problems and I'll talk about analog quantum simulators which is different like an analog version of a quantum computer and finally I'll talk about some ideas more theoretical ideas coming from the field of quantum information which is tensor networks and how tensor networks can help to solve some of these problems okay so let me start with quantum computer so how can how if we had a quantum computer could help to solve the problems that I was mentioning before and to overcome this exponential problem or super exponential problem that we have to deal with when we are solving these problems in physics or chemistry and that's something that was already realized by Richard Feynman many years ago when he had to give a talk in an island and this was at the time so people were concerned about classical computation and at the time people were building smaller and smaller computers because they wanted to have them faster faster and more powerful and they were concerned about what will happen if we make them very small and we touch into the quantum mechanics so classical computers will have to overcome quantum physics and quantum physics so it's always related to uncertainty so probably we'll have problems when we make these computers smaller and smaller and there is a conference to deal with that problem at that time and Richard Feynman came up with a completely different problem so he didn't talk about that but he said that is well known that many problems and problems that appear in physics are very difficult to solve and that's something that we will have to confront if we had classical computers however if we had a quantum computer that's something that we could solve and in fact in the paper that it's here that was not written by him but transcribed by somebody from his conference so he said that he was talking about high energy physics that if we have a high energy physics problem and we want to make predictions on this high energy physics problem so the first thing that we can make a lattice out of that so we can discretize and time put two level systems in spain in the lattice this is what we will call today qubits and develop a method to compute the evolution of the system just by making this space of qubits interact in something that we will call today two qubit case so he was describing a quantum computer and the message behind is that if you have n qubits on a lattice in a classical computer if you want to describe that you will have to use two to the power n coefficients that's what I said before that makes this exponential dependence on time however you have a quantum system itself then you can use n of these qubits just to store the state of your system just prepare the state of your system on that state so you have a superposition like that just take n qubits and put your quantum computer in the superposition and therefore you will have just to measure at the end and measure will tell you what are the physical properties that you're interested in it was something that was not taking over so I think until some few years later by some people who make formulate mathematically this quantum computer and in fact already in 2019-96 so Seth Lloyd came up with a quantum algorithm for a quantum computer to describe to perform the dynamics of a quantum anybody system so basically what he said that you have a Hamiltonian that describes your interaction of your systems of your qubits on a lattice or wherever and you want to compute the evolution operator just let's what we call trotterize it just like right in terms of small steps and each of these steps that you apply after each other would correspond in a quantum computer to different two qubit gates or quantum gates and therefore he translated the evolution operator of a quantum system into an algorithm in which is a sequence of two and one qubit gate and then he showed that the computational time in a quantum computer would increase like the number of qubits that you have square times I mean this would be just to give you the eight units of energy times the time for which you want to simulate the system everything squared divided by epsilon rd epsilon is the error that you tolerate for your system so he showed that the computational time would increase polynomially like n squared and this has to be compared with this what I told you before this super exponential because here you will have a memory that will be of the order of n qubits actually since the first paper of Seth Lloyd there have been many papers and the most efficient algorithm that we have so far for computing the dynamics in a quantum computer of a many body system is the one by these people here and it already gives an improvement so the computational time is now would be linear with the size with the number of qubits that you have linear with time and would only it tastes like logarithm of one divided by the error so it's a big improvement ok so what about now and this as I mentioned before this is polynomial time compared to this two to the n to the n that could be in a classical computer so what about the ground state so imagine that we have now a system and the system is in thermal equilibrium and we're interested in describing the physics of this system at zero temperature so what we say the ground state of you are interested in the ground state of our Hamiltonian and for example you're interested in computing the energy or to compute some physical properties related to zero temperature so with a quantum computer what we can do is to have an algorithm that prepares the ground state of this Hamiltonian and then measure there and with that we will compute the physical observables that we're interested in and so it turns out that this problem is difficult so this was shown in a paper by these people in 2006 that this problem in full general it is what is called a QMA hard problem and the QMA hard problem is people in computer science classify the problems with respect to their difficulty and this is a problem that's not even a quantum computer could solve but that a quantum computer can check efficiently so you want is the equivalent to the complexity class the classical complexity class called NP which are problems that they're difficult to solve they're exponentially long with a classical computer but they're very easy to check so it can be efficiently checked with a classical computer an example for that is factorization so it's very difficult if you do a big number and it tells you to find two factors then the best algorithm takes a very long time so most exponential time with the number of digits of the system however if I give you the factors it's very simple to check that they are factors because you can just multiply them and so there is a quantum mechanical version of that this QMA which are problems that are very difficult to solve even with a quantum computer but can be checked with a quantum computer so this problem of preparing the ground state of computing the energy of a many body system it turns out that this QMA hard so it's a problem that will take typically exponential time in a quantum computer so there have been some heuristic algorithms so some ones that maybe for some problems will work in practice and that's a very active area of research nowadays so to find for the third generations of quantum computers some problems that maybe a quantum computer could solve and with classical computer it will still take this exponential time but let's say with food generality this is a problem which is very difficult. Okay so what I want to tell you is now something about an algorithm that we just publish of how to find the ground state of a many body system that will prepare the ground state of the many body system giving a Hamiltonian which is a sum of local Hamiltonian so you can imagine that this is the lattice that I was talking about before there are qubits or higher level systems in order of the qubits. Somebody gives you what is the Hamiltonian which is maybe a sum of Hamiltonians acting on some small sites of your system and you would like to prepare the ground state and so since this problem is QMA hard it means that we cannot aim at something that is better than exponential so the computational time for any quantum algorithm that you have would scale exponentially with a number of the lattice size that you have here and so the question is which is this alpha and also which is depressive that the quantum computer still takes an exponential time with the size of your system but you have to compare this with a classical computer which will take you to the end of the power and if you limit the memory as I mentioned before if you had only a restricted memory like what you have in a quantum computer and so let me tell you very, very superficially so what is the idea of the algorithm so the idea is that you prepare a random product where you put all of your qubits in a random state so maybe what's in state 0 and the other one is 0 plus 1 0 minus 1 just take the random state and if you take a random state then you write now this random state as a linear superposition of the eigenstates of your Hamiltonian then typically there will be some overlap with respect to the ground state and since there are 2 to the power n states then typically this overlap will be of the order of 1 to the 2 to the n halves such that the absolute square is 1 divided by 2 to the n so you have 2 to the n terms in the sum so you want to have a normalized state then each of them will be of the order of 1 to the 2 to the n halves and now what you could do is to measure the energy and if you measure the energy then you will obtain some result and this of course will not be the ground state and only with probability 1 divided by 2 to the n will be the ground state so you measure of the order of 2 to the n times and you write down what is the energy that you have then at some point you take the lowest that you have and then you say there will be a good guess for the ground state energy this algorithm will take a time which will be of the order of 2 to the n because you will have to run it 2 to the n times to have a probability of the order of 1 to have the ground state however as I explained it doesn't work and it doesn't work because it's not possible to measure the energy of the system so the energy is this Hamiltonian which is a sum of Hamiltonians and in pieces we know how to measure a simple observable but not the sum of observables so this means that in practice what I would measure is one of these Hn maybe H1 here and if I measure this one then I will have the average of the energy I will not project the state into the eigenstate I will have the average the average doesn't tell me anything about what is the ground state of energy of the system so that's why you have to modify this algorithm to make it useful and I mean one of the idea is to use some Fourier method and that's I mean I will not say what it is but the basic idea is that out of the initial state which is this random state then you can add some auxiliary system use some auxiliary number of quids in such way that with some transformation with some algorithm then out of the superposition you can single out the ground state put a 0 these auxiliary quids and the rest so all the 2 to the n minus 1 rest turns out that this auxiliary state is in a state that is orthogonal and so now you can measure this auxiliary state and if you find 0 then this certifies that you are in the ground state and if you do that then at the end it will take you a little bit of an overhead because this I mean the step here requires some time but it only grows logarithmically with the phi 0 this phi 0 is 2 to the n half so it's just a linear overhead so this means that by using this trick then indeed you will be able if you repeat 2 to the n times to find the ground state and to measure the n ground state energy and so on actually there is a slight improvement of that and this to use what is called amplitude amplification people call it Grover algorithm and it turns out that Grover several years ago show that when you have when you are able to do a transformation that goes from some initial state to some other one and you can mark this state so this auxiliary is in a state that is orthogonal to the rest then it's possible just to prepare this state this state that is here and that with a number not with a time not with a required time that goes like this coefficient square but only like this coefficient so that's not so important but it means that in practice the algorithm that I just told you will have a computational time that will grow like 2 to the power n half and we believe that that's a tight bound so there is nothing better than we can prove for that and that again has to be compared with the classical algorithm then it means that if we had a quantum computer then in principle we could solve problems related to dynamic very efficiently we could solve problems related to ground state or a thermal equilibrium of systems in a not so efficient way however as you know we don't have a quantum computer and proof of that is that if you write in Google, so quantum computer you get pictures like that however as it was mentioned in the presentation actually during the last three or four years they have been a big effort in some companies and industries the European Union the American government the Chinese government to start building a quantum computer so there is a big effort worldwide and that's a tremendous challenge that so maybe at some point we will have a quantum computer and I have to say that the challenge that is to build a quantum computer is not only a technological one it's not only experimental one but actually there is a theoretical challenge behind that because now one of the main problems of scaling up the prototypes that we have of quantum computers to be quantum computers is that there are errors and in order to correct these errors in this small quantum computer you need error correction and in order to have error correction you have an overhead and this means in practice that for each qubit that you require in your quantum computer instead of 10,000 qubits in order to correct of the error so your quantum computer requires 10,000 qubits and in reality you need 10 to the 8 qubits and this overhead of the error correction then it could be decreased by two or three orders of magnitude just by theoretical, let's say methods so better error correcting codes this would boost actually the field of quantum computation okay nevertheless so this gives me the opens up the possibility of maybe trying to solve this many body problems not with a quantum computer we will have to wait a little bit for this quantum computer but with a different method that's analog quantum simulation so let me explain to you what it is and the idea of analog quantum simulation is that again so imagine that you have one of these problems that at the end you model in a lattice with some Hamiltonian that describes the interaction between the systems that are in the lattice now the idea is not to build a quantum computer that can't do gates and can't perform gates but to take another system another system that in the lab you can tame very well and where you can engineer the interactions and to make your system in the lab to behave in a way that is described by the Hamiltonian that you want to solve okay so it's a little bit like an analog version of a quantum computer in which you don't do gates anymore but you just tame your system in such a way that it behaves that the interactions in your system are the ones that you want to solve and actually there are many many proposals and experiments now of building this analog quantum simulator with atoms in optical lattice, trapions, quantum dots superconducting qubits, photons etc etc and the idea is the same as with Feynman so the idea is that if you build a quantum system to emulate the physics of a quantum system then you will never have to store the coefficients of your state because you just prepare your system your simulator in that particular state so then you go ahead you overcome the problem of the overhead that I mentioned before so what is the advantage or what are the advantages and disadvantages of analog quantum simulators with respect to quantum computers well the first advantage is that you don't have to be able to have all possible gates of quantum universal quantum computation but you have to engineer the interactions only in your system and so for example if your system is the problem that you want to solve as they appear in many systems are translation and invariant and homogeneous then you will have to be able to engineer a homogeneous system you will have to be precise and engineer a particular Hamiltonian for each of your particular components of your system but maybe one that is more important is that it's probably much more with respect to errors and you can qualitatively see it like that so imagine that this is the Hamiltonian that you want to simulate and then you have some errors and the errors are extensive meaning that at each lattice size then your Hamiltonian is not exactly the one that you have but it has some corrections so this means now that this part or your Hamiltonian may be big very big and huge because it scales with the number of size that you have in your system the Hamiltonian H that is in front is very small and so what happens is that in many of these problems then you're interested in some intensive quantities so you don't want to know what is the way function of the whole system but you don't want to know what is the magnetization per particle the energy per particle you're interested in the thermodynamic limit and so therefore you have to look at intensive quantities so things like that, observables that are intensive and so this means that very likely then the will accumulate in an intensive way so meaning that if this error here is 10% of the Hamiltonian that you have so typically here you will have that when you measure when you look at your observable then you will have a 10% of error in your observable in a quantum computer if you have an extensive error then your quantum computer will not be able to give you any sensible result or say without other words you have a quantum computer which has an algorithm and in the middle of the algorithm you flip a qubit and continue at the end the result that you have will have nothing to do with the result that you're looking for however in these systems I mean one is a hope that something that is not proven but then one has the hope that they are more reliable because they are looking at intensive quantities and so people have discussed this with computer scientists many times and it's very difficult to formulate in super mathematical terms but there is an image that shows that so one of the first quantum or the first quantum simulation that was performed was done with atoms in optical lattices and people were observing quantum phase transition between a super fluid and a multi-insulator so they wanted to simulate some Hamiltonian the so-called Bosch-Hauer Hamiltonian and they have errors and the errors I mean you can estimate where this epsilon was of the order of 10% there are huge errors and they were looking at the phase diagram and then changing the parameters and they wanted to see whether there was something which was super fluid changing the parameters it was going to something that is called a multi-insulator and changing the parameters maybe back to super fluid and even though they had very errors that's the plot I mean whatever they plot did you see that there was a phase transition here so they were measuring something and that's exactly what I mean that even if you add some the quantities that you want and maybe here they are not so precise you have a 1% error or 5% error but that's enough in order for example in this case what they wanted to do is to see whether there are quantum phase transitions and either to characterize some of these phase transitions so let's say that actually one of the systems in which people are doing these quantum simulations trying to solve many body problems the systems are called atoms in optical lattices as I mentioned before the atoms can tunnel I mean they are trapped in some external potentials created by light, by standing wave and then the atoms can hop from one minimum of this potential to the next one and also when they are close to each other then they interact and this emulates the physics for example of electrons that are moving in a solid and so in fact now you can have the experiments with this simulator to engineer the interactions between the particles maybe using the different layers using magnetic fields such way that at the end this Hamiltonian can simulate how our light Hamiltonians and it's as interesting this is very interesting because actually you can not only simulate bosons so these atoms could have bosonic character can be also fermionic character can also have some internal states some spins you can change the geometries you can have the square lattice this Kagome lattice the dimension can work in one dimension two dimensions and three dimensions so in fact this I think this experience have had a very big impact in condensed matter physics and so actually has brought together the field of atomic physics quantum information and condensed matter physics and during the last five years so our group and some other groups including Marcelo who is here have been extending this idea of simulation condensed matter using cold atoms in optical lattice to simulate condensed matter physics so problems also to try to simulate high energy physics problems and then the idea is that in high energy physics you have matter which are like the fermionic degrees of freedom and then you have the forces which are represented by gauge fields which are bosonic and so what you can do is to put this in a lattice a lattice Hamiltonian for any problem for problems in QCD, QED and the fermionic will be described now as fermionic atoms and then the interaction will be mediated by bosonic atoms that are also moving in the lattice and if you tune the interactions properly then they correspond to the Hamiltonian that would describe these high energy physics problems and I wanted to just to throw some ideas that we have developed lately in order to show how you can do analog quantum simulation of quantum chemistry problems very briefly and I have to say that the level of experimental difficulty goes exponentially up in this direction so that's something that people are doing in the lab there have been first experiments prepared for observing very toy models in high energy physics and that's something that will require much more effort but anyway so I'm a theorist and I've doing my life have seen the things that you proposed that look impossible today I mean in a few years look possible ok and so what is the main idea of this paper that we just submitted when they did that when you have a chemical problem so you can think in terms of Born-Oppenheimer approximation so that's one that the typical quantum chemists do very often in what you would like to do is for example to see what happens if you have a nuclear set at some positions and so what is the equilibrium of the nuclear so is there the nuclear at different positions then you have some electronic states for this nuclear and this will have certain energy so you fix like an infinite mass the nuclear and then you find what is the ground electronic state for your system and this gives you an energy now you change the position of the nuclear and then you will get a different energy so what you have is a landscape of the energy as a function of the position of the nuclear and now you look at what is the minimum of this landscape and this would be the configuration where the nuclear at the equilibrium position and that's something that people in quantum chemistry are interested and once you have this configuration this is the form that the molecule will have the geometry that the molecule will have then you can study what is the energy what are the excitations, what are the physical properties and so on and it is to do something similar with atoms in optical lattices and so in this Born-Oppenheimer approximation then what you have is that so there is a nuclear Hamiltonian which is classical because you fix the position of the nuclear then you have the kinetic energy of the electrons so the electrons can still move then the interactions the electrons interact with the nuclei so they are attracted by the Coulomb interaction but the electrons also interact with each other through the electron-electron interaction as I mentioned before what you would like to do is to find what is the ground state of this Hamiltonian for some fixed position of the nuclei and at the end we compute the energy and as you will find what is the minimum is the stable configuration of your system and so what's the idea with colatos in optical lattice so first of all to do it on a lattice you have a big lattice in which now you will have electrons will be represented in these lattice by some fermionic atoms the fermionic atoms can move in the lattice can hop in the lattice and would represent the kinetic energy and if the lattice is sufficiently big and then you go to the continuum limit then this would represent this term in your Hamiltonian the kinetic energy of the electrons then to have some external lasers that will attract electrons and this would correspond to the Coulomb interaction of the nuclei so how you can use holographic methods in order to create something that decreases like 1 over r in such a way that these atoms get attracted by this nuclear this would be this part of the interaction and then for the repulsion between the fermions to represent the electron-electron interaction so what we do is we copy nature and these atoms in optical lattice are neutral atoms so they will not repel each other with one Coulomb interaction so I mean maybe they have a van der Waals interaction 1 over r6 but the idea is to introduce a boson atom in such that the boson atoms somehow interacts with the fermion and interacts with the other fermion and mediates interaction in as much as the same way as electrons interact with photons and the photons mediate the Coulomb interactions so we figure out a way of doing that and that's what is our proposal okay so with that I would like to go to the third part of my talk so we're talking about many body problems and so how quantum computer can help to solve some of these problems we don't have a quantum computer maybe an analog quantum simulator and now I'm going to change and try to look for some other possibility and these are tensor networks so this is not a quantum computer it's not a quantum analog so this would be some mathematical methods to solve these problems and that are inspired in part at least by the quantum information theory and as you know so two of the revolution in science that occurred the last physics, quantum physics and information theory now can join together in what is called now quantum information theory and that's the theory that is behind the quantum computer, the quantum communication systems and so on so people especially in the 90s sat down and tried to translate the Shannon information theory to quantize it in order to describe now the states of quantum computers and so on and try to figure out how to use quantum computers, how to prove that quantum communication is secure etc etc and the idea is to be spire but by some of the developments in quantum information theory to try to attack many body problems with the different angles that the ones that have been used so far and so the idea is that in quantum information when you have a quantum computer so people were asking so if I have a quantum computer why is it much more powerful than a classical computer and so in fact if your quantum computer would be in a product state so the qubits in your quantum computer are never entangled then I mean it's very easy to show that you cannot get any advantage with respect to a classical computer because you could describe this product state of the order of n parameter you just have to say what is the state of each of your system so everything grows polynomially so there is no advantage so it seems that whenever you don't have product states namely when you have entanglement then is when you have some gain in a quantum computer so that's why people saw that entanglement is a resource in quantum physics so let's start let's study entanglement and let's ask questions like how entangled is a quantum computer in a quantum computation and now you can translate the same question to many of the systems not a quantum computer but one of these problems that we were talking about so you have now a problem in higher energy physics so you have a problem in quantum physics or atomic physics these that have this exponential coefficients so how entangled are the states and it's very clear that if the states are not entangled then you could have a method to solve that would not require this exponential overhead that I mentioned before and what happens is that if they are little entangled whatever this means maybe it's simple to solve this problem so that's a little bit the idea behind these tensor networks the idea is that in nature in some problems in nature especially if you are in thermal equilibrium and you have local interactions that's important to have the two things local interactions and thermal equilibrium then turns out that the states that appear in your system are not very general actually they can be parameterized by the possible Hamiltonians and since the Hamiltonians are local then the number of parameters that you can that you can describe all these problems is very little so this means basically that the states that appear in nature in local interactions and thermal equilibrium actually are in a corner of this exponentially big Hilbert space which is physically relevant and so the idea would be so what happens if we have now a description that allows to address very efficiently this corner of Hilbert space maybe not so good the rest of the Hilbert space but since this is the physically relevant then we could describe our system so how could you identify what is the relevant part of the Hilbert space where the dynamics occurs in this system and this has come by asking questions so how entangled are these states and it turns out that the states that will feel the condition that I told you before are very little entangled as opposed to what they could be so they fulfill the so-called area law so if you take a system in one of these states and you look at the entanglement between some region and the rest then the entanglement is not extensive it does not increase with the volume of your system which is what would happen if you take a random state in Hilbert space but only increase with the perimeter with the area it's the surface area and so this makes that the states that appear in nature and the condition that I mentioned before are very very special are in this corner of Hilbert space so maybe instead of treating all the states in Hilbert space on equal footing when we write wave functions there is another way of treating them especially and that's what tensor network aims at and so tensor networks what it tells you is that when you write your state in a superposition then the coefficients that appear here can be considered to be a tensor so for each configuration you have a complex value and the idea of tensor networks is to write this tensor with many coefficients so this has a huge dimension or due to the huge dimension of the Hilbert space in terms of smaller tensors that with few coefficients that are contracted among themselves so to write a big tensor with many coefficients it's a very small tensor that are contracted among each of them and so what you will have to store, what you have to compute in practice will be this small tensor so the number of parameters since there is one tensor per lattice size will scale only like n or polynomial except exponentially so I'm totally speaking what we want to do with tensor networks is to just take the state of our system which is represented like a big tensor and to write it in terms of these small tensors which have the lex, the original lex plus some other auxiliary lex that are contracted with each other such way that we can approximate the whole state instead of these tensors and the game that we have is that instead of storing and computing all these coefficients we will have to compute the coefficients of small tensors which are much less and of course if you try to do that for a state in the middle of the Hilbert space this will not be possible or in fact this small tensor will not be that small so these indices that are here will have exponential dimensions however the hope and actually what can be proved in some cases is that it is possible indeed to have descriptions that are very efficient with small tensors as long as you are in this corner of Hilbert space and this happens especially with this project the entangle per states which are tensors that have this configuration they have the geometry of your lattice and in this case it has been proven so lately in this paper here that you have a state thermal equilibrium at any finite temperature and with a geolocal Hamiltonian then you can approximate it in terms of one of these peps the number of coefficients that you have only scales polynomially with the size of your system not exponentially so that's something that as I mentioned has been proven so this gives a hope that maybe using this tensor description then we can address some of the problems in many body systems actually these peps it's nothing else but a generalization to higher dimensions of something that was well known in condensed metaphysics that are called matrix product states and I have to say that there are other tensor networks that are very interesting from some other points of view like this may be introduced by Gifre some years ago so people have now been able to do some computations using these tensor networks and in particular in one dimension let's say in one spatial dimension one plus one dimension this has happened already many years ago and it was a successful a success it's basically due to what is called the density matrix randomization group algorithm but the challenge was to go to higher dimensions but it seems that in the last couple of years there have been a big advance using this tensor, these peps in this two plus one dimensional system so already some time ago so Philip Corbott and collaborators were able to use these peps and to get very accurate results for the TJ model now during the last couple of years there has been these assignment collaborations in which they have taken different methods and tried to solve the Haber model for different methods and peps have played a very important role there so now they have a phase diagram for the Haber model and if you talk to Philip Corbott who is also the person who is behind these computations with peps now they can see the physics of stripes they can put nearest neighbor interactions they can do two band models, three band models so I think that now they are able to solve many of the problems that seem to be very difficult and also Philip Corbott with collaborators has been able to solve some other models and very accurate with these methods so it seems that indeed these tensor networks may help to solve some of the problems or is adding a new tool to solve many other systems as long as you are in thermal equilibrium and with local interactions so in fact for you want to compute dynamics there is no method that I know that will give you this polynomial overhead so it still will escalate exponentially with whatever you do even in practice so to finish let me say that these tensor networks are not only able to describe states but you can describe observable operators using this tensor network description you can describe maps for example you can describe maps that map bulk properties into boundary properties so you can have holographic principles described by these maps and also geometry so for example you can use a hyperbolic geometry put your tensor networks in your hyperbolic geometry and you can use it to describe bosons, fermions and even anions in these systems and that's why the field of tensor networks apart from being used to solve problems numerically they have been used also for example to introduce symmetry protected topological phases so I'm very happy to hear that Xiaowen Wen got this Dirac Middle this year and among the many things that he did is he introduced symmetry protected topological phases using tensor network language you can also use boundary correspondence people have lately applied these tensor networks to solve some lattice gauge theory models they have also made some toy models using this hyperbolic geometry of ADSEFT models these tensor networks are also using quantum information in one way quantum computing quantum error correcting codes and in terms of tensor networks people are using this language of tensor networks also to describe that some sequentially general state-state and also they have appeared in mathematics independently and so they were discovered independently many places so I think that tensor networks can be rediscovered over and over during the years and in particular in mathematics it seems that there is also a very important area of research ok so with that I would like to finish so saying that quantum science and technology promises an impact in society and industry but maybe also in science and in particular it can help in different ways to address some of the problems that are difficult in science especially when you have quantum many-body systems told you here so how we believe that quantum computers through quantum algorithms can help to some of these problems analog quantum simulators may also give help and in fact these experiments are very advanced so there are claims that people have been able to simulate problems that are interesting and that we cannot solve with classical computers already and also the field of tensor networks which is just putting together a little bit of quantum information with condensed metaphysics with the expertise of people also in high energy physics looking at area loss I think that has been very rich so if I have to say something that is very impressive from the point of view of quantum science and technology is the connection that it has with many areas of physics so I've been working in this area for the 90s and in the 90s quantum information theory had a very clear connection with quantum optics because people were trying to build quantum computers, quantum simulations with quantum optical systems and also with solid state physics because they also were building quantum computers and quantum simulations but it seems that this has extended a lot and it's not only let's say the quantum computers and quantum simulations which is very attractive but also the language so people in quantum information learn a lot from condensed metaphysics high energy physics, quantum gravity, quantum chemistry and for example last year I went to more meetings on let's say quantum gravity than to quantum information theory because there are many many connections and this is becoming really a great place so in Munich we have started a center for quantum science and technology as well not theoretical, it's more experimental but anyway one part of our, one important part of our research is to try to make connections between quantum information and high energy physics quantum cosmology even quantum chemistry and I think that that's something that is opening up for the students not only to work on the technology part but also on the scientific part and I would like to end just thanking the members of my department we are many people in our department so we are three staff members Marie Carmen Bagnuls who is not here but I want to also to highlight that we have several Italians people coming from Tieste so Lorenzo Pirolli did the PhD here, Tomaso White did his master's thesis here and Giuliano Giudici is visiting, he's from the group of he's from here actually for ICTP and he's visiting for some months and we also have some Italian people so with that I would like to thank CISA ICTP and also the university because of this new center congratulate for this new center and hope that they will still be educating students and bringing them abroad and we can take advantage from them as well thank you very much thank you very much, it was a great great talk actually, really enjoyed it do you mind taking some questions any questions comments please concerning your simulation of the electronic molecular problem in order to simulate the Coulomb interaction not only you need the bosons you also need massless bosons can you take out the mass actually there would be one possibility to have massless bosons if you would go to a part of your dispersion relation which is linear so I mean they are moving in lattice so in principle the energy of these bosons can not only be near zero momentum but could be in the linear part of the dispersion relation in this case would be massless but this is not what we use in terms of levels of the atoms then you can have some virtual interactions which is not exactly the same as what you have with, let's say, Dirac with quantum electrodynamics but with some other more tricks from atomic physics is how we do that thank you cheers so this appearance of hyperbolic geometry is it something optional or is there the appearance of hyperbolic geometry that you mentioned or is there some system where it naturally has a hyperbolic geometry well this was optional this was mathematical so you can just consider a lattice which has any geometry and some people have taken hyperbolic geometry that does not correspond to a physical system as far as I know but it's the way in which for example people have built in Stanford some ADSEFT toy models which they put these tensor networks in that geometry and then they represent the ADS in the bulk represents the ADS geometry hyperbolic geometry and the boundaries where the conformal theory should be but that's more from the mathematical point of view so there is no physical system that I know that has such a geometry so it's about the first part so can you map the ground state problem in a grover like database search or is something different well this is it's something different so because you have to be able for grover algorithm you have to be able to mark your state so how do you mark the ground state you don't know what is the energy so that's the first part of the algorithm it's an algorithm that will mark your ground state by putting some encila and when you have that in principle you could measure and then you could find the ground state and then you can put on top of that grover because you can mark, you have an encila now in a different way and then going back and forth then you can use grover in order to speed up so the square root speed up is grover the square root is grover so supposedly you want to analog simulate quantum analog simulate a quantum chaotic billiard what are the signatures of the fact that the classical limit is chaotic it's a good question so because I guess that in principle should be more difficult to simulate like it's even a classical simulation of a chaotic system you have an exponential spread out and so here I think that people in quantum you know this better than me but in quantum systems what happens is you have a small perturbation of your Hamiltonian then your vector will diverge very much with respect to the original one so I guess that what could do is to evolve with two different Hamiltonians make an interference effect one of them is a little bit perturbed with respect to the other one and this will tell you that the overlap decreases exponentially something like lochmit echo and I think that people show that in the classical limit so you have a system that the problem the h bar is going to zero the one that you define h bar is going to be an exponent for the classical system so there will be a way I mean now that's a very fashionable system to study quantum chaos because they also related to SWK models quantum gravity and so on and people have been proposing experiments in order to observe quantum chaos with those systems and I guess that some of the proposals are required to make this kind of interference effect if you change a little bit your Hamiltonian Questions? A couple of short questions What is this kind of a vain diagram with a polynomial in the QMA you had another branch called QBP or something What is that? So BQP is kind of the equivalent of P for quantum so it's bounded quantum polynomial so it means that these are problems that a quantum computer can solve efficiently probabilistically Quantum computer typically works probabilistically and so these are the problems that I would say roughly speaking the problems that a quantum computer can solve efficiently and they are different than NP so P is included that so the classical polynomial are included in a quantum it includes part of NP because factoring problem for example is there it probably does not include all NP because there is I mean NP hard problems so three problems probably cannot be solved by a quantum computer but it also has problems that are not even in NP even in what is called BPP which is the probabilistic classical problem so it's a very complex people call it a SU of complexity classes and when you put quantum physics it gets more complex Can you define which of the NP problems can be of this class? We only know examples so in fact it's not known whether NP it's equal to BPP is equal to P so that's one of the problem And in your analog simulations you were able I mean people are able to simulate gauge interactions for instance is it doable to think about simulating gravitational interactions? Okay so what you can do and people have been proposing is to use I mean to simulate quantum field theories in core space so with back reaction to gravity so the problem is that there would be back reaction then you should have something like a quantum gravity theory and so since there is no theory for that this would be this may be possible so there are people in which have been also proposing how to have semi classical gravity and this can be included and have been proposed as well but of course I guess that we would be able to put gravity there then this could maybe help us to understand some of the problems of quantum gravity Okay so let's thank you again for such a great talk