 Okay, the next speaker is going to be Maria Hita Pérez, who is talking to us about coupling three Joseph Johnson flux qubits for non-stochastic adiabatic quantum computation and she is from Madrid. Good morning everybody. Today, I'm going to present our work on three Joseph Sonjanci and Flasky with couplings for adiabatic quantum computation. Particularly, I'm going to introduce the work in these two papers. The first one, Ultrastrong capacitive coupling of Flasky bits provides a numerical and analytical study of the effective capacitive interactions between two Flasky bits and a Flasky bit on a resonator and the second one, three Joseph Sonjanci and Flasky bit couplings provides a complete numerical study of the interactions between three Joseph Sonjanci and Flasky bits. Among all of the different platforms that have been proposed for the implementation of adiabatic quantum computers, superconducting circuits stand for their flexibility in fabrication. They provide building blocks for quantum computations such as harmonic oscillators and qubits forming an harmonic multilevel system such as church qubits, Flasky bits or three Joseph Sonjanci and Flasky bits. And also multiple coupled mechanisms between them such as mutual inductance, capacitance, and other superconducting tunable elements. This allows for the realization of a large variety of effective Hamiltonians which are not only useful for adiabatic quantum computation, but also for quantum simulation and other quantum computations algorithms. For example, we have this circuit here which reproduces an isentype interactions, typical to that that they use for the waste quantum annealers, or this circuit here which exhibits a James Cumming dynamic. We are particularly interested in the implementation of non-stochastic Hamiltonians using superconducting circuits. I'm going to briefly review what a non-stochastic Hamiltonian is, even thought most of you already know this. A quantum Hamiltonian is stochastic in a certain basis. If its entries are real and its diagonal elements are non-positive. These definitions let us differentiate two different types of Hamiltonians. We have stochastic Hamiltonians which do not suffer from the same problem, are in principle solvable using Monte Carlo methods, and on the other hand non-stochastic Hamiltonians which suffer from the same problem and cannot be simulated efficiently. This makes them intrinsically interested because they can be used to simulate other quantum systems that suffer from the same problem, and these type of Hamiltonians are also interested because they have been shown to enable a universal adiabatic quantum computation. So what we do in our works is we propose several superconducting circuits to implement these non-stochastic Hamiltonians. Particularly, we study the coupling between three Josephson Janssen flash qubits. A three Josephson Janssen flash qubit consists on a superconducting loop Interrupted by three Josephson Janssen. Two of them identical and one of them alpha times smaller. The main property of this system is that when the external flux when the external flux equals half of the flash quanta it develops two opposite sign current states left and right, which can be connected through tunneling across the potential barrier. We are particularly interested in the tunneling inside of the unit cell which we call the one in this direction. The symmetric and anti-symmetric superposition of the two opposite sign current states form the low energy subspace of the system and are the ideal candidates to form a qubit, and hence the effective Hamiltonian of the system can be written in the form of a typical qubit Hamiltonian where delta represents the gap between the zero and one states and is also called the tunneling amplitude through D1. So we want to couple two three Josephson Janssen flash qubit. We first derive the full general Hamiltonian for this type of system following the rules for circuit quantization and circuit theory and we find that the general Hamiltonian has this form where h1 and h2 are the Hamiltonian of the two single elements the qubit 1 and the qubit 2 and the last terms give interactions between them. Then we derive the effective Hamiltonians. We do so by means of the C4W transformation which maps agonist basis in the full Hamiltonian basis to agonist basis in the birth Hamiltonian basis without the interactions. Provided that the energy subspace of the system can be divided into two subspaces, a low energy subspace which we call p and a high energy subspace which we call q. In our case of two flash qubit couplings we have that the low energy is composed by the four lowest energies of our system which is formed by the composition of the two qubit qubit states. We perform this study using two different procedures. First we perform an analytical study following the typical perturbation method but we also perform a numerical study for following the non perturbative method proposed here by Cosani and Hubert Barton. This method allows us to obtain the effective Hamiltonian simply in terms of the full Hamiltonian, the projectors in the low energy subspace and the C4W transformation u. The only issue with this procedure is that it is computationally expensive. Note that computing u in the full basis requires working with matrices whose size depends directly on the number of elements of the circuit and the dimension of the physical qubits or elements or either truncating the Hamiltonian to a certain number of states that has to be determined by convergence. We propose here a method to avoid this obstacle simply by computing P0up in terms of the operator a which comes directly from the singular value of the composition of the multiplication between the two low energy subspace, the VIR and the full projectors into the low energy subspace. Hence the computational cost of the process is reduced to that of obtaining the low energy eigenstates of the system which can be done efficiently with numerals methods. So when we perform this C4W transformation we find that the general effective Hamiltonian for this type of circuits of 2-flask qubit coupled by a coupler-situor and by a geosubsection has this form. When the two first elements are simply the effective qubit Hamiltonians where delta can be renormalized by the capacitive coupler and the last term gives interactions between the qubits in three different directions sigma s, sigma s, sigma y, sigma y and sigma c, sigma c and is modulated by the coupling strength which we call J. We first study the capacitive coupling between the flash qubits this means that we consider this circuit but without this geosubsection. When studying the effective Hamiltonian what we find is a strong renormalization of the qubit gap due to the active capacitance and interactions in the three different directions. We find for small values of the parameter gamma we find a strong sigma y, sigma y contribution which comes from the expected church interactions between the two flash qubits and when increasing gamma we see that the other two terms gain strength. We find an also strong sigma c, sigma c contribution which comes from interactions mediated by the high energy again states of the system and becomes equal in magnitude to sigma y, sigma y for a sufficiently large gamma and a residual sigma s, sigma s contribution which is directly related to intercell tunneling and comes with an enhancement of church noise. We then study the inductive coupling between three geosubsection flash qubits mediated by a geosubsection Janssen whose geosubsection energy is gamma times the geosubsection energy of the flash qubits. In this case we find an ultra strong sigma s, sigma s couplings which comes from the expected the polar magnetic interactions between the flash qubits and some residual sigma y, sigma y and sigma c, sigma c turns which are up to three times smaller and can hence be neglected. In both cases we see that the coupling strengths depend directly on the qubit parameters. The extensive analysis of the capacitive and inductive coupling suggests that combining them for a could allow for the obtention of non-stonquastic Hamiltonians where with interactions in the three arbitrary with arbitrary interactions in the three directions. One could, for example, couple the two flash qubits using a six-undit DC squid to find capacitive interactions fixed by design and an inductive interaction that can be modulated by changing the flux treated in the squid. So another interesting approach to the coupling of flash qubits is to use an oscillator to mediate the interactions. For example, in this work by Manuel Pino and Juan José García Ripoll, they study a quantum annealer where the qubit interactions are mediated by bosons in an oscillator. In this type of superconducting circuit architectures we find that the spin boson Hamiltonian encodes and spin spin Hamiltonian with interactions in a direction which is determined by the spin boson interactions. For example, in the work by Manuel Pino and Juan José García Ripoll that I have just mentioned, they couple a flash qubit and a resonator in the sigma x direction and what they find is that these Hamiltonian yields sigma s sigma s qubit-qubit interactions. In this respect we study the capacity coupling between a three Joseph Sonjancion flash qubit and an LC resonator. Again, following the rails for the quick quantization we find that the general Hamiltonian of this system is given by the Hamiltonian of the two single elements, the qubit and the resonator plus a term given interactions between them. Then using the superwall transformation we derive again the effective Hamiltonian and we find that it is given by the Hamiltonian of the qubit with a parameter delta which is also renormalized by the coupling plus the Hamiltonian of the resonator in terms of creation and annihilation operators plus a term given interactions between them in a direction which is perpendicular to that that one finds when studying the inductive coupling between a flash qubit and a resonator. These last terms provides evidence of ultra strong coupling which magnet use above 12 percent of the qubit and resonator energies and it's also dependent of the qubit and resonator parameters. To conclude I would like to make some final remarks. We have derived a complete set of Hamiltonian from the coupling of flash qubits finding evidence of ultra strong coupling and non-stop elasticity. We introduce an efficient method for obtaining the nonperturbative FFT Hamiltonian of theoretical superconducting circuit elements coupled such as a flash qubit and we saw ultra strong capacitive qubit resonator interactions which combine when they with the inductive interaction will allow for the study of new regincent light matter interaction. This way we take a step towards non-stochastic adiabatic quantum computation providing also tools for quantum simulation the study of new new ultra strong dynamics or novel Hamiltonian models. For future works we would like to study the tonability of these couplings by adding different circuit elements and it's some of the suitability of the presented Hamiltonians for quantum simulation, quantum annealing, adiabatic quantum computation, another quantum computation algorithm. With that I want to thank you all for your attention and I cannot leave without introducing our group. We are the quantum information and foundation group. We work at the Institute of Fundamental Physics in Madrid which is part of the Spanish Research Council and this project has been made in particular in collaboration with Gabriel Yawma, Juan José García Ripol and Manuel Pino who is now working at the NanoLab at Universidad Salamanca. This project is also part of the European funded project Abacus. And that's all. Thank you all for your attention. The questions are welcome. Thank you for your talk. Maria we are open for questions from the audience. Yes? Can you, can you, would you be able to extend your numerical analysis to four qubits, say three or four qubits coupled? Yeah, in principle that's the advantage of using this formula instead of this formula because you can like, you can like obtain the numerical Hamiltonian for the four qubits and then you only have to find like, in this case, if you want four qubits, two elevators to four levels of the system and then you can perform this analysis and obtain the effective Hamiltonian in principle. We have not tried yet. I actually have a question about this particular formula. I did not really understand what you are exactly transforming from this, from this standard Schiffer-Wald transformation to this parameter of this operator A. Can you please explain what this operator is, how is it different from the standard Schiffer-Wald transformation? Yeah, so what we do is that we use the property of the of the operator U, the Schiffer-Wald transformation, and we find that it can be written in terms of this parameter A, which is, which comes from the singular value of the composition of the multiplication of the projectors into the low energy subspace of the Hamiltonian and the very Hamiltonian. So what we do is we find the low energy eigenstates of the system without the interactions and with the interactions. We then find the operator B by multiplying the two projectors and then we perform the singular value of the composition. The properties of the multiplication P0, P or P Udaga P0 ensure that we can pre-write this form in terms of this operator A, which comes from the operators that we find when performing the singular value of the composition. So if you want like more details, it is explained on our, on this paper, on the one of the three Josephson Janssen Flasky with couplings. And also you can ask me and I can send you like the derivation of the formula. And thank you. Maybe today during the break I will ask you. Okay. Comment on, on what approximations need to apply and whether the, you know, how accurate the results are when you do the singular value decomposition? It is, in principle, it is the same if you use like the full formula, because you don't have to do any approximation, it's just like algebra, multiply and matrix and projectors, and you find this. So as you probably know, there was a paper by D-Wave a year or two ago where they experimentally demonstrated non-stequastic interactions with a capacitive circuit. Could you comment on similarities and differences between what you're proposing and what they did? Yeah. Like, I think that the circuit scan is like similar, but we in, in their case, they use like trussman qubits, I think, or composed by RFS quits, qubits. And we use these three Josephson Janssen Flasky qubits. I think that is like one of the main difference. And also, I'm not sure if they use like a perturbative method for the study of the FETF Hamiltonian, that's what I think. And, and I think that are, that are like the, the two like main difference, but also I think that it is true that the results that we find like coincide with their results. So I think that's a good thing. And that's it. We just wanted like to study another like qubit proposal for implementing this type of sequence. No? Okay. So what are you, what are your assumptions when you consider separately the capacitive coupling and the inductive coupling? Do you know what you are losing by making this separate analysis? We don't like, we simply like build the full Hamiltonian, taking only into account here in, when we consider the capacitive coupling, we only take into account the capacitor. And then when consider the Josephson Janssen, we only take into account the Josephson Janssen. We're really in this, in this graphic here, we also consider the capacity, the capacitance that is always coupled to the Josephson Janssen. But the interactions are much more stronger from, for Josephson Janssen down for capacitors, so it is important. Time for the last question. Thank you for the nice talk. So do, do these coupling values, how tunable are they as a function of that external flux value or the, the flux that would penetrate there kind of in the loop defined by the coupler? And then also, or do they tune at all with any application of offset charges to the circuit? Okay, so in principle, these two coupling cannot be, are not tunable in this form, but for example, you could replace this Josephson Janssen via a squid, which is a loop consisting onto Josephson Janssen. And then you could tune it using a flux written in the squid. And also, and also another study that we want to make is to find a way to make the capacitive interactions tunable, because in this form, with just a capacitor, it isn't tunable. An option could be the circuit we present here, for example. Okay, since we have to move forward with the schedule, I would say that we can leave further questions. You can, for further questions, you can contact Maria personally.