 Can you hear me and see me this right? Okay. Yes. Okay. And thank you German for the introduction I'm Tomohiro Hattori from Keio University Today I'm going to talk about performance comparison among various methods of fixing spins in Kontamani link This work was done in collaboration with Hirotaka Irie, Tadashi Kadowaki and Shutenaka At the beginning of my presentation, I would like to give an overview of our research One of the methods of fixing spins called HQA hybrid Kontamani link is proposed by Hirotaka Irie at all We have verified the performance of HQA using simulation and a natural Kontamani link machine We come to three conclusions through the qualifications First the longer the running time is The better solution we can obtain in HQA Second there is an appropriate number of fixed spins depending on the Kontamani link time So better solution can be obtained by HQA than by original QA Throughout the presentation, I will show you the results to arrive at these conclusions Next I would like to explain background There are many There are still many difficulties to use a Kontamani link machine The number of QBIT is one of them The number of QBIT in Kontamani link machines is still small compared to the number of spins in Ising model Which represents combinatorial optimization programs to be solved So we have to reduce the number of QS spins to be able to input into such a Kontamani link machine To deal with such large programs, many kinds of hybrid algorithms have been proposed Those methods can be categorized two types Decomposing and fixing spins Most of them are based on the idea of decomposing and original large programs into sub-programs So multiple iteration between crush curve and Kontam servers are required At that time one of the methods of fixing spins caused HQA hybrid Kontamani link is proposed like this HQA does not require the multiple iteration The purpose of this research is to investigate features of HQA Next I would like to introduce HQA The concept of HQA is illustrated in this graph The horizontal axis shows the state of spins The vertical axis represents energy landscape In HQA, first, we luckily explored the solution by pre-processing And fixed spins, whose value can be easily determined using a crush curve method Second, we precisely explored the solution by Kontamani link And we can obtain the final solution This is the algorithm Of course, we clarified spins into two categories HQA spins and frozen spins by a crush curve way We input the HQA spins into Kontamani link machines with effect of frozen spins And we can get final state This is the example when a program size is 6 We get pre-processed state by fixing spins And we can reduce the size from 6 to 3 And we can input into the Kontamani link machine and get the results Since the solution is obtained in this way, a multiple iteration between crush curve and Kontam servers are not required Next, I would like to introduce pre-processing We use main field annealing as a pre-processing And in main field annealing, we solve this equation tau MS represents main field annealing time We solve this equation with decreasing temperature of T and get a graph like this Horizontal axis is time and vertical axis is magnetization of spins We consider two components of solute value at T equals tau MS And the speed of convergence and categorize these spins into two types, frozen spins and QA spins As shown in this graph, the dynamics of QA spins magnetization is more complex than frozen spins In doing so, we extract it as spins whose values are not easily determined by main field annealing Hence, the loss of pre-processing is reducing the size and separating spins into two types as if separating a coarse grain and fine grain This is a pre-processing setup This equation represents annealing schedule in this research And initial temperature and final temperature is set like this We named the solution derived by main field annealing, MF solution This is a program setup We solved the annealing model on an undirected complete graph and choose magnetic field and spin interaction from standard normal distribution The model called a Schrington-Kick-Patrick model We named the solution derived by FX.amplify to be the FA solution We chose a difficult program for which a main field annealing could not reach FA solution The results were verified using QTIP or small size programs and D-Wave advantage systems 4.1 for large size programs In our simulation, we use these annealing schedules and tau QA represents a quantum annealing time Next, I'm going to explain the results This is a result about a small program in simulation The size of the program is 32 This is the graph which represents energy against the annealing time in HQA I use different colors for different numbers of QS pins Red is the case with the lowest number of QS pins And blue color is the case with the largest number of QS pins From this graph, in all cases, we can confirm that the longer the annealing time is the better solution we can get In the short annealing time area, we can get a better solution when the number of QS pins is small On the other hand, in the long time annealing area, we can get the lowest solution when the number of QS pins is relatively large And the number of QS pins is less than 6 at the red color area The solution sticks to the male-male solution On the other hand, when the QS pin is more than 8, we can get a similar solution to FAS solution This result indicates that by increasing the number of QS pins, we can get out of a local minimum and we can get a better solution We could obtain similar results with different scales To consider these results, we investigate minimum energy gap between ground state and fast excited state This is the graph in the case the number of QS pins is more than 8 Increasing the number of QS pins makes a minimum energy gap decreased So, increasing the number of QS pins would require a longer annealing time in QA We think that there is trade-off relationship between getting a better solution and reducing energy gap decrease This is the result about a small program using d-wave advantage These are a program setup With a confirmed result, these results are similar to the simulation results In all cases, we can confirm that the longer the annealing time is, the lower the energy we can get And by increasing the number of QS pins, we can get a better solution I assume that the energy values have not changed because there was enough annealing time for this program So, the energy is not changed but the QA is changed This is the result about the large program using d-wave advantage The overall trend was similar for the small scale program Longer the annealing time, we can get the lower energy And by increasing the number of QS pins, we can get lower energy But there is a different point The solution accuracy decreased when the number of QS pins was increased too much The blue line color, a case is that And this result, there is an appropriate number of QS pins when solving a quantum annealing machine We will examine these results in detail at the next page These are box and whisker diagrams of the previous results This is the result of the dependence between QS pins and energy in HQA The graph on the left hand represents the results in a short annealing time area at TQA equals one microsecond The graph on the right hand is the result in a long time annealing area TQA equals a 2000 second microsecond In short time annealing time area, we can obtain the lowest solution when QS pins, the number of QS pins equals 32 On the other hand, in a long annealing time area, we can obtain the lowest energy when the number of QS pins equals 96 From this result, we found that there is an appropriate number of QS pins depending on the quantum annealing time And furthermore, comparing the white-colored area and red-colored area From these results, the solution from HQA in the white area is always better than from the original QA in the red-colored area This is the conclusion of this research And the longer the quantum annealing time is, the better solution we can obtain in HQA There is an appropriate number of QS pins depending on quantum annealing time Better solutions can be obtained by HQA than by the original QA And I think these results support favorable quantum annealing If it was possible to reduce the size of programs using HQA and use favorable quantum annealing As far as the speed up and getting high accuracy solution, it would be possible, I think That's all of my presentation. Thank you for listening Thank you, Dr. Hattori Time for a few questions A quick clarification question After you find the fixed spins using the preprocessing step, do those spins that you found that way still evolve when you do quantum annealing or are they completely fixed? You are asking the effect of fixing spins, right? Yes, the lowercase n QA spins I'm sorry, the other ones, the frozen spins Do the frozen spins still evolve when you run the QA or are they completely frozen at that point? Yes, this is the result when the number of QS pins is small If we fix the opposite direction with the If we fix the spins different from FAS solution We can only The solution is stuck to Mx solution But we can Fix spins in the proper direction We can get FAS solution But the frozen spins are fixed forever This is the question that frozen spins are fixed, they don't move Is this the Yes, that's right Once I fixed and the spin is not changed Okay, okay. Now I think it's clear. Other quick questions So I actually have a question about how you are differentiating between which spins to freeze and How are you actually freezing the spins and differentiating between the frozen and unfrozen spins? I use this Hamiltonian I sold the spins and considering two components and The sold spins in the in the law and we choose spins from the Number of index is small and we This is the Hamiltonian a whole program Hamiltonian and we Once if we fix and sold we get this Hamiltonian and we We Input the this Hamiltonian into the Contaminated machine and solve the program and get the whole The solution is whole program I have a question Do you have any explanation of why the gap goes down with the number of frozen spins? So you had a picture of the gap you had a picture of the minimum gap with respect to the number of frozen spins So I don't quite understand why it goes down. So for example, where will you put The minimum energy gap when you don't have any frozen spins Why will it fit in that picture? this minimum energy gap is the modified in the fix Hamiltonian after fixed and I The reason why the minimum energy gap is decreased is I think The energy We can If the number of qs spins is rush the The many states of energy is There is a many states of there are many states of energy and if we but we fix spins and The energy we can the possibility The first excited state is changed the The other states I think so and the yes, the The We can many if the increasing number of spins and the we can get many different energy state and the And it makes the energy gap decreased. I think Okay, so let's let's move to the final question say please Yes, I have the same question at the previous one How how did you separate the frozen spin and movable spin? um I separate the spins using this dynamics and I The first I consider absolute value at t equals tau mf the If the The value of magnetization is small we fix we uh don't fix spin and we but the We fix spins the speed of convergence and is First I consider this value Like this and the final absolute value we can get the one or We can get this value and of the map. This is the magnetization spins magnetitions and we thought This value and we second I consider the speed of convergence if The convergence is earlier. We fix spins And we In this way Chose spins Okay, I think I think we should Close so let's thank Dr. Hathory again The coffee break the coffee break is upstairs. There is level and we reconvene here um around Five past eleven something like this. Okay