 Okay, so good morning everybody my name is Marcello and I've been working in Eastbrook in the past years and I just moved to ICTP and the topic I want to talk about today is actually slightly off the conference main goal which is RG. There will be something also about RG but we'll mostly deal with how we try to realize dynamics of lattice gauge theories in synthetic quantum systems and the conundrum of all of this is really try to approach lattice gauge theory is a very very broad concept using quantum information type tools and what I want to show you in concrete there are actually three examples let me just briefly mention what I will tell you at the end of the day one is how to build analog simulators analog quantum simulators of lattice gauge theories using ultra cold atoms trapped by light optical lattices the second thing I want to tell you which will be very brief unfortunately is how we can design novel algorithms to tackle exactly or quasi exactly the real time dynamics of lattice gauge theories using methods which are actually motivated from time-dependent density matrix normalization group techniques or gauge tensor networks and the last thing I want to show you is actually how we can use a quantum computer well a very small one to realize the dynamics of some interesting phenomena in in lattice gauge theory in particular in this case I will show you the first experimentalization of a Schringer model using trapped ions so these are the mostly the two topics I will talk about so the outline is then what I want to tell you very briefly what this synthetic quantum systems actually are they find a challenge and then use as a workforce a workers the Schringer model to tackle these three different paradigms okay so what are the synthetic many body quantum systems I think some of you most probably have heard of those these are systems which are composed of many degrees of freedoms these are many body systems where the interaction actually can be microscopically tailored by means of external fields so we can control this many body quantum systems and there are nowadays many experimental platform where this can be done the most prominent one I think is at least from my perspective is ultra cold atoms and molecules and here you see these are snapshots of a quantum system where people are able to image lattice models side by side quantum by quantum so there is really genuine single quantum control nowadays there are also many experiments using superconducting circuit architectures and these are just few pictures to illustrate that there are also trapped ions and there are many more okay photonic system cavity QED engineer materials and so on and so forth so this is really a huge field which is developing mostly thanks to technological advances so the point is which kind of physics once you realize this physical system you can explore I think there are two scenarios I mean one is that that I mean these engineer quantum system allow us to study phenomena that have been confined to theory only for a long while and the examples that I find most amazing are actually integrable systems was I mean physical realization are usually very hard to to get I mean in this in this cold atom gases for example one can realize those dynamics and one can also study the out of equilibrium dynamics very accurately which is in an isolated system which is not that easy for example incondescent metacentics and the second motivation for studying this quantum system actually traces back to Feynman and is to use them as quantum simulators of theories that otherwise we can't really solve either by analytical or numerical means so what is in few words the idea of quantum simulators I mean this is really Feynman's 1982 it's quite pioneering it said okay nature isn't isn't classical okay so if we want to simulate it at some point we need to use quantum hardware okay and this is in 82 it's quite amazing and the reason for a connexion method theories as I am it's actually quite simple I mean suppose that you want to study some Hamiltonian and which which works on a certain Hilbert space the dimension of the Hilbert space scales exponentially with the number of degrees of freedom so as far as it gets I can't fight this okay and imagine that you want to take for example a spin system exactly you are limited to 30 spins this is very far from thermodynamic limit okay so that's the essence of this of this intuition and obviously I mean a series many methods have been developed to undergo a circumvent this problem but I think there is still physics at equilibrium which is very hard to solve by control means and the typical example are fermions finite density because of the same problem and there is also the other problem that for time evolution real time evolution of many particle quantum system we do not know what I mean a controlled method that can be applied in a generic sense and just I want to give you this this kind of comparison and what a quantum simulator can try to do compared with one of the most powerful supercomputers on earth at the moment this actually is an experiment this dots are an experiment this is unobservable as a function of time is actually not important what this observable is and this dots here are a quantum simulation I wouldn't call it one simulation but I call that an experiment that traces this observable as a function of time and you see okay can go on forever I mean this is this is okay and it's an experiment you just measure and on the black on the black line here you can see the most accurate theory for this type of problem which is in this particular case time-dependent the energy run on the most powerful supercomputer and you see at some point this breaks down and this is really trying to fight an exponential scale yeah so classical computations I have in some cases limitations that quantum computers do not have simply not have so this this is in essence the the intuition of my man made real and just I mean these are a couple of slides that that illustrate briefly what the approaches are so what are analog and classical simulator I think most of you know I mean analog simulators ever varying only history form anti-kittera to current machines it's sorry analog simulators from current machines like wind tunnels where we cannot so for example I don't make equations like digital simulator are a bit more modern in terms of concepts because there we can control error I think the first worldwide example is probably Pascal in and nowadays we have supercomputers at the very same level you can do the same stuff with quantum technologies so you can define the same two columns with quantum technologies you can have analog simulators well you can have digital simulators first obviously they are the equivalent of our computers our classical computers and the basic idea there is that if you want to solve the evolution of some time-dependent problem you trotterize it you decompose into gates exactly as you do on a computer you decompose certain operations into to set of even more fundamental operations okay there is also the analog side okay the basic idea there is actually to do exactly like the wind tunnel the wind tunnel you want to simulate a very large plane you can't you don't want to build one for each time you actually do that and you don't want to have a wind tunnel which is I don't know under meters large so what you do is you take a small one we do the same year you want to simulate certain theory you do it you build it in a smaller in a smaller setting which you can control and then you started the time evolution of the properties or what you're interested in there is also a third option if you have 15 million dollars you can try to buy quantum annealer but the fact that this really works I don't know I don't I think it's recorded so I won't say anything about this so and just as a comparison this is with the analog line when you talk about when people talk about analog quantum computer really think about wind tunnels kind of experiments when people talking about digital quantum simulators or quantum computers think about really just adapting the computing strategies of classical simulation to quantum systems there are some highlights I will just flesh through I mean in this in this synthetic water system several things have been done at the experimental level some long standing field theories have been realized I mean other models have been realized as well and one has control of the single particle level but this is until now not of interest to us we're interested in gauge theories this is completely different from the point of view of gauge theories what has been done well not on gauge theories but on gauge fields what has been done is the realization of static so background classical gauge fields and just as a reminder I mean the typical example of a static gauge field is is the so-called of standard model you have particle running on a square lattice and what happens is that the tiling of this particle is actually assisted by a classical field so a phase and it's known that these phases actually I mean have certain importance in many condenser method phenomena and this has been realized for bosons a long while ago and for fermions recently and what has been observed is also the possibility of having skipping orbit and edge currents in this kind of in this kind of cold-atom setting however I want to emphasize this is classical okay these are classical fields this is there is there is no gauge symmetry so the challenge that that I mean we have is actually to identify a method of doing that obviously gauge fields are not important only in the context of particle physics even there I mean even thought this is the origin of all of that they have importance as we have seen also in the talks yesterday incognizant method system like QED3 theories they have also some importance in topological quantum computing I'm not I'm not gonna discuss these two I will focus mostly on this part but just to tell that I mean the impact is of gauge theories obviously enormous and why we want to do that I think one one one main motivation for designing actually quantum simulators for gauge theories that these are ubiquitous this we just said they exhibit extremely rich physics and they are challenging many body problems okay very challenging as most of you know much better than I do another perspective is actually to look at them as computing computing challenges and this is the DOE uses your super computer they think it's 2009 and you see that is QCD in terms of fundamental science is one of the most intense resource intense activities so I think it's also a good idea if you want to really build a quantum simulator which is not an easy deal to try to focus it on problems which are really really challenging there is a problem however I mean we are used to write like and field theories in general and continuum and here we want to create a create a connection with this synthetic quantum system which are usually made out of spins out of fermions out of bosons so the first challenge on the theory side is that we have to pick up a formalism to establish this link okay and I think there are quite some interesting options around also developed in recent years the one that we picked up is actually stimulated very much by the work of using his visa and other people also before him is a quantum link model and I'm gonna briefly explain what it is about in few slides so the first challenge pick up the formalism then we have to identify strategy to get a gauge symmetry because they are not there in the systems I mean at least naively thinking and then we have to find physical incarnations okay here we go okay just as I mean as I told you at the beginning I mean we will do that in the context of the Schringer model not because of limitation of doing dealing just with a simple theory just because I mean all the procedure is simpler most of these techniques that I want to tell you about can be extended to higher dimension and some of them not all of them can also be extended to non-abelian like this gives theories yeah so I'm a simulator no algorithms and quantum computers so brief review of the Schringer model this is QED in one plus one dimension so one special and one temporal dimension I mean you can see the Lagrangian I mean from the original paper here and the important parameter of the mass and the gauge coupling then this describes nothing but charges particles coupled to a human gauge field in 1D and the fundamental degrees of freedom if you think about I mean this is just electron and positrons and the electric fields there is no magnetic field there is no transfer of degree of freedom and if you're interested in its dynamics why is this interesting actually because this model like QCD actually shows confinement in a much simpler form but is still there can show dynamical stream breaking which is what I'm trying to illustrate here so if you try to pull two particles well a particle and antiparticle at some point the string just breaks and then you have creation of mesons and in addition if you want you can also study some non-trivial topology because you I mean differently from quantum metroponomics in other dimensionality is here a topological term can actually be introduced and as strong effects as was shown by Koleman so and this are just meson this is the stream breaking and I mean when I talk about this stuff of course you're still telling me okay keep your freedom on the ground and this is what I'm trying to do but just saying okay some of these phenomena are still extremely interesting and not fully understood and I mean these are remarkable simulation on the stream breaking of QED of the Schringer model done in a group of Jürgen Berges in Eiderberg and just to tell that this kind of phenomena even in the simple models are still to be fully understood especially the time dependent ones so we have this interesting physics and the point is we want to study a certain phenomenon I mean that we want to focus on a single one stream breaking is quite hard but so but let me tell you something different I mean we can study for example a Schringer mechanism okay we start from a vacuum if we quench the vacuum we start we will start having particle-antiparticle production because QED vacuum is unstable to that and what happens is nothing but this I mean this is axis of time and this is our vacuum state you will start having particle antiparticle creation and this is interesting not only from the theoretical side again but because in some intensity laser facilities people are actually trying to study this okay obviously this is the true theory this is QED in four in three plus one dimension but the Schringer model still will allow us to understand this mechanism if we can compute its time dependent dynamics or if we can quantum simulate that in another accurate way so now if we want to engineer that in a synthetic quantum system we have three main challenges the first is how to represent matter and antimatter I mean here we just have fermions or bosons how we do that the second one is out to do this the gauge fields and now how we can realize the proper dynamics most of these problems actually were solved by by other people and this is mostly taken from that is QCD literature so let me briefly tell you a few things the first problem is actually that we want to have the ingredients all in the same slide and this is what it is if we have a not for not so formal definition of a lattice gauge theory what we need as ingredients are a set of fields setting on vertices the fermions the matter fields and on bonds or links the gauge fields we will need a set of generators which define our gauge symmetry and they will have certain commutation relations obviously the physical ill-bred space will be defined by by Gauss's law or in our case it's just one Gauss law and then since we are interesting in more in Hamiltonian formulation that Lagrangian formulation we will need to define certain objects which are our gauge var and Hamiltonians that commute with all the generators and obviously they contain both the fermion fields and the parallel transporters now the first thing how we deal with the fermions this is easy we can just use staggered fermions that most of you probably know so we represent on odd sides electrons and even side positrons so that just a single tunneling of a particle over the bare vacuum creates an electron-positron pair and the corresponding mass here because of the staggering just comes with a factor minus one to the exit front staggered fermions the fermions are easy and this is easy in general the fermion part they are part of the gauge fields so the problem about the gauge field is that is extremely hard to get a quantum simulator of a parallel transporter actually I think there is still until nowadays not not an even single paper that points out the solution for this problem so we had to deal with another formulation of last is gauge theory and we use this quantum links introduced by uve introduced actually by orn first in the 80s then rediscovered by orland and then by uve in the 90s and there the idea let me tell you very briefly without any technical detail is to replace for you one theory is the parallel transporter with a lowering so with a increasing spin operator as plus and since the electric field has to be in another basis this will be nothing but sigma z operator or sigma 3 operator defective ameltonia for the for a lattice finger model in the corpus class sask formulation using this quantum links is actually quite simple here you have a term which couples the matter fields the time of the matter fields with the gauge field this will be you in a conventional lattice gauge theory you have a second term here and this is this is the matter field interaction the second term we have already seen this is the staggered mass of the staggered fermions and then you have a third term which is nothing but the electric field square term that's all you can have in one plus one dimension you cannot have any magnetic field and just as a cartoon I mean what happens imagine that this is your physical state that you engineer in your quantum system what you require is that every time a fermion tunnel from one particle to the other from one side to the other there is a spin flip of the corresponding degree of freedom that lives on the bond so what happens is that something like this is allowed while the single tunneling of a fermion is not allowed because of Gauss law obviously one can go on and work out the full gauge invariant illberg space this is not extremely interesting and this will depend most importantly on the representation of the spin one can do that and then there are sites which are not allowed so configuration which are not allowed configuration which are allowed so they are gauge invariant and gauge variant configurations but what is important for us is how to how we actually get these dynamics and in order to show you I mean I will just do it as a cartoon level and not tell you the full story if you're interested in the full story I will do that so imagine that now that you have a model of just two sites with the boson and a fermion degree of freedom so these guys the blue one are fermions and the orange one is a boson this microscopic very minimal system is actually described by the following Hamiltonian you have you can have tunneling of fermions tf this first step you can have tunneling of bosons second term here and then you can have interaction in case the boson and the fermions actually sit on the same site what happens in perturbation theory this is really simple I mean second or perturbation theory you is that the effective dynamics of the system whenever boson tunnels kicks the fermions on the other side because I have to pay energy price you so the process will not be resonant and if we write this boson degree of freedom using the Schringer representation what one gets just that this very many minimal building block level is exactly Hamiltonian that would need in term of gauge matter interaction in the Schringer model okay so this is exactly what we need and I mean this kind of I mean don't actually have been already realized using just boson so slightly different in many experiments especially in Munich at the JQI and what is important is that even out of the very simple example here you can understand there are local conserved quantities obviously here they are rather trivial so they're not associated to energy any gauge symmetry but once once you scale the system up so you really do it fully and you can do that using different ingredients coming from the optical lattice technology this becomes a true gauge symmetry you just need to have a boson degree of freedom that assists that lives on bonds and realizes the gauge field a firm in degree of freedom that just realizes the fermions staggered fermions and then you need a second bosons because second boson because of technical reasons okay so we know how to how to get very simple gauge theories in particular you one one plus one dimension using cold atoms in optical lattices and in this case for example we can try to observe stream breaking this is no problem really that's something that you can realize that you can see in real time and this was actually mostly motivated by the numerical experiments by Jurgen and we have also simulated what will happen in a cold atom experiments indeed you will have elected field relaxation starting from a very strong elected field in the middle as you have here this then relaxes obviously there will be light cones here and we can we could see the same doing numerical experiments on the cold atom model which is slightly different from this formulation so also the interesting phenomena are actually observable just a bonus light I'm thinking I'm running not that great with time just to tell you that okay this was a billion until now some people will say okay thanks you can do perturbation theory in some cases it's also possible to do non abelian for non abelian there are severe limitations so not all groups are accessible but what we found out and also other people found out is that for you and groups with then until 10 and for s u2 maybe s u3 groups s u3 is really very challenging there are ways of realizing the systems using in particular fermionics species which are called arcane earth atoms and these are just pieces which have two outer electrons in the shell and magics nature makes some magic and realizes naturally s u and symmetry is in this atomic systems which is then the basic building block to realize this non abelian gauge theories I will really just not discuss what is what our implementation are based upon their based upon the fact that we can actually use embedding algebras on the link degrees of freedom and then translate all this Hamiltonian and embed the algebra language to the cold atom settings and well okay that's it but if you're interested I will I will be happy to discuss you in more detailed in an abelian case so let me come back to abelian so what I told you is this analog simulation allowed to the finger model let me briefly fresh through this novel algorithms I mean based on this gauge tensor networks I mean the problem is that real-time dynamics we know is hard but in one dimension there are algorithms which are based on the normalization group time dependent normalization group that are actually allowed this allowed to do that for conventional models like easy models abad models and all that what we found out is actually that this model are very easy where can be extended in many cases to gauge theories we were obviously not the only ones also there was work at ikfo and in Vienna on these topics and what we have done is we have started a stream breaking dynamics in real time also including entanglement which is not possible to include in another kind of treatments and I just want to tell you that this is something in that is potentially interesting and if you're interested in discussing more about this I will be very happy to do that so last point okay is about the quantum computer okay so quantum computers we have seen already in the in the context of of experiments these are called digital quantum simulators and the idea is get an amiltonian trotterized so this mean decomposing to operation that you can actually perform and then observe the outcome of your computation of your experiment this approach is very powerful in the sense that you can solve generic problems is really like using a classic classical computer you can just arrange your set of operation in this case arrange your set of laser configurations and all that to design the problem that you're interested in the problem is that this is also very after realize okay you have to pay a price okay in the sense that the available resources are extremely limited okay I think until nowadays the best quantum computer that is on the market at six pins I think all of us here can code six pins very quickly I mean on a classical computer so our goal in order to somehow exploit this flexibility but at the same time do something which is has to be efficient was to define efficient quantum software that can realize then gauge theories and this again we did taking inspiration from from literature in energy context we focus on the now on the Wilson formulation of lattice gauge theories with this colleague in his book Christina Marcus Philip and Peter and the stringer model and there is some magic as you know in one plus one dimension one can integrate out the gauge fields analytically and what you get out of this you have to pay a price when you integrate out the gauge fields analytically you get extremely long-range interaction you know I'm going to be I don't know local and will be also I have to say quite fuzzy this was for example what was done in this paper here so you have translated the challenge from engineering the gauge symmetry to engineering an Hamiltonian which is I don't know local breaks a lot of symmetry is and all that but still is exactly is exactly the stringer model so the mapping is exact it's just an integration this strange flip flop term is nothing but the remuniscence of the gauge field coupling the second term will be the staggered mass of the fermion and the last one will be the electric field square that you have seen after integration obviously is I don't know local and so okay once we realize that we went to the group to run a block with group they have a power I mean one of the most powerful quantum computers I mean this is based on calcium 40 ions and this is their physical qubit and this is their imaging technique I mean this is not the interest to us and we told them they can they can look at spin models and we design a precise protocol to do that efficiently and the first data were actually quite interesting they could do that they could study four spins which is equivalent to four to two electron two positrons and three gauge fields and then we told them okay look since the preliminary results were so good try to study the stringer mechanism and this is what they did they're really engineer a state in an initial vacuum I have to say that this is not that the non-interacting vacuum is the strong coupling vacuum so it's slightly different from the original stringer paper and what they study is austra I mean what is as a function of time this axis how many particles are actually produced okay and obviously this is this depends on the value of the mass of the electron in the stronger the mass the less particles are produced and this was our numerical experiment theory very simple to do and they could observe the same stuff which is quite quite amazing I mean they had the limitation in the mass they couldn't go to very large masses for technical problems but in the regimes that they could assess the agreement was very good obviously once you have a quantum computer you are not limited to measure just local observable you can measure a lot of stuff well a lot some stuff and what they can measure for example is what in the context of the original stringer paper is called vacuum persistent amplitude that now a day has been rediscovered in quantum information is called Loshmit eco it's basically the the overlap of the initial vacuum with a time evolved vacuum and this they could also measure and they saw that as predicted by Schringer original this is actually perfectly matched with the matter with the particle production rate finally what they could also do is to study entanglement propagation this was a bit harder and cannot be scaled up once the system will become larger so I will not discuss this but since they there are I mean a lot of connection now they between entanglement theory and and I energy I think it was an interesting proof of principle that is at least in some in some cases can be done good so obviously we were not the only one we are not the only one working on this there are many other groups working on these ideas especially on this first topic on this second one a bit less and here you can find a I mean something which is already I maybe is also partially missing some references I apologize for that just to say okay obviously we are not the only ones I am done we are at the open questions thank you so there are many questions that still need to be set in these fields and this is just a flash I think another important point is that from our side there is really kind of need of getting new formulations of gage tiers this is something we are extremely interested in because we have really problems serious problems working with recent years recent theories are extremely challenging to be adapted to experiments quantum links are very good but there may be other way of formulating gage tiers which are simple enough that are even easier than quantum links and in that case this would really speed up experimental realization what we are working on I mean there is also another problem which is very serious is the continuum limit I think in this case we know how to address it at least for specific models and there are other things that we are still trying to tackle from the theory side and these are just very partial answers to some of these questions okay so let me thank again my collaborators in Eastbrook the group of Uwe Jens in Bern Ricky and Marcus and thank you for your attention