 My voice and display showing well. Yes, it is fine. Okay, then I'm going to start. Okay, hello everyone. And I first want to thank to the organizer and everyone for giving us today the opportunity to present our study. My I am Yuko Kamishima from Yasukawa Electric Corporation. The title of the presentation is Domain Role Behavior on Is it Machine? The study was conducted with Shu Tanaka of Keio University. The presentation outline is as follows. First, the summary of this presentation, then the introduction and objective of this study. Next, encoding and embedding methods used in this study. Then the explanation of traffic optimization problem considering crossings, the problem formulation, the conditions of the comparison and the results of the comparison. And then the discussion of the results. At last, the conclusion and outlook of this study. So the summary of this presentation is showing this slide. We compared two encoding methods, one hot encoding and domain wall encoding on a GPU based Is it Machine using a traffic optimization problem considering crossings. And perform a comparison with and without embedding. Results show that domain wall encoding gives equally or better solutions than one hot encoding when embedding exists. The table below gives a brief overview of the results. The results obtained in this study encourage the use of different encoding methods depending on the type of Is it Machine. So the introduction. It is well known that combinatorial optimization problems are difficult to solve with conventional methods. To solve this problem, specialized computers like Is it Machines are studied. Is it Machines required an encoding method to express discrete values. This is because only Cuba models which consists of quadratic binary variables can be solved with the Is it Machine. One hot encoding is a common method used. And also domain wall encoding is another method that we can use. Domain wall encoding has good results in quantum annealing. And it's usefulness in classical Is it Machine is yet to be shown. The objective of this study is to show the behavior of domain wall encoding in Is it Machines. To achieve this, we compared one hot encoding and domain wall encoding on a GPU based Is it Machine. As I have shown, encoding is a way to express discrete values. And one hot encoding is generally used in Is it Machines. One hot encoding is characterized by the use of one bit per one discrete value. And the cubo for it is as follows. One hot encoding's way of expressing discrete value is simple, but it requires a constraint which needs a dense connection of bit as shown in the figure right. That recently, domain wall encoding was proposed. The discrete value of domain wall encoding is set between bits. And number of bits required for discrete value. I'm sorry. The number of bits required for discrete value is one less than one hot encoding. The cubo for domain wall encoding is as follows. As shown in the figure right, the position between two bits which have different values is expressing the discrete value. Now I'm gonna explain about embedding. Embedding is the mapping of the problem graph to the Is it Machines graph. Sparsely connected Is it Machines, for example, D-Wave Advantage require embedding as shown in the figure below. Embedding makes chains of physical bits to represent the logical bits in the cubo. In this study, a Pegasus graph was virtually created to see the effect of embedding. The embedding we use is the clique embedding. And the clique embedding has regular increase of chain length in Pegasus graph as shown in this formula. Now I'm gonna explain about the optimization problem using this study. The traffic optimization problem considering crossings is a problem minimizing the total cost which is the travel time for all the vehicles. Vehicles in this problem would try to avoid certain entries into non-priority crossings. As shown in the figure right, the crossings to be avoid in this problem has two vehicles having colliding routes. The problem size of this problem is the number of vehicles times the number of possible routes. As shown in the figure below, each vehicle is given several candid routes and each vehicle can only choose one route. The cubo formula of traffic optimization problem considering crossings for one hot encoding is as shown in this slide. Each vehicle has a certain number of candidate routes and when one route is chosen, the cost increases for the cost of the chosen route. So that's this age dist. And for each colliding route costs increases and this is the age call. And for constraint function, a one hot constraint is given for each vehicle. So this makes that each vehicle has to choose only one route. Next, I'm showing the cubo formula for the domain wall encoding. The difference with one hot encoding is it required two adjacent variables. By using two variables, domain wall encoding can express the same way as 100 coding. So it can have the same meaning as 100 coding. In this formula, the coefficient D was fixed to one and the penalty coefficient P was set to the maximum coefficient of cubo's objective function times two. Now to the conditions for the comparison are shown below. We performed a comparison of a domain wall between one hot with and without embedding. The problem size was changed by number of vehicles and number of candidate routes and also the calculation time was changed. The Ising machine used was the fixed start amplifier annealing engine, which runs on a GPU. And 100 calculations were performed for each condition. Now to the results. Here in these figures, I show the comparison between the results obtained by the calculation. The vertical axis indicates the cost value of the main wall encoding and the horizontal axis indicates the cost value of one hot encoding. And each calculation result is shown as a black dot. And when the black dot is on the red line, it means that the results of domain wall encoding and one hot encoding were having the same cost value. And beware that in this figure only feasible answers, what feasible results are shown. So we can see that one hot encoding had better results without embedding. And when calculation times are in is one millisecond, we can see that one hot encoding had better results in domain wall encoding. And we can also see that domain wall encoding had better results with embedding. So in the figures below, we can see that domain wall encoding had better results than one hot encoding. We can also see that longer calculation time improves results. By comparing the one millisecond and 10,000 millisecond results, we can see that the solutions have improved, especially with embedding. When it's only, when we just gave a one millisecond, most solutions of one hot encoding were on feasible in larger problems, but when we give 10,000 milliseconds, some solutions of the larger problems in one hot encoding became feasible. So next, I will discuss about the results obtained. Here I show a histogram of the hamming distance from local solutions to the best solution. For this figure, I had translated the answer obtained in one condition to bit values expressed by each encoding. And calculated the hamming distance from each local solutions bit value to the best solutions bit value. The line in the figure is the mean value of hamming distance of one hot encoding and the main wall encoding. So by showing several conditions, I show that encodings with good results have short hamming distance from the local solutions to the best solutions. So by seeing other conditions, we can see that when we obtain good results, the hamming distance is shorter. This could be due to the effect of the chain made by the embedding. Here I show a visualization of greedy transition of the main wall encoding and one hot encoding from the same local solution to the same best solution when embedding exists. This visualization is not the real transition done in the calculation, but it gives us the image of it. In this figure, we see that both the main wall encoding and one hot encoding, this DW and OH, is making two logical bit flips. But also we can see that the physical bit chain, the physical bit chain made by the embedding has increased the total bit flip required for it. This suggests that the main wall is easier to search due to its short chain link. Now we'll discuss about the number of bit flips required to transition from a local solution to the best solution. Without embedding, a domain wall encoding requires FAA flips for each agent. The agent in this case is the number of vehicles. So then when we see the one hot encoding, we can see that it just requires two flips per agent. So if FAA is not less than two, the bit flip required is greater for domain wall coding than one hot encoding. It can be imagined that for problems that requires many descriptive expressions, one hot encoding will be easier to search. With click embedding, the bit flip requires multiply by the length of chain of the embedding. So in this case, the M. And the chain length is shorter in domain wall encoding than one hot encoding because less bit flips, sorry, because less bits are required in domain wall encoding than one hot encoding. So we can say that when embedding is present, the bit flip required for one hot encoding may become greater than domain wall encodings. So for conclusion, one hot encoding, we have done a comparison between one hot encoding and domain wall encoding based on a GPU base, sorry, on a GPU base is in matching. And domain wall encoding was more accurate than one hot encoding when embedding was present. And we think that length of the chain may have affected the search. The take-home message of this presentation is use the appropriate encoding for your ISAM machine type. And finally, the outlook. Domain wall encoding may be useful on machines where embedding is needed. For example, the B-Wave systems quantum annealing machine and Hitachi Shima annealing machine. Although in this study we use clique embedding, other embeddings embedding methods may change the effect of domain wall encoding. So that's all for the presentation and thank you for your attention. Time for questions, comments. Thank you for the nice talk, it's very, very clear. My first question is kind of more technical. So I guess one of the key differences between one hot encoding and domain wall encoding is that to get between valid configurations, so you can do a single bit flip for domain wall encoding and on both sides of the bit flip, you satisfy the constraint. But for one hot, I would imagine that you'd have to do, a single bit flip gives you an invalid configuration and then you have to have a second bit flip to get back to a valid configuration. Does this, first of all, is this a correct observation? And second of all, is this important that you, that this kind of having a single bit flip retain a valid configuration is important for these systems? Okay, thank you for your question and that's a very interesting question. And first I want to answer that a single bit flip for domain wall encoding is just a transition from a feasible answer to another feasible answer. That's true. And for one hot encoding, it's a transition from a feasible answer to an unfeasible answer. So it means that there will be a large energy change when a bit flip occurs on a one hot encoding. And that first, when we started this study, we thought that that's very important. But then we have observed that when there's no embedding, although the bit flips makes a larger energy difference on one hot encoding, the number of bit flips affects even more in this case. So we may say that in this machine, the number of bit flips affected much more, but this may change in other machines. Right, thank you. Just one final quick question. How close to application relevance do you think that this type of compute is? Is it many, many years away or do you think that it could be possibly application relevant in the near term? Well, that's very interesting. And yes, I think we can use this domain while encoding in edge applications like on factories, on machines in the near term, not in such a far distant future. But sadly, we're still trying to test out real applications so I think that will be a future investigation or a study to show. Okay, thank you. Thank you very much. I have a question, may I ask? Yes, please. Your figures were those results that satisfied condition constraint. And I wonder what was the ratio of feasible solution in each case? Have you compared those numbers? Sorry, I don't have the exact number of the ratio of the feasible answers, but I can say that when there's no embedding, all the answers were feasible. So I'm in this figure of the top two, these two figures are showing all the answers that I have calculated. But in the case of when embedding exists, we have seen that there is many times of curious on feasible answers. So I'm gonna show this supplementary result here. This is showing every answer that I have obtained with embedding when I just gave one millisecond. So in this figure, the feasible answer is just the answers that are on the left side in the various left side, we can see that there are some answers that are feasible. And the rest, most of the answers are unfeasible. And we can see that in the main voting coding, almost all the answers were feasible, but in 100 coding, we observed that the opposite thing, most of the answers were unfeasible when the problem size was large. Is this okay? Oh yeah, yeah, okay. I understand, thank you. Thank you very much. Really interesting, nice to see this being tested out in different settings. Are there any plans to publish this? Oh yes, of course. Sorry, this study is yet to be, we're planning to publish this in the near future if it's possible because as you can see, we have this idea that bit flips are very important, but still we cannot see if the energy transition itself, the difference of energy is affecting how much is the energy affecting to the transition of local solutions to the other local solutions. So if we can, if we finish that, I think we could make a good paper of it. Cool, really great, look forward to it. Okay, thank you. One last quick question. Okay, thank you for the nice presentation. So my question is, I suppose that choosing the coefficient of the penalty term may affect the conclusion. So my question is, so have you tried to do the calculation for the values combination of the coefficient of the penalty term? Yes, I think that's a very important part of this study. This time the coefficient D was just fixed one, but the penalty coefficient P was set to maximum coefficient of Cuba's objective function times two. So why did I set this coefficient to two? It's times two is that we haven't done a pre-investigation by changing the value one by one. And we have observed that it doesn't affect so much to the results trend. Yes, the results vary a little bit, but the trend itself didn't change. So in this case, in this study, we have just set it to the maximum coefficient of Cuba's objective function times two. Okay, thank you. So let's thank the speaker once more. So the next stroke of the session is... Recording.