 ... in efektivne premafetorizacije v konto manjelji z modula v lokalnih stručičnih vsev. Interaktion. Česepes-Balita je vsev. Česepes-Balita, Marko, Roberto, Sabastiani z Universitetu of Toronto in Katie Boothby z D-Webb System ink. Jin-Wen. If you have to keep the mask, you have to shout. OK. The microphones are not recording. OK. This work is supported by QATN. So what's the prime factorization? It's a problem of decomposing a composite number into a product of prime factors. It is computationally hard and often used in cryptography to design algorithms. So far no polynomial time, classic algorithm is known yet. Solving this problem via quantum annealing was first proposed by Roth et al. in 2017. They proposed a method for encoding the prime factorization problem into Kubo and demonstrated that integers up to just 200,000 can be solved by 2x processors. And then in 2018, John et al. proposed another encoding approach and they claimed a large number up to about 250,000 can be embedded to D-Wave 2000q processors. Both state of the art use the global embedding, which is encoding the given problem first into the Kubo without considering the hardware architecture and then embed the Kubo to the architecture with generic embedding algorithms. Instead, we propose another approach for prime factorization via quantum annealing based on locally structured embedding, which we proposed for solving cell problems via quantum annealing. We produce easy models that are directly compatible with Paxes and we solve them via D-Wave advantage processors. In this talk, we will present modular encoding of bitwise multiplier into Paxes, which is synthesized offline via optimization modular series. We will show that this modular encoding can be used in combination with QB sharing to encode up to 21-bit times 12-bit multipliers on Paxes topology, ideally enabling advantage system to factor 33-bit numbers and most of this large number, 8 billion. However, in real architecture exist 40 QBs and couplings. In our experiments, we tested the multiplier encoding of up to 17-bit times 8-bit for D-Wave advantage 4.1 to do prime factorization. We demonstrate that the pulse annealing and reverse annealing to performance enhancement techniques of quantum annealing can be find solutions of the prime factorization. These two integers are the two examples factorized with non-zero energy and with zero energy respectively. Here is our encoding approach. First, we represent the multiplication as a conjunction of small Boolean functions. This is implemented by representing the bitwise multiplication with this control further and representing the connection between control further with equivalence constraints. For example, C out of the current control further equal to the carryout, equal to the carrying of the next control further. You can consider this conjunction as the end function of all components shown on the left graph. The next step is to transform this conjunction into the sum of kubos or easy models of the corresponding small Boolean functions. For control further function, we compute easy model penalty function that is compatible with PAXES via optimization module series offline. In this definition, notice that the graph VE represents the subgraph of the PAXES topology and A represents the encelers we introduced to find a valid penalty function that is existing positive gap such that for all x, the minimum of the penalty function equal to zero if x satisfies f, otherwise greater than or equal to the minimum gap. The omt problem here is to maximize the minimal gap subject to the range constraints of bias and couplings. The variable equivalence constraints is encoded in the same way. The penalty function equal to zero if the variables are equivalent, otherwise equal to the maximal minimal gap here for alternative to change, where the control further is mapped to the disjoint subgraphs of the PAXES. The equivalent variables here carry out of this control further and carrying of the next control further are forced to be equivalent with this strong coupling. We map them to the same kubit, making this blue kubit encoding two variables simultaneously. To prevent the bias of this shared kubit exceeding out of the range, we add extra constraints to the omt. Through this kubit sharing, we get two more kubits in gray for the use of ancillary variables and moral couplings, which is expected to get a larger minimal gap. These kubit sharing techniques can be also extended to compensate the lack of continuous chain in 30-degree or 60-degree direction of the PAXES. For example, if we want to propagate input A in red node here, in 60-degree there is no continuous chain in this PAXES topology, but we can introduce a fresh variable seen out here and append this variable constraint to the control further, just the same as the carryout into the carrying. Alternatively, this encoding can be used to encoding up to 21-bit times 12-bit multipliers, which has minimal gap of 4 over 3. The other multiplier encoding with larger minimal gap is 2. The size is 22-bit times 8-bit. In our experiments, we targeted at a d-wave advantage, 4.1, as shown in orange, there are some 40-kubits and couplings. So we encode up to 70-bit times 8-bit using a unified penalty function of control further with exact gap. This exact gap is guaranteed by fixing the position of the enable variable by cubic sharing and imposing strict constraints on the shared qubit and the cost of left coupling. Here is some preliminary results that encode the correct factors of the input integers. We used two initialization methods to fit inputs to the annealer. One is to substitute the inputs directly to the penalty function. The other is to first substitute the inputs into the corresponding control further and re-encoding the control further. In order to factor in large numbers, we used pulse annealing and reverse annealing. We set the pulse annealing time to 800 microseconds and total annealing time to 40 microseconds and two different time intervals for forward and reverse annealing to find the optimal starting point of the pulse. For each problem instance, we read 1000 times. As you can see, after the forward annealing, we can get a very few samples that encode the solution. And no zero energy samples. But if we reverse annealing from the local minima, we get more solution samples and even zero solution samples. Here we plot the energy distribution of the found solution samples in red and one lowest energy non-solution samples in blue. Through reverse annealing, we get lower energy. Even zero energy for bottom three cases. On the left, we plot the hamming distance from the local minima to a new solution samples through reverse annealing and extract the exact solution samples that all control further functions correctly. As you can see, the hamming distance is less than 25. For the other initialization approach, we can obtain samples with energy less than gap only with forward annealing. That is because re-encoding the control further increase the minima gap. Correspondingly, the hamming distance is smaller than the previous. To conclude, we propose another approach for solving prime factorization problems via quantum annealing base on locally structured embedding. Our approach maximizes the minima gap of the local computation via optimization series. We can produce chance of almost equivalent length or less chance, which is expected to bring better performance and global embedding. We show the QBIT sharing techniques can be used to encode up to 21-bit times 12-bit multipliers. And we demonstrate the pulse annealing and reverse annealing can be used to find the solutions of the prime factorization. Our future work goes into three-direction investigating how to make the most of reverse annealing in combination of the forward annealing to reach zero energy, try with less 40 annealing if possible, and extending our approach to the next hardware architecture, Zepha. Last time, I want to show you the energy distribution of the samples evidented from quantum annealing for single control further. The red represents the solution samples and the blue represents the non-solution samples. Among 100 samples, we can get 90 samples that encode the solution and one encode the solution but with large energy. For the control further of the 21-bit times 12-bit, we can get 84 and 17-8 samples that encode the solution. I want to leave this to you to imagine the performance of the 43-bit advantage system to factor 33-bit numbers. That's all. Thanks for your attention. Thank you. One of your collaborators wants to... Yes, second participants. Artur Sebastiani here. No, no, I don't want to speak. Just... Ah, OK. Good. I thought you wanted to speak. Is there any question? So, it seems like the embedding for Pegasus is a lot better than the embedding for the 2000q. But one thing that I remember from some detailed studies of 2000q was that as you anneal the multiplication circuit, you have qubits whose dynamics freeze at very different times based on their role in the circuit. And so, I'm wondering if you've looked at anneal offsets as a parameter for homogenizing the passage through the quantum phase transition. Yes, I tried to offset the annealing time. Since our encoding, we almost have no chain problem. We don't need to offset the annealing time for a chain. But I tried to anneal time of the ancillary bits and the variables of control parameters. It seems like if we do that way, we broke the structure of the control parameter. And we also tried to offset annealing time of border control parameters and the inner. But we haven't got a better solution, so maybe we will investigate more detail in the future. OK, thanks. Any other questions? If not, let's thank the speaker again. Thank you.