>> My name is Franco Nori. Is a great pleasure to be here and I thank you for attending this meeting and I'll be talking about some of the work we are doing within the NTT-PHI group. I would like to thank the organizers for putting together this very interesting event. The topics studied by NTT-PHI are very exciting and I'm glad to be part of this great team. Let me first start with a brief overview of just a few interactions between our team and other groups within NTT-PHI. After this brief overview or these interactions then I'm going to start talking about machine learning and neural networks applied to computationally difficult problems in quantum physics. The first one I would like to raise is the following. Is it possible to have decoherence free interaction between qubits? And the proposed solution was a postdoc and a visitor and myself some years ago was to study decoherence free interaction between giant atoms made of superconducting qubits in the context of waveguide quantum electrodynamics. The theoretical prediction was confirmed by a very nice experiment performed by Will Oliver's group at MIT was probably so a few months ago in nature and it's called waveguide quantum electrodynamics with superconducting artificial giant atoms. And this is the first joint MIT Michigan nature paper during this NTT-PHI grand period. And we're very pleased with this. And I look forward to having additional collaborations like this one also with other NTT-PHI groups, Another collaboration inside NTT-PHI regards the quantum hall effects in a rapidly rotating polarity and condensates. And this work is mainly driven by two people, a Michael Fraser and Yoshihisa Yamamoto. They are the main driving forces of this project and this has been a great fun. We're also interacting inside the NTT-PHI environment with the groups of marandI Caltech, like McMahon Cornell, Oliver MIT, and as I mentioned before, Fraser Yamamoto NTT and others at NTT-PHI are also very welcome to interact with us. NTT-PHI is interested in various topics including how to use neural networks to solve computationally difficult and important problems. Let us now look at one example of using neural networks to study computationally difficult and hard problems. Everything we'll be talking today is mostly working progress to be extended and improve in the future. So the first example I would like to discuss is topological quantum phase transition retrieved through manifold learning, which is a variety of version of machine learning. This work is done in collaboration with Che, Gneiting and Liu all members of the group. preprint is available in the archive. Some groups are studying a quantum enhanced machine learning where machine learning is supposed to be used in actual quantum computers to use exponential speed-up and using quantum error correction we're not working on these kind of things we're doing something different. We're studying how to apply machine learning applied to quantum problems. For example how to identify quantum phases and phase transitions. We shall be talking about right now. How to achieve, how to perform quantum state tomography in a more efficient manner. That's another work of ours which I'll be showing later on. And how to assist the experimental data analysis which is a separate project which we recently published. But I will not discuss today because the experiments can produce massive amounts of data and machine learning can help to understand these huge tsunami of data provided by these experiments. Machine learning can be either supervised or unsupervised. Supervised is requires human labeled data. So we have here the blue dots have a label. The red dots have a different label. And the question is the new data corresponds to either the blue category or the red category. And many of these problems in machine learning they use the example of identifying cats and dogs but this is typical example. However, there are the cases which are also provides with there are no labels. So you're looking at the cluster structure and you need to define a metric, a distance between the different points to be able to correlate them together to create these clusters. And you can manifold learning is ideally suited to look at problems we just did our non-linearities and unsupervised. Once you're using the principle component analysis along this green axis here which are the principal axis here. You can actually identify a simple structure with linear projection when you increase the axis here, you get the red dots in one area, and the blue dots down here. But in general you could get red green, yellow, blue dots in a complicated manner and the correlations are better seen when you do an nonlinear embedding. And in unsupervised learning the colors represent similarities are not labels because there are no prior labels here. So we are interested on using machine learning to identify topological quantum phases. And this requires looking at the actual phases and their boundaries. And you start from a set of Hamiltonians or wave functions. And recall that this is difficult to do because there is no symmetry breaking, there is no local order parameters and in complicated cases you can not compute the topological properties analytically and numerically is very hard. So therefore machine learning is enriching the toolbox for studying topological quantum phase transitions. And before our work, there were quite a few groups looking at supervised machine learning. The shortcomings that you need to have prior knowledge of the system and the data must be labeled for each phase. This is needed in order to train the neural networks . More recently in the past few years, there has been increased push on looking at all supervised and Nonlinear embeddings. One of the shortcomings we have seen is that they all use the Euclidean distance which is a natural way to construct the similarity matrix. But we have proven that it is suboptimal. It is not the optimal way to look at distance. The Chebyshev distances provides better performance. So therefore the difficulty here is how to detect topological quantifies transition is a challenge because there is no local order parameters. Few years ago we thought well, three or so years ago machine learning may provide effective methods for identifying topological Features needed in the past few years. The past two years several groups are moving this direction. And we have shown that one type of machine learning called manifold learning can successfully retrieve topological quantum phase transitions in momentum and real spaces. We have also Shown that if you use the Chebyshev distance between data points are supposed to Euclidean distance, you sharpen the characteristic features of these topological quantum phases in momentum space and the afterwards we do so-called diffusion map, Isometric map can be applied to implement the dimensionality reduction and to learn about these phases and phase transition in an unsupervised manner. So this is a summary of this work on how to characterize and study topological phases. And the example we used is to look at the canonical famous models like the SSH model, the QWZ model, the quenched SSH model. We look at this momentous space and the real space, and we found that the metal works very well in all of these models. And moreover provides a implications and demonstrations for learning also in real space where the topological invariants could be either or known or hard to compute. So it provides insight on both momentum space and real space and its the capability of manifold learning is very good especially when you have the suitable metric in exploring topological quantum phase transition. So this is one area we would like to keep working on topological faces and how to detect them. Of course there are other problems where neural networks can be useful to solve computationally hard and important problems in quantum physics. And one of them is quantum state tomography which is important to evaluate the quality of state production experiments. The problem is quantum state tomography scales really bad. It is impossible to perform it for six and a half 20 qubits. If you have 2000 or more forget it, it's not going to work. So now we're seeing a very important process which is one here tomography which cannot be done because there is a computationally hard bottleneck. So machine learning is designed to efficiently handle big data. So the question we're asking a few years ago is chemistry learning help us to solve this bottleneck which is quantum state tomography. And this is a project called Eigenstate extraction with neural network tomography with a student Melkani , research scientists of the group Clemens Gneiting and I'll be brief in summarizing this now. The specific machine learning paradigm is the standard artificial neural networks. They have been recently shown in the past couple of years to be successful for tomography of pure States. Our approach will be to carry this over to mixed States. And this is done by successively reconstructing the eigenStates or the mixed states. So it is an iterative procedure where you can slowly slowly get into the desired target state. If you wish to see more details, this has been recently published in phys rev A and has been selected as a editor suggestion. I mean like some of the referees liked it. So tomography is very hard to do but it's important and machine learning can help us to do that using neural networks and these to achieve mixed state tomography using an iterative eigenstate reconstruction. So why it is so challenging? Because you're trying to reconstruct the quantum States from measurements. You have a single qubit, you have a few Pauli matrices there are very few measurements to make when you have N qubits then the N appears in the exponent. So the number of measurements grows exponentially and this exponential scaling makes the computation to be very difficult. It's prohibitively expensive for large system sizes. So this is the bottleneck is these exponential dependence on the number of qubits. So by the time you get to 20 or 24 it is impossible. It gets even worst. Experimental data is noisy and therefore you need to consider maximum-likelihood estimation in order to reconstruct the quantum state that kind of fits the measurements best. And again these are expensive. There was a seminal work sometime ago on ion-traps. The post-processing for eight qubits took them an entire week. There were different ideas proposed regarding compressed sensing to reduce measurements, linear regression, et cetera. But they all have problems and you quickly hit a wall. There's no way to avoid it. Indeed the initial estimate is that to do tomography for 14 qubits state, you will take centuries and you cannot support a graduate student for a century because you need to pay your retirement benefits and it is simply complicated. So therefore a team here sometime ago we're looking at the question of how to do a full reconstruction of 14-qubit States with in four hours. Actually it was three point three hours Though sometime ago and many experimental groups were telling us that was very popular paper to read and study because they wanted to do fast quantum state tomography. They could not support the student for one or two centuries. They wanted to get the results quickly. And then because we need to get these density matrices and then they need to do these measurements here. But we have N qubits the number of expectation values go like four to the N to the Pauli matrices becomes much bigger. A maximum likelihood makes it even more time consuming. And this is the paper by the group in Inns brook, where they go this one week post-processing and they will speed-up done by different groups and hours. Also how to do 14 qubit tomography in four hours, using linear regression. But the next question is can machine learning help with quantum state tomography? Can allow us to give us the tools to do the next step to improve it even further. And then the standard one is this one here. Therefore for neural networks there are some inputs here, X1, X2 X3. There are some weighting factors when you get an output function PHI we just call Nonlinear activation function that could be heavy side Sigmon piecewise, linear logistic hyperbolic. And this creates a decision boundary and input space where you get let's say the red one, the red dots on the left and the blue dots on the right. Some separation between them. And you could have either two layers or three layers or any number layers can do either shallow or deep. This cannot allow you to approximate any continuous function. You can train data via some cost function minimization. And then there are different varieties of neural nets. We're looking at some sequel restricted Boltzmann machine. Restricted means that the input layer speeds are not talking to each other. The output layers means are not talking to each other. And we got reasonably good results with the input layer, output layer, no hidden layer and the probability of finding a spin configuration called the Boltzmann factor. So we try to leverage Pure-state tomography for mixed-state tomography. By doing an iterative process where you start here. So there are the mixed States in the blue area the pure States boundary here. And then the initial state is here with the iterative process you get closer and closer to the actual mixed state. And then eventually once you get here, you do the final jump inside. So you're looking at a dominant eigenstate which is closest pure state and then computer some measurements and then do an iterative algorithm that to make you approach this desire state. And after you do that then you can essentially compare results with some data. We got some data for four to eight trapped-ion qubits approximate W States were produced and they were looking at let's say the dominant eigenstate is reliably recorded for any equal four, five six, seven, eight for the ion-state, for the eigenvalues we're still working because we're getting some results which are not as accurate as we would like to. So this is still work in progress, but for the States is working really well. So there is some cost scaling which is beneficial, goes like NR as opposed to N squared. And then the most relevant information on the quality of the state production is retrieved directly. This works for flexible rank. And so it is possible to extract the ion-state within network tomography. It is cost-effective and scalable and delivers the most relevant information about state generation. And it's an interesting and viable use case for machine learning in quantum physics. We're also now more recently working on how to do quantum state tomography using Conditional Generative Adversarial Networks. Usually the masters student are analyzed in PhD and then two former postdocs. So this CGANs refers to this Conditional Generative Adversarial Networks. In this framework you have two neural networks which are essentially having a dual, they're competing with each other. And one of them is called generator another one is called discriminator. And there they're learning multi-modal models from the data. And then we improved these by adding a cost of neural network layers that enable the conversion of outputs from any standard neural network into physical density matrix. So therefore to reconstruct the density matrix, the generator layer and the discriminator networks So the two networks, they must train each other on data using standard gradient-based methods. So we demonstrate that our quantum state tomography and the adversarial network can reconstruct the optical quantum state with very high fidelity which is orders of magnitude faster and from less data than a standard maximum likelihood metals. So we're excited about this. We also show that this quantum state tomography with these adversarial networks can reconstruct a quantum state in a single evolution of the generator network. If it has been pre-trained on similar quantum States. so requires some additional training. And all of these is still work in progress where some preliminary results written up but we're continuing. And I would like to thank all of you for attending this talk. And thanks again for the invitation.

>>Hello. My name is Isaac twang and I am on the faculty at MIT in electrical engineering and computer science and in physics. And it is a pleasure for me to be presenting at today's NTT research symposium of 2020 to share a little bit with you about programmable quantum simulators theory and practice the simulation of physical systems as described by their Hamiltonian. It's a fundamental problem which Richard Fineman identified early on as one of the most promising applications of a hypothetical quantum computer. The real world around us, especially at the molecular level is described by Hamiltonians, which captured the interaction of electrons and nuclei. What we desire to understand from Hamiltonian simulation is properties of complex molecules, such as this iron molded to them. Cofactor an important catalyst. We desire there are ground States, reaction rates, reaction dynamics, and other chemical properties, among many things for a molecule of N Adams, a classical simulation must scale exponentially within, but for a quantum simulation, there is a potential for this simulation to scale polynomials instead. >>And this would be a significant advantage if realizable. So where are we today in realizing such a quantum advantage today? I would like to share with you a story about two things in this quest first, a theoretical optimal quantum simulation, awkward them, which achieves the best possible runtime for generic Hamiltonian. Second, let me share with you experimental results from a quantum simulation implemented using available quantum computing hardware today with a hardware efficient model that goes beyond what is utilized by today's algorithms. I will begin with the theoretically optimal quantum simulation uncle rhythm in principle. The goal of quantum simulation is to take a time independent Hamiltonian age and solve Schrodinger's equation has given here. This problem is as hard as the hardest quantum computation. It is known as being BQ P complete a simplification, which is physically reasonable and important in practice is to assume that the Hamiltonian is a sum over terms which are local. >>For example, due to allow to structure these local terms, typically do not commute, but their locality means that each term is reasonably small, therefore, as was first shown by Seth Lloyd in 1996, one way to compute the time evolution that is the exponentiation of H with time is to use the lead product formula, which involves a successive approximation by repetitive small time steps. The cost of this charterization procedure is a number of elementary steps, which scales quadratically with the time desired and inverse with the error desired for the simulation output here then is the number of local terms in the Hamiltonian. And T is the desired simulation time where Epsilon is the desired simulation error. Today. We know that for special systems and higher or expansions of this formula, a better result can be obtained such as scaling as N squared, but as synthetically linear in time, this however is for a special case, the latest Hamiltonians and it would be desirable to scale generally with time T for a order T time simulation. >>So how could such an optimal quantum simulation be constructed? An important ingredient is to transform the quantum simulation into a quantum walk. This was done over 12 years ago, Andrew trials showing that for sparse Hamiltonians with around de non-zero entries per row, such as shown in this graphic here, one can do a quantum walk very much like a classical walk, but in a superposition of right and left shown here in this quantum circuit, where the H stands for a hazard market in this particular circuit, the head Mar turns the zero into a superposition of zero and one, which then activate the left. And the right walk in superposition to graph of the walk is defined by the Hamiltonian age. And in doing so Childs and collaborators were able to show the walk, produces a unitary transform, which goes as E to the minus arc co-sign of H times time. >>So this comes close, but it still has this transcendental function of age, instead of just simply age. This can be fixed with some effort, which results in an algorithm, which scales approximately as towel log one over Epsilon with how is proportional to the sparsity of the Hamiltonian and the simulation time. But again, the scaling here is a multiplicative product rather than an additive one, an interesting insight into the dynamics of a cubit. The simplest component of a quantum computer provides a way to improve upon this single cubits evolve as rotations in a sphere. For example, here is shown a rotation operator, which rotates around the axis fi in the X, Y plane by angle theta. If one, the result of this rotation as a projection along the Z axis, the result is a co-sign squared function. That is well-known as a Ravi oscillation. On the other hand, if a cubit is rotated around multiple angles in the X Y plane, say around the fee equals zero fee equals 1.5 and fee equals zero access again, then the resulting response function looks like a flat top. >>And in fact, generalizing this to five or more pulses gives not just flattered hops, but in fact, arbitrary functions such as the Chevy chef polynomial shown here, which gets transplants like bullying or, and majority functions remarkably. If one does rotations by angle theta about D different angles in the X Y plane, the result is a response function, which is a polynomial of order T in co-sign furthermore, as captured by this theorem, given a nearly arbitrary degree polynomial there exists angles fi such that one can achieve the desired polynomial. This is the result that derives from the Remez exchange algorithm used in classical discreet time signal processing. So how does this relate to quantum simulation? Well recall that a quantum walk essentially embeds a Hamiltonian insight, the unitary transform of a quantum circuit, this embedding generalize might be called and it involves the use of a cubit acting as a projector to control the application of H if we generalize the quantum walk to include a rotation about access fee in the X Y plane, it turns out that one obtains a polynomial transform of H itself. >>And this it's the same as the polynomial in the quantum signal processing theorem. This is a remarkable result known as the quantum synchrony value transformed theorem from contrast Julian and Nathan weep published last year. This provides a quantum simulation auger them using quantum signal processing. For example, can start with the quantum walk result and then apply quantum signal processing to undo the arc co-sign transformation and therefore obtain the ideal expected Hamiltonian evolution E to the minus I H T the resulting algorithm costs a number of elementary steps, which scales as just the sum of the evolution time and the log of one over the error desired this saturates, the known lower bound, and thus is the optimal quantum simulation algorithm. This table from a recent review article summarizes a comparison of the query complexities of the known major quantum simulation algorithms showing that the cubitus station and quantum sequel processing algorithm is indeed optimal. >>Of course, this optimality is a theoretical result. What does one do in practice? Let me now share with you the story of a hardware efficient realization of a quantum simulation on actual hardware. The promise of quantum computation traditionally rests on a circuit model, such as the one we just used with quantum circuits, acting on cubits in contrast, consider a real physical problem from quantum chemistry, finding the structure of a molecule. The starting point is the point Oppenheimer separation of the electronic and vibrational States. For example, to connect it, nuclei, share a vibrational mode, the potential energy of this nonlinear spring, maybe model as a harmonic oscillator since the spring's energy is determined by the electronic structure. When the molecule becomes electronically excited, this vibrational mode changes one obtains, a different frequency and different equilibrium positions for the nuclei. This corresponds to a change in the spring, constant as well as a displacement of the nuclear positions. >>And we may write down a full Hamiltonian for this system. The interesting quantum chemistry question is known as the Frank Condon problem. What is the probability of transition between the original ground state and a given vibrational state in the excited state spectrum of the molecule, the Frank content factor, which gives this transition probability is foundational to quantum chemistry and a very hard and generic question to answer, which may be amiable to solution on a quantum computer in particular and natural quantum computer to use might be one which already has harmonic oscillators rather than one, which has just cubits. This has provided any Sonic quantum processors, such as the superconducting cubits system shown here. This processor has both cubits as embodied by the Joseph's injunctions shown here, and a harmonic oscillator as embodied by the resonant mode of the transmission cavity. Given here more over the output of this planar superconducting circuit can be connected to three dimensional cavities instead of using cubit Gates. >>One may perform direct transformations on the bull's Arctic state using for example, beam splitters, phase shifters, displacement, and squeezing operators, and the harmonic oscillator, and may be initialized and manipulated directly. The availability of the cubit allows photon number resolve counting for simulating a tri atomic two mode, Frank Condon factor problem. This superconducting cubits system with 3d cavities was to resonators cavity a and cavity B represent the breathing and wiggling modes of a Triumeq molecule. As depicted here. The coupling of these moles was mediated by a superconducting cubit and read out was accomplished by two additional superconducting cubits, coupled to each one of the cavities due to the superconducting resonators used each one of the cavities had a, a long coherence time while resonator States could be prepared and measured using these strong coupling of cubits to the cavity. And Posana quantum operations could be realized by modulating the coupling cubit in between the two cavities, the cavities are holes drilled into pure aluminum, kept superconducting by millikelvin scale. >>Temperatures microfiber, KT chips with superconducting cubits are inserted into ports to couple via a antenna to the microwave cavities. Each of the cavities has a quality factor so high that the coherence times can reach milliseconds. A coupling cubit chip is inserted into the port in between the cavities and the readout and preparation cubit chips are inserted into ports on the sides. For sake of brevity, I will skip the experimental details and present just the results shown here is the fibrotic spectrum obtained for a water molecule using the Pulsonix superconducting processor. This is a typical Frank content spectrum giving the intensity of lions versus frequency in wave number where the solid line depicts the theoretically expected result and the purple and red dots show two sets of experimental data. One taken quickly and another taken with exhaustive statistics. In both cases, the experimental results have good agreement with the theoretical expectations. >>The programmability of this system is demonstrated by showing how it can easily calculate the Frank Condon spectrum for a wide variety of molecules. Here's another one, the ozone and ion. Again, we see that the experimental data shown in points agrees well with the theoretical expectation shown as a solid line. Let me emphasize that this quantum simulation result was obtained not by using a quantum computer with cubits, but rather one with resonators, one resonator representing each one of the modes of vibration in this trial, atomic molecule. This approach represents a far more efficient utilization of hardware resources compared with the standard cubit model because of the natural match of the resonators with the physical system being simulated in comparison, if cubit Gates had been utilized to perform the same simulation on the order of a thousand cubit Gates would have been required compared with the order of 10 operations, which were performed for this post Sonic realization. >>As in topically, the Cupid motto would have required significantly more operations because of the need to retire each one of the harmonic oscillators into some max Hilbert space size compared with the optimal quantum simulation auger rhythms shown in the first half of this talk, we see that there is a significant gap between available quantum computing hardware can perform and what optimal quantum simulations demand in terms of the number of Gates required for a simulation. Nevertheless, many of the techniques that are used for optimal quantum simulation algorithms may become useful, especially if they are adapted to available hardware, moving for the future, holds some interesting challenges for this field. Real physical systems are not cubits, rather they are composed from bolt-ons and from yawns and from yawns need global anti-Semitism nation. This is a huge challenge for electronic structure calculation in molecules, real physical systems also have symmetries, but current quantum simulation algorithms are largely governed by a theorem, which says that the number of times steps required is proportional to the simulation time. Desired. Finally, real physical systems are not purely quantum or purely classical, but rather have many messy quantum classical boundaries. In fact, perhaps the most important systems to simulate are really open quantum systems. And these dynamics are described by a mixture of quantum and classical evolution and the desired results are often thermal and statistical properties. >>I hope this presentation of the theory and practice of quantum simulation has been interesting and worthwhile. Thank you.