 Professor Yamamoto from NTT Research who was kind enough to actually come to Trieste and he will give a talk on recent progress in coherent icing machines okay Professor Yamamoto please. Recording in progress. Morning again. Do you hear me? Okay. So let me first thank the organizers for inviting me to this very important meeting. And today I am going to talk about recent topics and progress in coherent icing machines. Do you hear well? Do you hear me? Not really. Well maybe now it's better. I can speak loud. Here is an outline of my talk today. I will start the state of the art coherent icing machines and then I will describe the principles of two types of coherent icing machines are optical delay line based and the measurement feedback based CIMs. Then I will describe some benchmark results against quantum computing and quantum inspired algorithm on digital platform. I will also talk about energy to solution in all optical CIM. Then I switch the gear a little bit and the talk about the fair sampling property of this device for all degenerate ground states and low-energy excited states. If I have a time I will briefly describe the new machine called coherent sattmasin using chaotic non-linear dynamics. Then I will conclude. The research on CIM has started around 2010 so the first proposal is actually to use injection locked lasers or both Einstein condensate to represent spins. In this case Kraskal XY spin model is implemented. Then a few years later use of regenerate optical parametric oscillator has been proposed in this case a Kraskal icing spin is implemented on this device. As for the hardware development the 2016 100 spins all to all connected with 10,000 weights this machine was demonstrated at the Stanford at the same year NTT demonstrated the slightly larger machine with 2,000 spins all to all connected with 4 million weights. As for the prototype last year NTT announced the 100,000 spins again all to all connected with 10 billion weights. If CIM differential equation ODE is implemented as a heuristic program either on FPGA or GPU we can construct a machine on cyberspace and this cyber CIM with 100 spins with sparse connection was also demonstrated at NTT. There are two OPO coupling schemes employed for those machines. The first type is optical delay line based co-herentizing machine and first of all the single sort of a ring cavity can support n identical and defect-free OPO pulses pumped by the model of the laser pulse train in degenerate optical parametric oscillator and the part of those OPO pulses is extracted by the output coupler and the use of optical delay lines with EOM modulators we can introduce dynamically using time division multiplexing technique all to all coupling. This scheme has advantage of high speed and low energy operation but external optical circuit and necessarily very complex. The second type is called measurement feedback based co-herentizing machine and in this case external optical delay line circuit is replaced by optical homodyne detector analog to digital converter FPGA and digital to analog converter again and then derive the EOM modulator. This second scheme has advantage of all to all amplitude control coupling can be easily implemented by a single measurement feedback circuit but of course the speed and energy cost of such an approach is limited by the digital platform. Here it's our incomplete list of CIM research groups at present time as for the hardware research variety of different physical systems such as optical parametric oscillator, second harmonic generator, atomic cavity QED, various laser systems and superconducting circuit under the investigation. The theory of CIM has been developed from the perspective of quantum optics, condensed matter physics and the neuroscience. There are also various algorithms sort of development based on chaotic search or continuous time dynamical system, ODE, mixed integer linear programming and the neuro-inspired algorithm and finally application areas are several groups have been working on machine learning, compressed sensing, drug discovery and communication network areas. So let me start with our operational principles of optical delay line based coherentizing machine. The left top panel shows the degree of entanglement new minus tilt smaller than one represents the existence of internal entanglement among OPO pulses in this particular configuration and as you can see that the necessary and sufficient condition for inseparability is satisfied in the entire complete but the maximum entanglement is formed at the OPO oscillation threshold. The second step of this device is quantum correlation induced collective symmetry braking at threshold. Each OPO has a simple harmonic potential for which the electric field amplitude zero is a stable point but at the threshold this harmonic potential is deformed to bistable, zero phase, pi phase are potential and the symmetry is broken. Instead of random braking of symmetry at OPO threshold the device actually collectively breaks the symmetry for instance if OPO1 breaks a symmetry to zero phase, OPO2 actually breaks a symmetry to downspin and so on because of the existence of quantum entanglement. The right bottom panel shows the success probability for N equal 16 anti-phenomagnetically coupled ising spin model and for this case the success probability of random guess is about 10 to the minus 5 and then immediately after a few round trips of OPO success probability is increased by two orders of magnitude and this enhancement of the success probability originate from the formation of internal entanglement. Then at the threshold region collective symmetry braking sets in and one of the two degenerate ground state is selected and its amplitude is exponentially increased and the unselected ground state amplitude is exponentially suppressed and this exponential amplification and deamplification is actually the last step of CIM search process. This sort of physics is somehow related to the so-called concept ion selection which appears in physics of quantum to classical transition. It is a typical sort of physics often observed in open dissipative quantum system and as shown in the left bottom figure the OPO network system is dissipatively coupled to vacuum state reservoirs and during this dissipative coupling internal quantum correlation or entanglement formed among OPO pulses at the same time external quantum correlation between system and the reservoir is actually formed and the right bottom panel shows the quantum uncertainty relation for each OPO pulse. It starts from vacuum state at the beginning of the computation and then at OPO threshold maximum entanglement is formed between OPO pulses then at above threshold those entanglement decreases and eventually the computation finishes with quasi-coherent state and this sort of a dynamical sort of a change of internal quantum state is so-called vacuum induced ion selection namely the here when system wants to sort of a transit to a classical system it actually must eliminate any quantum correlation between system and the reservoir and in this way such a classical system can broadcast its internal state without any disturbance and this is our definition of classicality or objective reality and this type of CIM actually indeed are features this kind of ion selection. The next machine the measurement feedback CIM operational principle is much more subtle as shown on the left panel the here incident squeeze the vacuum state into the external beam splitter create internal external quantum correlation and that induces sort of a measurement induced state reduction for the major the state xj and this sort of a positive amplitude measurement result make our displacement for the target OPO pulse xi and in this way the two OPO pulses xi and xj are positively correlated in this our ferromagnetic coupling case and so in the end in this type of second type of CIM actually convert each time internal external quantum correlation into internal classical correlation and the right top panel shows the success probability as a function of saturation parameter g squared and this saturation parameter is a key parameter for CIM it actually decides the classicality and the quantumness of the operation and with increasing g squared toward one quantum correlation increases and the machine becomes more quantum mechanical as you can see that the exact density operator master equation model is well described by two types of Gaussian quantum model are based on this quantum measurement theory but not by the classical measurement theory neglecting the state reduction induced by homodyne measurement are the next concept chaotic amplitude control is most advanced heuristic are attached to a measurement feedback CIM the whole point of this new heuristic is that the normally ising coupling term is linearly sort of are implemented into the machine which is called linear feedback or via nonlinear filter such as tangent hyperbolic function this is called nonlinear feedback both linear feedback nonlinear feedback system is gradient descent and often trapped by local minima and can never reach the true ground state the new operational mode chaotic amplitude control introduces the error signal e of i and this error signal is exponentially increased or exponentially decreased depending on the present time intensity is lower or higher than the target amplitude and this sort of are dynamical or exponentially varying error signal introduces a correlated sort of noise injected noise external noise is in correlated internal state and also introduces asymmetric coupling and this asymmetric coupling introduces a new sort of a physics into the machine which is chaotic search as you can see on the top panel the here xi and xj are problem variables and its space and the e of i the vertical axis is a new sort of a dimension introduced by the error signal whenever the machine approaches a local minimum or global minimum a stable point the machine internally generate error signal e of i and make this sort of a stable point unstable and therefore the machine can escape or so in this way you can see that the right bottom panel the trajectory of amplitude control cim the here internal opo pulse are constantly flip its face forever even though it is actually identify global minimum this stable point is destabilized by the amplitude control feedback so that it permanently explore the new state and this is in sharp contrast to simple gradient decent so let me show you a few benchmark results are here are the first one is co-herentizing machine versus quantum computing and we actually picked up sharington kark patrick all two all coupling spin glass model our left panel shows the time to solution the ground state versus problem size square root of problem size and there are two sort of our quantum computing model are assumes ideal hardware namely no decoherence no gate error therefore there is no quantum error correction needed and also all two all qubit coupling somehow realized and even such a ideal quantum computing hardware grover such and discrete adiabatic compute computing such actually features exponential scaling why a chaotic amplitude control cim the time to solution actually scales as a square root of problem size and this sort of a scaling difference is traced back to our linear amplification in unitary system versus exponential amplitude amplification in dissipating system namely if you look at the right top figure in the case of grover such we prepare the linear superposition of all candidate states and one step of grover iteration increase the amplitude of the ground state by one over square root of two to n therefore in order to actually realize 100 percent success probability or amplitude probability of ground state equal one we have to repeat square root of two to n times for this grover iteration and that leads to exponential scaling of quantum computing approach right bottom panel we show that the amplitude of the ground state is exponentially increased at the opo threshold corrective symmetry breaking point and this comes from open dissipative nature of this particular device second benchmark result are actually describe the performance of cyber cim on cpu versus state-of-the-art digital heuristic in this case a breakout local search one of the best heuristic which constantly features our best sort of a solution for max cut our three panels for program size n equal 100 200 500 spins actually demonstrate the general trend our cyber cim is less performant than bls when the given program instances are easy but if the program instance are hard then the cyber cim is more performant namely with appropriate computation time it can always report and return the correct optimum solutions while bls does not our last benchmark result is cyber cim are implemented on gpu versus our discrete simulated bifurcation machine this discrete simulated bifurcation machine is very similar to cim open dissipative system digital platform even though it was originally invented as a superconducting unitary device but now it is our digital heuristic and the left bottom panel uh compares cim and the discrete simulated bifurcation machine and the trend is actually similar to the previous slide when the given program instances are easy uh discrete simulated bifurcation machine is faster but when the program instance is really really hard then cim is actually better to find the solution uh right panel shows our time to solution for sparse graph so-called g-set problem uh from program size 800 to 2000 as you can see in the histogram of the right most two panels they are uh discrete simulated bifurcation machine uh actually shows the slowest speed are and their uh chaotic feedback control which is similar to previously describe the chaotic amplitude control is actually fastest uh among the solved programs and the three machines are tested here uh cannot few actually are g-set programs which is shown by uh black sort of a history this is actually they are all optical implementation of chaotic amplitude control as i said their chaotic amplitude control actually normally implemented in digital platform consisting of adc fpga dac but uh if we can construct such optical non-linear optical uh circuit uh then we can actually are uh implement these are algorithm all optically are and their uh red solid sort of our uh device uh all thin film lithium niobate degenerate optical parametric amplifier device except few sort of our uh passive elements all sort of active device are constructed by a different opa device and if we employ those approach as shown on the right bottom panel are energy to solution or two approach one is cyber cim on gpu the other is all optically implemented cim the energy to solution is different three orders of magnitude so if we multiply time to solution uh to this result probably all optical uh sort of approach uh enjoys five to six orders of magnitude lower a sort of delta e delta t product compared to cyber cim let me here change the subject a little bit uh now i would like to talk about fair sampling uh of degenerate ground states and the low energy excited states normally izing machine is designed to find any one of the ground states uh one optimum solutions are good enough but some other applications such as Boltzmann sampling for machine learning or a structure based virtual screening for drug discovery uh or decomposition of large optimization programs into our sub programs to be solved separately uh we need actually to get the reasonable number of optimum and sub optimum solutions and as shown on the left bottom panel adiabatic quantum computation is poorly suited for such application because the final state is connected to the initial ground state are in this adiabatic transition so that one particular ground state has a exponential bias over other degenerate ground states or of course are excited low energy uh states uh on the other hand uh coherentizing machine the here are oscillation property are as a function of external pumping uh into opa opo uh is shown on the right panel and the first bifurcation happens at maximum eigenstate of jij matrix this is not the ground state and the ground state actually appears are very uh later uh pump plate but all those sort of our ground state and the low energy excited state have more or less a similar sort of a threshold pump plate uh gain threshold so that the stochastic nature of the such process of cim can actually sample uh those good solutions with uh fairly equal probability and the next slide actually confirms this theoretical prediction uh as you can see uh from the here uh left bottom panel uh here it's our uh sort of our uh if we perform the here single run uh computation uh of cim uh 1000 times uh for particular uh problem instance for which are eight degenerate ground states and the six degenerate first excited state and the four degenerate uh second excited state exist and the lower panel shows a histogram are how many times out of 1000 triumphs the single run actually the uh 1000 run actually report those state and as you can see that the lowest uh probability to report particular state is larger than 10 percent which means that at most if we can run 10 times uh cim we can exhaust all of those uh ground state and the first excited state and the second excited states single run can report many sort of desired states uh because are uh as I have described already where never machine approaches are local minimum uh or global minimum it generate internal error signal and destabilize that state and migrate to the next one so uh right panel uh shows some surprising result normally the here uh quantum device uh improves its performance when the system is more closed in the case of our cavity device high q cavity is always preferred uh compared to low q cavity but if I plot the sampling time are as a function of cavity q value or cavity decay time per round trip you can see that the time to sample uh decreases monotonically with our uh decreasing the here uh cavity decay time uh and if cavity decay time is one lower panel which means that uh internal field decays one over e even after one round trip it's a extremely extremely low q cavity device but the performance is still improved and the optimum point sweet spot is actually t decay is a point to this is ridiculously low q cavity device and that is actually uh demonstrate the power of external dissipation uh for some time of computational task uh so let me uh uh uh finish my presentation this morning uh by introducing new machine which solves our satisfiability problem or maxat problem uh with continuous variable as you know the case at the problem uh for instance k equals three three set problem that each clause has three problem variables uh x one or x five bar something like that and this given problem is defined by coefficient c i j uh if c i j is one it means that uh problem variable x i is included j screws and if it is minus one which means that the x i bar is included in the j's clause and if c i j equals zero then both x i and x i bar are not included so once we uh introduce c i j matrix then we actually sort of define completely the given three set problems then the j's clause state is represented by capital k j which is a product of one minus c k j x k over two and this is zero if j's clause is already closed and one if j's clause is not satisfied okay and uh we sort of relax the problem variable continuous variables from discrete to continuous namely introduce soft spins then the set problem and the maxat problem is uh respectively defined by our summation k j equals zero or minimize k j and the next slide shows our how to sort of uh define this machine the first equation differential equation described here each opo amplitude x i last term is an important amplitude error collection feedback term which is actually are determined by the right sort of uh figure if x i is included in clause j one j two and j three are the two other variables except x i is coupled to x i through this our feedback term f i to sort of uh satisfy each clause and the error function e of i is as i said already are targeted to stabilize to target intensity otherwise it is exponentially increasing or exponentially decreasing as shown on the right lower panel our and the left lower panel shows a problem uh variable amplitude and you can as you can see that they are mostly clamped at either plus one or minus one are true or false are and then features chaotic or non-linear dynamics and then benchmark result uh summarized in this slide the first are two top panels actually are compared the two strategies the first first strategy is a three set problem is mapped to ising model and then the ising mass problem is more solved by the chaotic amplitude control c im and as you can see that those sort of three set to ising mapping strategy is four to five orders of magnitude slower than the than the direct use of uh set cfc machine uh by the way uh two figures the horizontal axis is our number of clause normalized by number of program variables uh so-called alpha parameter and their program size capital n lower two panels shows the state of the art set solver based on continuous variables ctds and as you can see on the left lower panel ctds cannot actually find the satisfying solution for alpha equal 4.26 this is a phase transition point and the most difficult point for any solver encounters are only set the cfc can find the satisfying solution for this are phase transition point and then are even at this phase transition point they are set cfc features polynomial scaling if the program size is smaller than one thousand if larger we don't know the answer so let me conclude my talk our the present day i think a physical computing research is somehow combination of our four or more computational concept they are either analog computing optical computing quantum computing neuromorphic computing they are definitely not the mainstream of computing technology mainstream is of course based on seamos hardware and application specific heuristic are in the mainstream these are computing uh nice thing about this uh technology is hardware development and algorithm development are completely independent so uh each expert group actually uh worked on that but the physical computing it's early stage still so or hardware algorithm core design is indispensable and are also those sort of a different concept of a novel computing skin can be nicely combined towards the future professor Yamamoto for the beautiful talk with impressive results so we have time for a few questions from the audience and then maybe from the from the online participants as well thank you for the very nice talk i had some questions about the benchmarking results i think it was slide 11 or 12 yes 11 actually i guess it applies to both but the plot on the left so are these actual um runs on on an on a cim uh or or is it um this is all numerical the american yes our uh this is our uh the here uh cim cac case is implemented on gpu and the global search and the discrete adiabatic computing is probably cpu i don't know the here uh one qubit can actually answer the what the digital platform gpu cpu okay because the sharkman carpatrick is is np hard so it would seem like you have a polynomial solution for an np hard problem uh with the uh the cim oh the are you asking why uh cim features a square root exponential for such np hard problem right we believe this is our finite problem size effect as you can see problem size is really small less than 1000 and that's why this is just a couple of slides back i think on your results for an antiferromagnetic chain uh yes this one i'm wondering about the the blue versus the green data uh did did you put a bias on one of the spins or did you post no there is no bias uh this is our our the purely ising hamiltonian with no german term are n equals 16 antiferromagnetic spinning are but we post selected okay we are one up down up down one of the two ground state uh when it is reported as a final state then we selected and if the other one is selected then we discard it that's why other questions questions from the audience maybe the audience uh online also thank you for the nice talk um so i was just wondering where do you see the first application area for this particular compute technologies there are specific you know time scales or you know variable number that would be well matched compressed sensing are drug discovery and the communication network they are present cim uh cyber cim particularly uh on gpu is actually i think are ready for practical use but again i think we are not really expert on the application area uh so the real market exists or not i don't know but technically uh cyber cim is ready i think but uh physical cim in particular all optical cim is actually uh only sort of idea exists and hardware development actually takes more than 10 years okay i think it's time to move uh to the next speaker so let's thank professor yama motte again