 Full text:
 PDF (eReader) / PDF (Download) (1174 kB)
An online advertisement optimization, which can be represented by a combinatorial optimization problem is performed using DWave 2000Q, a quantum annealing machine. To optimize the online advertisement allocation optimization, we introduce a generalized version of the Markowitz meanvariance model which is a basic model of portfolio optimization. The obtained optimization performance using DWave 2000Q is higher than that using the greedy method which is a conventional method. Additionally, to conveniently use Ising machines including a quantum annealing machine, new software called PyQUBO is developed. The first half of the paper gives a review of several combinatorial optimization problems and how to represent them using the Ising model or the quadratic unconstrained binary optimization (QUBO) form. We show the results of the online advertisement allocation optimization and the explanation of PyQUBO in the last half of the paper.
Ising machines have actively developed since the birth of commercial quantum annealing machine, DWave.^{1}^{–}^{6}^{)} Recently developed Ising machines specialize in searching better solutions to combinatorial optimization problems represented by an Ising model or a quadratic unconstrained binary optimization (QUBO) form.^{7}^{,}^{8}^{)} Some earlier studies reported the advantages of using Ising machines for specific combinatorial optimization problems.^{5}^{,}^{9}^{–}^{12}^{)} However, it is still unclear whether Ising machines are more efficient than conventional computing technology. Thus, it is necessary to do physicsbased theoretical studies and develop hardware and software for various applications.
Combinatorial optimization problems exist in our daily life, e.g., optimization of shiftplanning, traffic flow optimization, and delivery optimization. As the amount of data traffic increases due to the development of more sensing devices, better highperformance computing technology for combinatorial optimization processing is required. Additionally, new combinatorial optimization problems are expected to emerge in the internet of things (IoT) society because of improved global network connectivity. Thus, Ising machines have attracted much attention as a new computing technology for highperformance combinatorial optimization processing.
Kadowaki and Nishimori in 1998 proposed quantum annealing as an alternative to simulated annealing.^{13}^{)} They investigated the dynamic behavior of the Ising model with the timedependent transverse field using the Schrödinger equation. In the quantum annealing, we prepare the ground state of the Ising model with a large transverse field as the initial state. After that, we gradually decrease the transverse field to zero, and we observe the final state. When the transverse field is decreased with a sufficiently slow schedule so that adiabatic condition is satisfied, we obtain the ground state. Reference 13 investigates the difference between the dynamic behavior of simulated annealing and that of quantum annealing. The authors concluded that the quantum annealing could obtain the ground state with higher probability than the simulated annealing when we use the same scheduling function which is a faster schedule that satisfies the Geman–Geman theorem for the simulated annealing.^{14}^{)}
In the original version of quantum annealing proposed by Kadowaki and Nishimori,^{13}^{)} the Ising model is used as a classical Hamiltonian, while the timedependent transverse field is used as a quantum driver Hamiltonian. Quantum Monte Carlo simulation can be used to simulate the Ising model with transverse field by applying the Suzuki–Trotter decomposition,^{15}^{,}^{16}^{)} where the Ising model with transverse field (quantum Ising model) in ddimension is transformed to the Ising model without transverse field (classical Ising model) in
Before the birth of commercial quantum annealing machine, theoretical and numerical studies on the performance of quantum annealing were done. After quantum annealing machines are available, the comparisons of experimental quantum annealing performance and that of other methods using a digital computer have been made. Recently, Maezawa et al. proposed and developed a new type of quantum annealing machine.^{25}^{,}^{26}^{)} Additionally, several Ising machines with operating principles not based on quantum annealing have been developed.^{2}^{–}^{6}^{)}
Toward the practical usage of Ising machines, it is necessary to develop software for Ising machines and discover various applications. A highlevel programming language for the DWave machine based on Julia,^{27}^{)} Pythonbased software tools for annealing machines,^{28}^{)} and virtual hardware embedding suite for adiabatic quantum computing^{29}^{)} are examples of software developed recently.
In this paper, we review approaches to use Ising machines. Furthermore, we show our result of an application of Ising machines and software for Ising machines developed by one of the authors (K.T.). The organization of the paper is as follows. In Sect. 2, we briefly review combinatorial optimization problems. In Sect. 3, we explain the relation of the combinatorial optimization problem and the Ising model. In Sect. 4, we show the experimental result of online advertisement allocation optimization using DWave 2000Q. In Sect. 5, we introduce software called PyQUBO. In Sect. 6, we provide a summary of this paper.
Optimization problems pertain to finding conditions that minimize a given function, called cost function, under some given constraints. A mathematical expression of these problems are given by
We assume that equalities and inequalities represent the given constraints. Suppose that
Next, for the case when only an inequality constraint exists,
Herein, we concentrate on the case where the decision variables are integers, i.e.,
To perform combinatorial optimization using Ising machines, we should prepare the Hamiltonian of the Ising model or the QUBO which corresponds to the cost function of the combinatorial optimization problem. In this section, we explain methods to construct the Hamiltonian of the Ising model or the QUBO from the given combinatorial optimization problems. Before showing methods to represent the cost function of the combinatorial optimization problem using the Hamiltonian of the Ising model or the QUBO, we explain the relations between the Ising model and the QUBO. The Ising model and the QUBO are defined on an undirected graph. Consider an undirected graph
Let
From the above, when the QUBO represents the cost function, we can reformulate the combinatorial optimization problem using the Ising model. In the penalty function method (see Sect. 2), the equality/inequality constraints which are imposed in the original combinatorial optimization problem are included in another combinatorial optimization problem [see Eq. (4)]. Thus, if we represent the equality/inequality constraints in the original combinatorial optimization problem using only linear and quadratic terms of 0–1 binary variables, we can reformulate the combinatorial optimization problem using the Ising model or the QUBO. In principle, even if kbody (
Here, we explain how to express some types of combinatorial optimization problem using the Ising model or QUBO. According to the existence of equality/inequality constraints, we categorize combinatorial optimization problems into three types. References 7 and 8 provide more detailed information.
Here, we consider combinatorial optimization problems without constraints and represent them using the Ising model or QUBO. A typical combinatorial optimization problem without constraints is the number partitioning problem described as follows. Given a set of N positive integers,
Here, we consider combinatorial optimization problems with equality constraints and represent them using the Ising model or QUBO. In this section, we show two examples, i.e., the graph partitioning problem and traveling salesman problem. The graph partitioning problem has an equality constraint in the original form of the combinatorial optimization problem. Conversely, in the traveling salesman problem, equality constraints do not appear in the original form of the combinatorial optimization problem but are introduced when we use the QUBO expression.
The graph partitioning problem is described as follows. Consider an undirected graph
To represent the cost function using the Ising model, let
From the above discussion, we prepared the Hamiltonians representing the cost function and the equality constraint. As in the penalty function method [Eq. (4)], we introduce a new Hamiltonian as
Next, we consider the traveling salesman problem. The traveling salesman problem for a graph
We can express the cost function as
Here, we consider combinatorial optimization problems with inequality constraints and represent them using the Ising model or the QUBO. Here, we explain how to represent the knapsack problem using the QUBO, which has an inequality constraint in the original form of the combinatorial optimization problem.
The knapsack problem is described as follows. There are N items and the weight and price of the ith (
Next, we consider a method to represent the inequality constraint,
In the previous subsection, we explained standard methods to represent typical combinatorial optimization problems by the Ising model or the QUBO depending on whether equality/inequality constraints exist. At the stage of thinking about the representation of the combinatorial optimization problem using the Ising model or the QUBO, we do not take care of the graph structure of Ising machines. Let
In this section, we show the result of our study on the optimization of online advertisement allocation using a quantum annealing machine. In general, advertisements need to be served to appropriate users both from the viewpoints of the clients and users. In the case of online advertising, an indicator of the performance of serving online advertisement is the clickthrough rate (CTR) which is the rate at which displayed advertisements in websites are clicked. Here, the CTR fluctuates with time. It is necessary to increase the average value of CTR and to reduce the variance of CTR. The situation is similar to the portfolio optimization in economics. Thus, we construct a model based on the Markowitz meanvariance portfolio theory which is a prototypical problem in economics.
Before showing our study on online advertisement allocation optimization, we explain the Markowitz meanvariance portfolio model (Markowitz model).^{34}^{)} We assume that there are n financial assets in the financial market. Let
The above equation is the most basic form of the Markowitz model. Reflecting that the minimum unit of investment exists, however, a discretized version of the Markowitz model, where the nonnegative real variable
To use Ising machines, we have to represent the nonnegative integer variable
Based on the previous study,^{35}^{)} we constructed a model to optimize online advertisement allocation. The optimization of online advertisement allocation aims to find the matching of advertisements displayed to users so as to maximize the CTR and minimize the variance of the CTR. A smaller variance of the CTR is preferred because the future profit can be more predictable when the fluctuation of the profit keeps low. The optimization we consider is regarded as the matching problem on a bipartite graph (see Fig. 1).
Figure 1. Schematic picture of our model of online advertisement allocation optimization given by Eq. (38).
Here the number of advertisements is m. Let
The first term in Eq. (38) is the sum of the normalized expected CTR. The formulation with the normalized expected CTR
We investigated our model with artificial data reflecting practical situation. We denote the CTR which is the advertisement j to the user i at time step t as
Since the definition of CTR is a ratio of the number of clicks to the number of advertisement displays, CTR can be modeled using the binomial distribution
We compare the performance of DWave 2000Q and that of the greedy method with the artificially generated data as mentioned above. In our experiment, the following parameters are used. The number of advertisements is
Now, let us look at the optimization with a quantum annealing machine, DWave 2000Q. The quantum annealing machine used in our experiment does not have a complete chimera graph because of the qubit yield. Thus, we employed the virtual hardware graph solver named VFYC (Virtual Full Yield Chimera)^{36}^{)} which enables us to solve the problem on the abstract complete chimera graph by applying postprocessing to the solution obtained by the quantum annealing machine with missing qubits. We solve the QUBO of our problem with 64 fully connected logical spins realized by an embedding method for complete graphs.^{37}^{)} The coupling strength in the embedding is set to
When we obtain the solution with the optimization mentioned above method, we calculate the Test CTR at each time step by Eq. (41). Furthermore, the average and the standard deviation of the Test CTR for all time steps are calculated using Eqs. (42) and (43). We also sampled the parameters μ and Σ of the multivariate normal distribution from the Beta and inverse Wishart distribution respectively as explained before. The expected value of Avg. Test CTR, STDEV. Test CTR and the objective function of Eq. (38) were calculated using 5 sampled parameters.
We compared the dependencies of Avg. Test CTR and STDEV. Test CTR on the coefficient of the variance α with different optimization methods (Fig. 2). In Fig. 2(a), we observed that the Avg. Test CTR obtained by DWave 2000Q is larger than that obtained by the greedy method at
Figure 2. Dependencies of (a) the average, (b) the standard deviation and (c) the objective function of Eq. (38) on the variance coefficient α, using the greedy method and the quantum annealing performed using DWave 2000Q.
Next, let us look at the objective function
From the above results, we conclude that the profit of the online advertisement allocation and the standard deviation of the CTR is improved by the proposed method with a quantum annealing machine.
One of the authors (K.T.) developed software called PyQUBO that helps to abstract codes.^{30}^{)} The advantages of using PyQUBO are readability of the code which leads to the prevention of errors in the code, reduction of compilation time, automatic validation whether the given constraints are satisfied. In this section, we provide a short explanation of PyQUBO.
To utilize Ising machines, we need to represent combinatorial optimization problems using the Ising model or the QUBO as described in Sect. 3. Here, we consider the case where the cost function and the equality/inequality constraints of the combinatorial optimization problem are formulated in terms of polynomial functions with binary variables. If products of three or more binary variables exist in the polynomial function, we need to reduce the higherorder interactions by introducing auxiliary variables, as explained in Sect. 3. After the reduction of higherorder interactions, we obtain the QUBO matrix corresponding to the combinatorial optimization problem we want to solve.
Practically, we need to write a program to prepare the QUBO matrix corresponding to the combinatorial optimization problem. To obtain the QUBO matrix, we rewrite the Hamiltonian such that the linear and quadratic terms of the spin variables or 0–1 binary variables are separate. The code of a program written to achieve such program created by the above concept tends to be complex and unreadable. Let us see an example of this process for the number partitioning problem. As described in Sect. 3.1.1, the number partitioning problem aims to partition a set of given numbers into two groups such that the sums of the numbers in each group are equal. The Hamiltonian of this problem can be written by a format similar to Eq. (13).
To write a program for preparing the QUBO, we need to manually expand the Hamiltonian as

The code is unreadable since the original form of the Hamiltonian, i.e., before the expansion of the Hamiltonian, cannot be directly seen in the code. No clear correspondence between the Hamiltonian and its expression in the code exists. Thus, it may be difficult to fix the code if one wants to modify the Hamiltonian partially. Additionally, this program is susceptible to software bugs from miscalculations because the expansion of the Hamiltonian needs to be manually calculated when there are many equality/inequality constraints and the form of the cost function is complicated. The serious situation often occurs when we consider a complicated combinatorial optimization problem in the real world. To consider the complicated combinatorial optimization problem, we start with a simple Hamiltonian to obtain the first stage of a proofofprinciple result. Next, we modify the Hamiltonian to be acceptable for realworld applications based on the proofofprinciple result. Since the above strategy of program development has been adopted in many industrial products, we should write readable and extendable code. With PyQUBO, we can define a Hamiltonian with spin or binary objects in a more straightforward way. We demonstrate construction of the QUBO matrix for the number partitioning problem with PyQUBO (Table II).

Firstly, we define the array of spin variables (line 6) and the Hamiltonian of the problem (line 9). Then, we compile the Hamiltonian to get the model (line 12). Finally, the QUBO is generated by calling the to_qubo() method of the model (line 15). The code becomes much straightforward and readable than the conventional one in Table I, which leads to the extendable code.
The Hamiltonian in PyQUBO is internally represented as a tree. For example, the Hamiltonian for the number partitioning problem is represented as a tree shown in the left panel of Fig. 3. The compilation process of PyQUBO is shown in the right panel of Fig. 3. Firstly, the Hamiltonian is converted to a higherorder polynomial by folding the tree. Then, a higherorder polynomial is reduced to a secondorder polynomial by introducing auxiliary variables. Finally, the QUBO matrix is created from the coefficients of the secondorder polynomial. In the next section, some features of PyQUBO are introduced, including placeholder and automatic validation of constraints.
Figure 3. Tree representation of the Hamiltonian for the number partitioning problem in PyQUBO (left). Compilation process of PyQUBO (right).
To obtain better feasible solutions with high probability, we should update penalty coefficient parameters in the Hamiltonian. If we compile the QUBO every time we change the penalty coefficient, the large number of compilations is needed, and as a result, we spend much compilation time. However, using a placeholder can reduce the number of compilations. The actual value of the parameters defined by the placeholder can be specified after we compile the QUBO, which means that we can obtain QUBOs with various parameter values with only one compilation. Let us see an example of using the placeholder for the graph partitioning problem (Table III). For this case, we need to tune the coefficient λ in the Hamiltonian given by Eq. (14). We define the penalty coefficient λ by the placeholder (line 14) so that we can change the value even after compilation. The actual value of the penalty strength is specified when the to_qubo() method is called (line 27).

When we study combinatorial optimization problems with equality/inequality constraints using Ising machines, we should prepare additional functions in the code to judge whether the obtained results satisfy the given constraints. However, we do not need to write additional functions in PyQUBO. We can tell the compiler which terms in the Hamiltonian represent equality/inequality constraints using the Constraint class in PyQUBO. By just using Constraint, we can extract the information about unsatisfied constraints in the obtained results. In the example for the graph partitioning problem (Table III), the Hamiltonian for the problem is contained in a Constraint object (line 17). The information about the unsatisfied constraint is obtained when you decode the solution (line 29). If the broken object is empty, it means that all constraints are satisfied.
In the first half of this paper, we reviewed combinatorial optimization problems and how to express the combinatorial optimization problem using the Ising model or the QUBO formulation. In Sect. 4, we showed our experimental result for online advertisement allocation optimization using DWave 2000Q. We calculated the CTR that measures the benefit of online advertisement. We observed that the average value of CTR obtained using the greedy method is larger than that obtained using DWave 2000Q, and the variance of CTR obtained using DWave 2000Q is less than that obtained using the greedy method. Thus, we confirmed that quantum annealing outperformed the greedy method for our problem. In Sect. 5, a software developed, called PyQUBO, which conveniently helps to utilize Ising machines was explained.
Acknowledgment
One of the authors (S.T.) was supported by JST, PRESTO Grant Number JPMJPR1665, Japan and JSPS KAKENHI Grant Numbers 15K17720, 15H03699.
References
 1 M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, Nature 473, 194 (2011). 10.1038/nature10012 Crossref, Google Scholar
 2 M. Yamaoka, C. Yoshimura, M. Hayashi, T. Okuyama, H. Aoki, and H. Mizuno, IEEE J. SolidState Circuits 51, 303 (2016). 10.1109/JSSC.2015.2498601 Crossref, Google Scholar
 3 T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K. Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Takesue, Science 354, 603 (2016). 10.1126/science.aah4243 Crossref, Google Scholar
 4 P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, H. Mabuchi, and Y. Yamamoto, Science 354, 614 (2016). 10.1126/science.aah5178 Crossref, Google Scholar
 5 S. Matsubara, H. Tamura, M. Takasu, D. Yoo, B. Vatankhahghadim, H. Yamasaki, T. Miyazawa, S. Tsukamoto, Y. Watanabe, K. Takemoto, and A. Sheikholeslami, Conference on Complex, Intelligent, and Software Intensive Systems, 2017, 432. 10.1007/9783319615660_39 Crossref, Google Scholar
 6 Web [https://annealingcloud.com]. Google Scholar
 7 A. Lucas, Front. Phys. 2, 5 (2014). 10.3389/fphy.2014.00005 Crossref, Google Scholar
 8 S. Tanaka, R. Tamura, and B. K. Chakrabarti, Quantum Spin Glasses, Annealing and Computation (Cambridge University Press, Cambridge, U.K., 2017). Google Scholar
 9 V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Babbush, V. Smelyanskiy, J. Martinis, and H. Neven, Phys. Rev. X 6, 031015 (2016). 10.1103/PhysRevX.6.031015 Crossref, Google Scholar
 10 F. Neukart, G. Compostella, C. Seidel, D. von Dollen, S. Yarkoni, and B. Parney, Frontiers in ICT 4, 29 (2017). 10.3389/fict.2017.00029 Crossref, Google Scholar
 11 K. Terada, D. Oku, S. Kanamaru, S. Tanaka, M. Hayashi, M. Yamaoka, M. Yanagisawa, and N. Togawa, Proc. Int. Symp. VLSI Design, Automation and Test (VLSIDAT), 2018, 2018. 10.1109/VLSIDAT.2018.8373233 Crossref, Google Scholar
 12 K. Kitai, J. Guo, S. Ju, S. Tanaka, K. Tsuda, J. Shiomi, and R. Tamura, arXiv:1902.06573. Google Scholar
 13 T. Kadowaki and H. Nishimori, Phys. Rev. E 58, 5355 (1998). 10.1103/PhysRevE.58.5355 Crossref, Google Scholar
 14 S. Geman and D. Geman, IEEE Trans. Pattern Anal. Mach. Intell. PAMI6, 721 (1984). 10.1109/TPAMI.1984.4767596 Crossref, Google Scholar
 15 M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976). 10.1143/PTP.56.1454 Crossref, Google Scholar
 16 H. F. Trotter, Proc. Am. Math. Soc. 10, 545 (1959). 10.1090/S00029939195901087326 Crossref, Google Scholar
 17 T. Kadowaki, Ph. D. thesis, Tokyo Institute of Technology (1999). Google Scholar
 18 K. Tanaka and T. Horiguchi, Electron. Commun. Jpn. 83 [3], 84 (2000). 10.1002/(SICI)15206440(200003)83:3<84::AIDECJC9>3.0.CO%3B2N Crossref, Google Scholar
 19 R. Martoňák, G. E. Santoro, and E. Tosatti, Phys. Rev. E 70, 057701 (2004). 10.1103/PhysRevE.70.057701 Crossref, Google Scholar
 20 D. A. Battaglia, G. E. Santoro, and E. Tosatti, Phys. Rev. E 71, 066707 (2005). 10.1103/PhysRevE.71.066707 Crossref, Google Scholar
 21 K. Kurihara, S. Tanaka, and S. Miyashita, Proc. 25th Conf. Uncertainty in Artificial Intelligence (UAI2009), 2009. Google Scholar
 22 I. Sato, K. Kurihara, S. Tanaka, H. Nakagawa, and S. Miyashita, Proc. 25th Conf. Uncertainty in Artificial Intelligence (UAI2009), 2009. Google Scholar
 23 I. Sato, S. Tanaka, K. Kurihara, S. Miyashita, and H. Nakagawa, Neurocomputing 121, 523 (2013). 10.1016/j.neucom.2013.05.019 Crossref, Google Scholar
 24 M. Ohzeki, S. Okada, M. Terabe, and S. Taguchi, Sci. Rep. 8, 9950 (2018). 10.1038/s41598018282124 Crossref, Google Scholar
 25 M. Maezawa, K. Imafuku, M. Hidaka, H. Koike, and S. Kawabata, IEEE Conf. Proc. 16th Int. Superconductive Electronics Conf. (ISEC2017), 2017. Google Scholar
 26 M. Maezawa, G. Fujii, M. Hidaka, K. Imafuku, K. Kikuchi, H. Koike, K. Makise, S. Nagasawa, H. Nakagawa, M. Ukibe, and S. Kawabata, J. Phys. Soc. Jpn. 88, 061012 (2019). 10.7566/JPSJ.88.061012 Link, Google Scholar
 27 D. O’Malley and V. V. Vesselinov, IEEE High Performance Extreme Computing Conference (HPEC), 2016, 2016. 10.1109/HPEC.2016.7761616 Crossref, Google Scholar
 28 Web [https://docs.ocean.dwavesys.com/]. Google Scholar
 29 T. D. Goodrich, B. D. Sullivan, and T. S. Humble, Quantum Inf. Process. 17, 118 (2018). 10.1007/s1112801818634 Crossref, Google Scholar
 30 Web [https://github.com/recruitcommunications/pyqubo]. Google Scholar
 31 J. Cai, B. Macready, and A. Roy, arXiv:1406.2741. Google Scholar
 32 D. Oku, K. Terada, M. Hayashi, M. Yamaoka, S. Tanaka, and N. Togawa, submitted. Google Scholar
 33 N. Dattani and N. Chancellor, arXiv:1901.07676. Google Scholar
 34 H. Markowitz, J. Finance 7, 77 (1952). 10.1111/j.15406261.1952.tb01525.x Crossref, Google Scholar
 35 G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, K. Wu, and M. L. de Prado, IEEE J. Sel. Top. Signal Process. 10, 1053 (2016). 10.1109/JSTSP.2016.2574703 Crossref, Google Scholar
 36 Web [https://docs.dwavesys.com/docs/latest/c_postprocessing_5.html]. Google Scholar
 37 V. Choi, Quantum Inf. Process. 10, 343 (2011). 10.1007/s1112801002003 Crossref, Google Scholar
 38 S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Nat. Phys. 10, 218 (2014). 10.1038/nphys2900 Crossref, Google Scholar
 39 Web [https://docs.dwavesys.com/docs/latest/c_postprocessing_1.html]. Google Scholar
 40 Web [https://github.com/dwavesystems/dimod]. Google Scholar
Author Biographies
Kotaro Tanahashi received the M.Eng. from Kyoto University in 2015. He currently works for Recruit Communications Co., Ltd. as a machine learning engineer. He is also a project manager of MITOU Target Program at Informationtechnology Promotion Agency (IPA).
Shinichi Takayanagi received the M.Sc. from Hokkaido University in 2006. He is presently an data scientist at LINE Corporation. His research interests are quantum annealing, statistical mechanics, and machine learning. He is a member of the Physical Society of Japan and Information Processing Society of Japan.
Tomomitsu Motohashi received the M.Eng. from Sophia University in 2009. He is presently a CTO at digital medical startup Susmed Corporation. His research interests are quantum annealing, blockchain, operations research, and machine learning. He is a member of Information Processing Society of Japan. He was awarded the encouraging award in 2009 from The Institute of Systems, Control and Information Engineers of Japan and 2nd prise of KDD Cup 2015.
Shu Tanaka received the B.Sc. degree from Tokyo Institute of Technology in 2003 and the M.Sc. and Dr.Sc. degrees from The University of Tokyo in 2005 and 2008, respectively. He is presently an associate professor at Green Computing Systems Research Organization, Waseda University, a PRESTO researcher at Japan Science and Technology Agency (JST), a project manager of MITOU Target Program at Informationtechnology Promotion Agency (IPA). His research interests are quantum annealing, Ising machine, statistical mechanics, and materials science. He was awarded the 9th Award for the Encouragement of Young Physicists by the Physical Society of Japan in 2015. He is a member of Physical Society of Japan.
Cited by
ManyQudit Representation for the Travelling Salesman Problem Optimisation
114002, 10.7566/JPSJ.90.114002
Derivation of QUBO Formulations for Sparse Estimation
034801, 10.7566/JPSJ.89.034801
Deeper Understanding of Constrained Quantum Annealing from the Perspective of the Localization Phenomena
10, 10.7566/JPSJNC.17.10