So thank you. Thank you very much for the for the introduction. So, yes, exactly. So today I'm going to give you a non-exhaustive and perhaps a bit biased overview of the developing field of quantum simulation. Right. So as you would see, it is not accidental that this talk sits in between. After we are stuck on the relativistic heavy ion collisions and the black is stuck on quantum matter of correction.
In fact, quantum simulation is something in between a physics experiments and and apart from computation. So let me. So to walk you through, walk you to the to the thoughts, the philosophy and the practice of quantum simulation. So let me let me start from the very basic. So science science is about making quantitative observations of phenomena, then formulating a theory or a mathematical model to explain these observations and making new predictions out of out of the model.
So an important part of our job as theoretical physicists is to be able to compute the physical behaviour that results from a model. Okay, So usually the simplest predictions can be obtained by explicitly solving a model, right? However, in the general case, explicit solutions are not available and we have to resort to a cascade of less and less, less and less pretentious strategies to solve to solve the problem.
For example, we can we may want to do a brute force simulation of the model, usually using a computer. Or we may want to resort to a controlled approximation schemes or two if none of this work. We can try to replace the model with some simpler phenomenological model. Right. So let me take a minute to remind you how this works in the context of classical classical mechanics. Right. So one of the early drives to develop classical mechanics was to explain planetary motion.
Right. So. So the motion of a single planets orbiting around the sun is described by Newton's equation. This equation can be explicitly solved, and this exact solution tells us a great deal about the astronomical observations. Now, when we move to the problem of many celestial bodies, so call and call and their number we deal with and Newton equations. Right. We deal with anything and equations, and this problem is famously not solvable, even for an equals three.
Right. What we can do instead is performing a brute force computation of the trajectories of the of the bodies. So how do we do that? Basically, we use Newton input so that the system, state of the system is described by six times ten variables. So all the positions and all the velocities of all the bodies. Right. And we use Newton Equation to update this 610 variables for the next time step.
So for a short time step and we repeat this process over and over to obtain to compute the full trajectory. Now, using modern computers, this can be done very efficiently for thousands or even millions of bodies. Right. So it's fair to say. Sorry. So it's fair to say that this that this problem is said that the brute force, numerical simulations are extremely powerful in this problem. And here in this this animation, I hope you see it.
Yes. You see a computer animation of a of the trajectories computed for our many by the computational problem computed using a computer. Okay. So this is this is the situation. Now, let's move to quantum mechanics, right? So quantum mechanics, the original drive to develop quantum mechanics stemmed from from the from the problem of atomic spectra, for example, hydrogen. Right. So we deal with the motion of a single electron orbiting around the nucleus.
Right. And this is described by Schrodinger equation. Now, Schrodinger equation in this case is explicitly solvable. And the exact solution of this problem tells us a great deal about the observation of atomic spectra. So great. So far, so good now. And here you see the illustration of some some of the atomic orbitals the sent from the solution. Now problem comes when we go when we move to the problem of many, many quantum particles like for example, and problem of and electrons.
Right. So in this case, we deal with the Schrödinger equation with the embody Schrodinger equations equation. And and this problem, of course, is famously not solvable even for n equals three. Just like in the classical case. But here there is something, something much worse happening. Essentially, the NYU Schrodinger equation describes the evolution in time of the embody wave function of the system, of the wave function of electrons. Right.
So this wave function is nothing but a complex number assigned for every possible classical configuration for the end for the end particles. Right. So imagine these particles. Let's be generous and say that this particles can be in a given number of states, say ten or even five or even two. Right? Just every particle can be in two states. Of the number of configurations of the system is two, is two by two, by two by two, two end times.
Right. So it's two two power and. So essentially we have the system is described by two power and numbers. Complex numbers. Right. And we have two. This is the the many body weight function. Right? Now, this this wave function, as you see the complexity of this object explodes exponentially with the number of particles. And I'm sure I think that in the recent pandemics, pretty much everyone got a sense of how quick and exponential growth is.
Right. So this is a we we could try to estimate how bad this is for a real problem of interest. So we could try to convince ourselves. But then here I instead I reported the words from a Nobel lecture of Walter Kohn, 1999, who argued, even provocatively argued that the many body wave function, the quantum anybody wave function is not a legitimate scientific concept.
So the the essential the essential point here is that the amount of information that you would need to describe the wave function of many electrons, for example, quickly exceeds the amount of matter in the entire universe. So even for a thousand, even for a thousand electrons, this would be this would exceed the amount of matter in the entire universe. So this is the bottom line here, is that so you may agree or not with this strong viewpoint, but that the problem is real.
So the problem of the exponential world is real, right? So it's not about it's not really about technological advance advantage. Advance is not about building better. It's not about building a better and stronger supercomputer. So there is a fundamental limitation in the capability of computers to deal with quantum physics. Right. And this is something new is different from from classical mechanics.
So I think it's fair to say that brute force numerical computations of the quantum antibody problem are fundamentally out of reach. Hmm. Okay. And this is actually why if you try to Google quantum, anybody physics or quantum, anybody would function as you as you could do, for example, for the other for the previous, in contrast with what you did with the previous images. Here you find just a bunch of funny and incomprehensible drawings that we theoretical physicists like.
And this is because essentially it's very hard to even visualise this massive, huge amount of information encoded in the many body wave function. Okay, So basically, quantum theory calls for countless approximation schemes and phenomenological theories. And this includes, for example, what you I think what you have seen in the previous talk by Dr. Brewer. Okay, So this is this is the this is it.
But then you may legitimately wonder at this point why Why do we even care about solving the quantum anybody problem? Right. So after all, for example, if football is made of many quantum particles, as far as we know, but we can predict this behaviour extremely well. Right. So why why do we even care about solving them? Anybody, quantum anybody problem?
So the point here is that many important phenomena in nature are intrinsically quantum and intrinsically anybody in the sense of ignoring quantum effects or ignoring correlation effects between the particles is not a good idea for describing. Right. And here I will give you an exhaustive list of some some problems that would be very nice to solve if we could deal with the quantum anybody problem.
Right. So in the domain of chemistry, for example, if we had the solution of the quantum anybody problem, this would give us this would allow us to perform a first principle computation of molecular structure, for example, or molecular reaction rates of chemical reactions. Or this would allow us to predict this, would allow it to give us the opportunity to predict new molecules, for example, with obvious applications, for example, for drug design and so on.
So here in the illustration you see an example. This is basically a so-called light harvesting complex that plays a role in the process of photosynthesis. And it's an example of a process where quantum quantum mechanical effects are important. Now let's move to the domain of condensed matter physics to discuss close to what they actually do.
Right. So here solution of the them anybody problem would give us first principle computation of the properties of materials, for example, of their structure, of their out of equilibrium response and so on and so forth. And this would give us the opportunity to predict new materials or new phases of matter. For example, the conditions for the for having high temperature superconductors. And this would obviously be extremely important for applications.
Right. And finally, also in the context of high energy physics. So the solving the problem would give us first principle computation, for example, of a hydraulic structure or nuclear reactor or nuclear reaction rates. And this would give us the opportunity to predict the behaviour of matter in extreme conditions of temperature and pressure.
For example, in the early universe dynamics or in the inner stars or in relativistic heavy collisions have, just as you heard in the previous stuff by by Dr. Brewer. So this is would obviously be extremely interesting also in this context. And these are all examples coming from natural sciences. But actually, there are examples that go beyond the domain of physics. I think Dr. Blackett will briefly touch upon this aspect in the in the next stock.
Okay, so the summary of the situation is the following. So basically solving the quantum anybody problem would lead to a huge progress, to a huge leap forward along in several branches of science. Right. So this would be really great. However, we have said that brute force numerical computations are fundamentally impossible. On the other hand, controlled approximations in many cases are known to suffer badly. Right. Otherwise, we would have sort of all these problems. And then.
And phenomenology is not always satisfactory for us. So this seems like a seems like a bad situation. So that's why our question is, is there a way out from from the simulation? And as you can guess from the fact that I'm giving this talk. The answer is a possible answer is yes, in the sense that the that there is a possibility that was suggested in the early eighties by my research.
Richard Feynman, that is actually the contents of the stock, which is the possibility of performing quantum simulations. So basically the following fundamental idea is the following. So if we can't if we can't store and process in anybody, we function in a classical computer which is made of units that can state that can be in two states, either zero or one. So the point is, because this would say for the for the problem of a thousand electrons, this would require a computer as big as universe.
But then how why don't we store and process the information contained in this way function, for example, a thousand electrons? Why don't we use it and process it using a processor that is made of quantum units, that is made of quantum particles, like, for example, a thousand atoms. Right? So provided we can operate with this objects, manipulate them quantum mechanically, right? Not classically, but quantum mechanically, then we only need a thousand particles to stimulate thousands of particles.
We don't need something as big as bigger than the universe. Right. So this is this is the central idea That was central insight that was before the proposed. Right. And this goes under the name of of quantum simulation. Right. So the goal is to is to simulate quantum mechanics, Quantum anybody problems using quantum and using a controllable quantum, anybody's system. Right. So this is. And this looks like a strange idea.
I just invite you to pay attention to the rest of the talk. What I try to make make it a bit bit clearer. So how do we how do we simulate a quantum system using a quantum machine, using a set of quantum devices? So the most important thing is that we need a new type of hardware, right?
So basically what we need is a highly controllable quantum anybody system where we can access, control, manipulate, observe individual individual quantum particles, which which operates in a way that is not somehow affected or not disturbed or not destroyed by the, for example, by environmental processes. Right. Okay. So we need we need this kind of machine or hardware. Right. And suppose we are given such a machine.
So suppose we are given a quantum antibody system that we can control, maybe not perfectly, but we have a good degree of control on. So what do we do with it? How do we perform a quantum simulation of our system of interest? For example, our favourite molecule, our favourite material or our favourite nucleus or whatever, whatever you like. So what do we have to do? So for first thing first you have to essentially encode the states of the system of interest in the states of your hardware.
So you need a mapping of correspondence between the states of the system of interest and the states of your of your quantum particles. So this mapping can be more or less obvious, more or less direct. But in some cases, it's very it's very direct. In some cases it's very convoluted, very abstract. And you need some imagination to to to build this this correspondence. But it doesn't matter. You just need one correspondence between the two, between the two.
Right. Second, we need to be able to prepare the system in the states that we need in the state that we like for. And this is where the control enters. Right. So we need to prepare in a wave function that represents the wave function that we care about. Then and this is usually the bottleneck. We need to be able to design the forces or technically the Hamiltonian that governs the system in a way that mimics or resembles the the forces or the actual evolution of the system of interest.
Right. And here is where where most of the pain comes. Right, because it's your system has its own native infractions and doesn't want to do what you want. Right. So you have to be somehow drive it into into the into doing what you want. Right. And finally, you have to be able to measure the observable something there. So there is some observable that you care about in the actual system, and you have to be able to map it back to your hardware and be able to measure this precise quantity.
Okay. So basically, all this set of requirements can be summarised in a single equation, which is the one that you find that you find in the bottom of the slides, which expresses the equivalence between the quantum anybody dynamics happening on one side, on the actual system and on the other side something in your artificial hardware.
Right. So basically this, this correspondence is about building is about building, constructing a dictionary between your system of interest and your and your hardware. If you are able to do this, then you can perform your quantum simulation. And of course, this is this is not easy in general. And that's that's already in the in the paper by Feynman, you see a remark on how difficult this can be. But of course, this is one one possible way to proceed with.
Okay. So before before going on to tell about tell you about examples. Right. So to tell you about where we stand in this enterprise and and what the what what can we do now, what we can hope to do next. Right. I want to give you a little piece of theory and draw an important distinction between two approaches to quantum simulation. So one that is called analogue quantum simulation and one that is called digital quantum simulation.
So I don't know, quantum simulation is close to what I described so far. Right. And is actually in the examples that I would give that I will give I will talk about analogue quantum simulation. So analogue means essentially that your hardware, your quantum system has some native interactions. Right? And basically the cards that you can play, you can play two cards essentially.
Right. So you can addition, you can use additional engineering in the system to tweak the native interactions to to take the form that you want. Right. Or you can be somehow device clever mapping some clever counterintuitive mappings that map your your native system to some system that superficially looks very different. But is this what you what you would like. So you can play you can play these two games essentially.
And this is. And so basically you have your system. It functions more or less in the way it wants to function up to some up to some messaging. Right. This is an analogue quantum simulation. Now, digital quantum simulation is a priori something very different. So here the principle is different. So the idea is that you want to switch on and off.
You want to be able to switch on and off your units, your your individual units of your quantum hardware such that you can address them individually and perform individual operations on them. So on every. On each one of them. Or maybe pairs of them. Right. You want to be able to perform this building block operations that we call unitary gates. Right. And then the idea is that an arbitrary quantum dynamics for the full system of particles can be approximated arbitrarily.
Well, provided you are given a large enough set of of gates of any target. So the idea is that you have to form a set of gates that you can efficiently implement, right. In such a way that these gates form a universal gates set right in universal gates. That means that an arbitrary quantum dynamics can be approximated by a sequence of these building blocks. Right. So you can you can decompose it in a sequence of unitary gates. Right. That is also called a quantum circuits.
So basically, you know, a lot of quantum simulation, you don't need to devise some strange or clever mappings. All you need to do, all you need to have is a software. Right. So you need a program that compiles your actual your target evolution that you want in your actual system. It is complicit in terms of a sequence of gates. Or a lot of circuits. So in this illustration, you see what the quantum circuit looks like. This is an illustration. So in this case, time flows from left to right.
Right. So on the left we have our initial quantum units that are prepared in some reference state. Let's call it zero. Okay, so this is our initial state. Right. Then we have our digitised evolution. So our sequence of unitary gates. Right. And this is created by a software that creates it in such a way that this sequence of gates reproduces exactly the dynamics that we want. Right. So this funny boxes and drawings are the are drawn from some set of gates, some special set of gates.
Right. And then at the end on the right, we measure the cubits. And the answer to our problem is encoded in this measurement. So you see that the the the power of the digital machine is that is hardware independent. Somehow it can approximate arbitrarily well any any quantum dynamics in principle. So it's universal in this sense. This is kind of the ultimate goal of the field of of the field of, uh, of the field.
And this is, this is why the the analogy with of between this things and the classical computer is the reason why we call this a quantum computer. Right? So you will hear more about this in the talk by Dr. But I can let me just I can help mention in the fact that in the development of this field of universal quantum computation, the researchers in Oxford played really a protagonist, protagonist role. So, you know, it's an art record and interesting and so on and so forth.
So this is a very important topic. What way then here in particular? Very good. So, right, so this is this is what we would like to do, right? So if you've never heard before or thought about or thought before too deeply about quantum simulation or quantum computation, I hope I conveyed the message that this is a very exciting theoretical idea. Right. But then where do we stand in the implementation, in the implementation of this idea? Right.
So there are several competing experimental platforms that are participating in the race to build somehow a more and more powerful quantum machine. Right. So there are there are very, very many by now. Here I reported two pictures that illustrate to two kind of families of this of this quantum device. This on the left, we see the so-called Ammal family, so the atomic molecular and optical systems. And on the right, we see solid so-called solid state systems.
So in the pictures give you a rough idea of what what these devices look like. But the important, important point to stress here is that this device are at the moment that these devices are kind of imperfect. So they are they are called MSC, MSC devices, meaning noisy intermediate scale quantum devices. And the problem is that the size of these devices is limited. And they the time, the coherence time. So the time where this machine works in a quantum mechanical way is also limited. Right.
So, roughly speaking, you may say that the boundary, the frontier of current quantum devices is more or less at the boundary of classical computation. So on the task where we think whether this objects would outperform classical computers at the moment, they can compete with classical computers, right? And it's kind of a active topic of research to try to make a proof of principle experiment that that provably beats what what quantum, what the classical computer can do. All right.
So this is this is the current state of the art. Of course, the Holy Grail. The holy grail of the field. The ultimate goal is to build a scalable and tolerant quantum computer. So this would give us unlimited, unlimited size and unlimited coherence time. Right. So really, you will hear more about this in the next talk by Dr. Plucker. Now, I just want to say that this, of course, many, many of us believe that this that this will come right.
And many and huge amounts of people are working to make this possible. This this, of course, will need a lot of a lot of work and a lot of effort. Good. So so okay, So what I'm going to do in the remaining part of the talk.
It's to essentially give you an overview, a selected overview of some of the some of the experimental platforms that are considered today to some of the best analogue quantum simulators that we use to somehow to to study quantum anybody physics and try to solve quantum anybody problems. And then I will describe at the end, I will describe one successful example of of a quantum simulation performed using this devices that is drawn by is drawn from my from my own work.
Okay, so let's let's go there. So I will focus on family of quantum simulators that is called the Go under the name, of course, not on quantum simulators. So that all quantum simulators are essentially the effort to produce them originated from the from the from the effort to to produce to essentially to the efforts in low temperature physics. So to produce and the the coolest somehow system that that that that functions in a quantum mechanical way.
Right. So what what they have done essentially is to isolate or to trap a gas of atoms, to confine them in a region of space and to cool them to extremely low temperatures in the order of nano kelvin. Right. Using various very experimental techniques that were developed for this purpose. Right. So disclaimer, I'm a theoretical physicist, so this is not what I, it is not what I do personally. Right. But I will, I will try to give you somehow a flavour of how these things are are done.
So okay, so essentially these gases are trapped and cooled for extremely low temperature using several techniques. And in this cooling process where this is where the number of particles is extremely reduced compared to microscopic samples, but still it's a we obtain a very cool, a very cold gas which is still contained, which still contains a lot of atoms. Right. And the crucial point here is that the the the temperature is lowered to such an extent that the the the wavelength.
Right. So the the wavelength associated with the quantum mechanical nature of the motion of this particles is comparable to the size of the confinement region. Right. So when this happens, it means that the quantum mechanical nature of these objects cannot be ignored, of the motion of this object cannot be ignored anymore. Right. And this is how people realised, for example, the first Bose-Einstein condensate.
So the first an exotic phase of matter that was not observed so far by was not observed before by cooling and cooling this gas of bosonic atoms. Okay, So this is this is what you do. And then, of course, when when people successfully realised this, this very cold gas is then they realise that these systems somehow they have all the control techniques that they have developed to do this somehow could be pushed forward and maybe used to realise this.
Feynman's dream of a fully controllable, isolated one from anybody's system. Right. So this is how it how it originated. Right. So, so people try to develop other experimental techniques to control the interactions between these particles, for example, the strength of the interaction between particles or the the shapes of the range of interactions or the selective interactions so that certain particles preferentially interact with certain others and so on and so forth.
So all this kind of engineering can be achieved with atomic molecular optical techniques. Right. And this is an important step in the realisation of a of a quantum simulator, because remember what I said before, we have to be able to tweak the native interactions of the Hamiltonian to mimic, to, to, to mimic the behaviour of our, our preferred system of interest, because the atoms are just atoms, they don't know that they have to simulate the relativistic heavy collision.
Right. So we have to do something for that. Okay, So we have we can control interactions and we also experimentalists develop very advanced imaging techniques. So to somehow measure remember in this list of requirements, that is also the requirements of being able to measure individual particles. And this is something that today people can actually do.
So there are very advanced imaging techniques that basically give us access to two spatial, temporally resolved measurements in the in the system. Okay. So this is a this is the playground. Okay. So one one development that stemmed out of this development after this efforts is the so-called optical lattices. You can do optical lattices as a sort of analogue quantum simulator for condensed matter physics in the following sense. So condensed matter physics is about electrons moving in a solid.
So a solid is a crystal, right? Is a periodic structure or regular structure. And electrons move in this periodic structure, right? So what people managed to to realise in the field in the context of cold atoms is essentially to realise periodic potentials by using lasers. Right. So basically arranging counter propagating lasers, we can form sending waves. Right. And extending waves generate the potential that is felt by the atoms.
So the atoms now move periodic potential pretty similarly to what electrons do in the in materials, right, but with important differences. So one crucial difference is that this this is just a potential. So the the crystal is not dynamical itself. So as we say, there are no vibrations on the crystal, There are no phonons. And this makes our system much more coherent, much more isolated. Right. And second, it's there is a difference of scale.
So electrons are tightly packed together in materials, and you can hardly imagine to visualise, to be to pick one single electron and control it and and visualise it. Right. Well here this is completely programmable. Right. So we can decide the, the lattice spacing of this of the system. Right. So we can basically realise called neutral items that are very heavy, very slow and very distant from each other. Right. And jump from one potential well to the next of this sort of egg box.
Egg box. Right. And interact with each other. So basically and this, this is fully programmable in the sense that the geometry of this lattices is completely customisable. Right. As you see in this representation where people manage to trap food, to confine the atoms in an arbitrary pre-defined geometries. Right. Performance of these figures. So the Hamiltonian, the kind of the forces that govern the system, the the the native Hamiltonian takes in this form.
So it's not important to really understand what it is. But what I want to stress is that this term proportional to J is associated with the tunnelling of of neutral atoms from one side of the lattice, so from one bottom of the well to the next, and this other instead of corresponds to the interactions when two atoms sit on the same lattice side. So this is kind of a simulator for is very closely resembles the kind of hamiltonians that we have in condensed matter physics.
And this, this, this is why this created a lot of excitement because we can now simulate. Yeah, we can very directly simulate how many functions of interest in contacts in condensed matter physics. For example, the Hamiltonian that is thought to govern high temperature superconductors. Okay. So now there is, as I said, there is additional engineering tools that we can apply right here to the creativity of the people. That was really stunning, right?
For example, we can use a speckle pattern for our lasers to create a quasi random to superimpose a quasi random potential on this lattice. Right. And this. This mimics the fact the behaviour moves the motion of particles of quantum particles in a disordered potentials. So in this way we can study phenomena like understood localisation phenomenon, right? That that was originally proposed for the motion of electrons in the, in the in the impurity bands of a semiconductor.
Okay. Then the other interesting thing is that you can shake, right? You can periodically vary your optical lattice. Right. Just let me open it in time. You can shake it in a in a single direction or you can shake it circularly. And interestingly enough, essentially, that this periodic shaking gives rise to artificial, for example, to artificial magnetic fields. Now, this neutral, these atoms are neutral, so they are not sensitive to magnetic fields in the sense of orbital magnetic field.
But you can engineer, you can engineer such artificial magnetic fields by this additional additional toolbox, essentially. And this this was this was realised in previous years. And finally, another example that I report is that we can use internal spin states. So the atoms, the atoms have internal states, right? And this internal states can be used to mimic an extra dimension of the system. So what they call a synthetic dimension.
So we can we can confine we can basically simulate the motion of a higher dimensional system by using internal internal levels of the atoms. So you see that you can really play with your creativity and engineer very interesting mappings between your native hardware and your target point of interest, your target system of interest to, to to study many more or less obvious quantum anybody problems.
Okay. And of course, as I said, one of the ultimate goals in this game is to study open products in condensed matter physics, for example, the elusive mechanism for high temperature superconductivity. Okay, So this is so, so much for optical lattice is another very interesting platform that is gaining a lot of traction in recent years is the so-called so neutral atom processor. So in particular what is called Gap memory's right.
So here essentially we have still called reactive atoms, in particular in alkali atoms of a hydrogen like atom. Right. And here the particular thing here is that atoms are trapped in in what is called optical tweezers. So essentially very tightly focussed lasers that form micro traps for the atom. So basically we can we can trap individual atoms in individual traps. And the advantage here is that then we can move, move around these traps and fully, fully reconfigure the geometry of the system.
So this is one. So the geometry is one piece of, of, of the analogue simulation. The other super important piece of this analogue simulation is essentially the ability to fully program the strength of interactions between these particles. And this is done by exciting the atoms to so-called reverse states, so highly excited with most states. But essentially we we program our lasers, right to drive transition between the ground state of these atoms and the very highly excited states.
Right. What is that? The essentially the electron that move that orbits around the atom occupies a very a very high orbit, so a very distant orbit from the nucleus. Right. So this the wave function of this electron is very big. But the advantage of this is that the atom now has a giant polarised ability, so it has giant interactions that let me show you how I would type before that. The Hamiltonian that describes the dynamic of the system basically as these pieces.
So this is a piece that drives the transitions, basically that the atoms are described as two level systems, the ground state and the reverse state. Right. And this this piece describes the transitions between the states. Then we have the energy difference between the two states. And then we have this interactions, which I'm going to describe. So these interactions are basically dipole dipole interactions or minor bars, interactions that are controlled by the distance.
Right. And the special properties of this of the system is the following. Essentially, if we picture the ground states of the wave function of the electron in the ground state of this atom, it looks like this small blue dot, right? This is the the electron orbiting near the nucleus. Now, when the atom is excited to the red states or to a highly excited state that its wave function looks more or less like that.
Right. So basically this is electrons is moves and forms the giant wave function and has a giant polarised ability. Now when we try to simultaneously excite nearby atoms for the ribbon states, the dipole dipole interaction between them is enormous. So this this is energetically prevented somehow. So we have the phenomenon of reverse blockade when where the when the these two giant functions overlap. Essentially it becomes impossible to simultaneously inside these two atoms.
And this mechanism can be very efficiently exploited to entangle, to create entanglement between atoms. Right. And this is described by this this piece in the. So here to show you the high degree of control with the geometry, essentially, this is this is a movie that was created by the Mitchell looking slab in Harvard.
Essentially, these are these are nothing but rather got them trapped in this micro, trapped in this optical tweezers and reconfigured to in arbitrary shapes through to create essentially arbitrary geometries and arbitrary animation that you see in this old video games. Okay. So this is another platform. And finally, another. Last but not least, another platform that I wanted to mention is from Niles.
Right. This, of course, plays a special role here in Oxford, here downstairs, because building we have one of the world leading. What leading labs realise interrupt quantum computing. Right. So led by the Lukas and Fritz Balancing and also the Oxford also play a leading role in in the sort of inspiration of the of this technology. Right. So coming from dating back to art record and Rustin and so on. So this is this is really something important here here in Oxford.
Right. So I am from that as a similar principle, except that the these are not atoms in the sense that these are ionised atoms. So they are, they are charged. Right. And they basically. Okay, let me skip the details here. But they are they are still trapped in some using some some special traps for charged particles.
And the point is that we end up with some especially in this, uh, geometry given by the pole trap, we end up with a lined with a crystal one dimensional crystal of atoms of io's sorry that are trapped electromagnetically, Right. And the atoms, the internal. So we can use the internal levels of these atoms to store quantum information. Right. So these atoms can be in two states, right? And the these atoms are able to interact. So to talk to each other via the vibrations of this crystal.
So this is like imagine a system that is floating in the vacuum, trapped with this electromagnetic field, right? Then there are some vibrations of this crystal, right? And this vibration coupled to the transition between the internal level of the system. And basically, if you go through the math, you end up with the Hamiltonian of this form, but you have some some type of easy model, right? With this is spin spin interactions.
And again, this is how we view it as an analogue simulator for condensed matter models, right? So some, some, some sort of some sort of quantum magnets in the most obvious sense. In the most negative sense. Very good. So this is a no. This is some of the leading experimental platforms that realise analogue quantum simulation and also hopefully at some point digital quantum simulations. So in the remaining time, actually, I have less time than I thought.
But in the remaining time I want to tell you one just one example of of how you can use this simulator, this quantum simulator, to realise somehow, you know, in a slightly non-obvious but not slightly non-obvious way to realise quantum simulations of problems of interest that are in other fields like in other contexts. So in particular, I want to present one example coming from going in the direction of high energy physics, right?
So what I want to talk about simulating quantum field theories, right? Sort of like the theories of fundamental interactions using these quantum machines right now.
This is a very, very, very difficult problem like condensed matter physics, where I showed you that there is a more or less direct analogy between the kind of hamiltonians you can realise in quantum simulators and the kind of hamiltonians you would like to realise, right, because they mimic the interest in quantum in high energy physics, this is much more difficult and the problem is the following.
So there are many problems. So one is that theories of fundamental interactions have usually two two kinds, at least two kinds of particles. So there are the matter particles, the fermions, right? Like the quarks or the electrons. And then you have the degrees of freedom, like h fields, like the electromagnetic field or the the gluon field. Right? So we have different kinds of particles. Right? And so so we have to have different kinds of species.
And on top of that, these particles can exist in many in many, many species of this particle exist. For example, they can have different charges, spin, colour, flavour and so on and so forth. Right. While for example, your beryllium atoms of your iron trap are all these properties are all the same. So you have to be able somehow to encode this thing in your, in your simulator.
And finally, most importantly, actually the most important problem is that this method met their degrees of freedom and degrees of freedom interact in a very complicated way that is dictated by what particle physicists call gauge symmetry. Right? So there is some some local symmetry constraints that the the motion of the combined motion of matter and the edge fields have to be right.
And of course your your poor atoms or photons or particles or whatever, they know nothing about the symmetry, right? They have their own Hamiltonian and you have to find a way somehow to encode this complicated interactions in your in your native hardware. Right. Okay. So this is, as I said, this is very difficult problem. This is very challenging both for analogue and digital quantum simulation.
So what do we do? So there is there was some recent progress in one in one example which which is the example of electrodynamics, right? So electrodynamics is one of the one of the one of the fundamental quantum field theory is, well, it's not fundamental, but it's it's one of the quantum field theory you may like to realise. Right. So this is the the describes the theory of electrons and positrons in interact interacting with photons.
So in particular the progress was achieved for a simplified version of one of quantum electrodynamics, which is fundamental dynamics in one dimension, right in one plus one spacetime dimension. So what goes under the name of Schwinger model, right? So this is of course, on the one hand, of course this is bound because it's easier, right?
So you have to start somewhere. But actually, I think most, most interestingly is the fact that this simplified version of Quantum through the right mix actually shares some some nontrivial features that are with the higher dimensional theories of interest, for example, quantum dynamics, for example, the phenomenon of core confinement. I'm sure you've you heard it in the previous talk by a by Dr. Brewer.
Right. So this phenomenon of poor confinement plays a crucial role in in the theories of strong interaction. And it is present in this lower dimensional version of quantum electrodynamics. So there are there are reasons why you would like to perform quantum simulations of this simplified theory, right? So. Okay, so the monster that you would like to simulate is the following. So this is the Hamiltonian of quantum electrodynamics. You know, one dimensional in one plus one dimensional space times.
Right. So this this has three pieces. One piece describes the mass of the of the electrons and positrons, the mass of the fermions. One piece describes the invariant interaction, so the minimal coupling between the formulas and the electric field. Right. One dimension. We have only electric field. We don't have a magnetic field. And the last piece is the energy associated with the electric field. Plus possibly a big round, a big, round electric field. Okay, so this is the Schwinger model.
The latest version of the Schwinger model of quantum electrodynamics. So we have we want to put this this complicated matter and gauge the dynamics in the in the way we want to include it in our quantum simulator. Right. So basically, before that, let me show you try to visualise what kind of processes happen in this, in this theory. So imagine we start from the vacuum, right? So there are no that are basically in the in the links of this one dimensional chain. We represent the fermions. Right.
And on the sorry, on the sides of this chain, we represent the furnace and on the links of this chain, we represent the electric field. The electric field is like a tower can take many values. Right. So this is like the vacuum configuration where there are no particles and the electric field, zero is zero everywhere. Right. Now, one of the some some of the processes described by the dynamics of this Hamiltonian look look like the following.
So when we act with this interactions on the on this link, basically we create a particle antiparticle pair. So an electron positron pair. Right. And the value of the electric, the electric field between these particles, the particles is correspondingly a just adjusted. So in such a way to respect nature in variance. So this this obeys essentially the Gauss law. When you when you cross a particle, the electric field changes. There is some electric field created by the particles.
Now, we can repeat this process in other locations. For example, here on the left, we create another particle antiparticle pair you can recreate, and then we can annihilate this particle antiparticle pair here and create a longer string of electric field. Right. And we can go on and create another particle antiparticle pair, annihilate and create an even longer string. And then we can even create another particle antiparticle pair on top of that and raise the electric field even higher.
And so on and so forth. So basically you should imagine the quantum anybody dynamics generated by this Hamiltonian as a complicated superposition of all sorts of processes of this form. Right. This is in a way that is described by the parameters of this Hamiltonian. So this is what we want to encode in our read Megatons or Trapped ions or optical lattice or whatever, whatever your imagination can can achieve.
Right. So basically what we were able to figure out is how to encode a simplified version of this quantum nature, the dynamics using Redbook atoms. So I say simplified version in the sense that we allow our electric field to have to take only two values instead of infinitely many values. So this is kind of a, if you wish, a low energy approximation of this of this theory.
But in the in the limit where you allow for more and more values of this electric field, you recover the full quantum electrodynamics. Right? So in this in this simplified case, essentially we realised. That you can you can map the configurations of of of particles of fields in the in this quantum link model, which is the name of this truncated, truncated version of quantum electrodynamics. You can encoded in the configurations of which we got from arrays in the blockade regime.
So in this regime, where to to rework atoms cannot be simultaneously excited, Right? This is why I mentioned this before. And the point I want to stress here is that the symmetry constraints or the constraints that matter particles and fields should move together in a is mapped is exactly mapped onto the constraints the two Friedberger atoms cannot be simultaneously excited. So this is somehow a non-obvious mapping. So it's not very it's not. Somehow not.
So another unknown logic, if you wish, as the condensed matter mappings for two optical lattices. Right. But it doesn't matter. It's a mapping, it works and you can map all the quantities from one side to the other and study and study dynamics. Quantum electrodynamics Looking at the evolution of your LEDERBERG atoms. Right.
So here you have to believe me. There is a dictionary between the states in such a way that every possible local configuration from method and electric field maps to a given configuration of there is MacArthur's. Okay. Once you map the states, you can ask what is the corresponding Hamiltonian? And here we have essentially a strike of luck in the sense that the native Hamiltonian of the God is already what you want is already upon suitably identifying the parameters,
is already the Hamiltonian of the quantum link model. So here, somehow we were with Lucky, right? So we have a mapping between the states and mapping between the forces. And finally, we want to be able to access the observables. Right? The corresponding observables. And here you can do that. So this this figure shows numerical simulations. Right on the left, you see the dynamics of the river. Got them. SHANE You see that to two bright spots can never be beside each other.
Because. Because of the river blockade. Right. So this is the evolution. So horizontally you have space vertical, you have time. This is the spacetime evolution of of your river. Gotham chains prepared in a state that corresponds within the mapping corresponds to a uniform string of electric field. So in the vacuum of particles, in presence of an electric field, and this is the corresponding enemies mapped 1 to 1 to the dynamics of Friedberger with the guidance.
And also here, the simulation shows the full string their models. So the truncated string, their model which signifies that the the the this somehow this truncated version is more or less doing what the full quantum electrodynamics problem would do. So you have in this case at least you have sort of a full dictionary, right between the Hamiltonian, your target Hamiltonian and your in your quantum hardware, in this case your readable gamma rays.
Okay. So I just want to close. Can I take another minute? Yes. So my final slide is basically is okay now, now that you have a way to encode your your your quantum field theory, if you wish on on your on your reader Gotham train. Right. So the question is what what kind of physics can we and we simulate. Right. So as you as you heard in the previous talk by Dr. Ruhr, essentially one one very interesting thing to do with quantum field theory is, is to study collisions between particles.
Right. So basically. Okay, So this is I want to repeat the story. So here essentially is a very important problem in particle physics. And essentially, because it's a very difficult problem, it's it's kind of an outstanding goal for for quantum simulation. Like, I don't see any long term goal for quantum simulation. So this is this is what you get. You can essentially wonder if some some cartoon version of this problem.
Of course, this is what we can do in the with the in the setting that they previously illustrated is very far from what our colleagues in particle physics would like to do presently. Right. So this is kind of there is a huge gap cover for there before we can actually simulate something that solves some actual problem in in particle physics. But it's a legitimate question whether at least the principal ingredients can be realised.
We present we present day machines, right? So what we were curious essentially is to study images on collisions. So those are like bound states, confined bound states of electrons and positrons. So I told you that in this one dimensional version of punching at the dynamics, electrons and positrons are confined pretty much like quarks in in higher in theories of strong interactions. Right. So you can imagine to prepare to wave packets of this mesons.
Right. So to to most it's prepared with a given momentum. Right. And so this is the initial stage you want to prepare and then you want to run the real time evolution. Right. So you want to encode this in your. But Gotham's run the real time evolution and measure for things that you would be interested in, in particle physics, for example, the scattering matrix of this collision process. Or you can even do more as I as I'm going to argue. Right.
So this is kind of the the protocols that you want to realise. Right. And in this paper, we the collaboration with Federica Saraceno, we have shown essentially how to perform, perform this mappings and this building block steps, right. So how to prepare in particular the initial states. So which is the, this chart is on with buckets, with a with an arbitrarily well-defined momentum. Right. And then to run the evolution and to measure the relevant observables.
So remember the list of requirements that are needed for a quantum simulation, right? And there is initial stage, there is the how many Estonian and there is the observable right. So a possible observable of interest is the scattering matrix. Right. And we have shown how to essentially how to use the capabilities of the quantum simulator to access this one thing.
Right now I want to just close with a remark that, of course, this is, as I said, there is a huge gap to bridge for the to do something that is of interest for for current particle physics, for example, to simulate a relativistic heavy collision. Right. But on the other hand, the the reason why we think this is interesting is, is that there are things that you can do using a quantum simulator, which you just cannot do in nature.
Right. So one of the things that once the simulator naturally gives you access to, for example, is that it allows you to watch inside the collision. Right. So this is something that is completely unthinkable in a actual in nature.
So natural particle physics experiment right here, we have full control and we can even stop that evolution and look, look what's happening and understand the processes that are involved, the complicated processes that are involved in the complicated collision, for example. And second, we can play with the parameters. So in nature, the the the value of the forces, the coupling constants are just given, right? We can play with it. So here instead we can change, right?
This is our simulator, right? So we can do whatever we want, or maybe not exactly whatever we want, but we have a large degree of tuned ability, right? So we can play with the parameters and explore the the predictions of the theory beyond beyond what would be accessible in actual nature. And this perhaps would allow us to discover some other effects or some other physics.
Right? So there are reasons to believe that this this kind of approach someday will will be very extremely valuable in for study. And even in this visit, that is probably the most difficult goal for quantum simulators. Okay, I think I'm done. So let me skip this drop dance and just jump to the final slide.