We study the ground state phase diagram of the twisted three-leg spin tube in magnetic fields by the density matrix renormalization group (DMRG) method. The twisted spin tube is composed of triangular unit cells and possesses strong quantum fluctuations under geometrical frustration. We apply the sine square deformation method to remove strong boundary effects and obtain smooth magnetization curves without steps of finite systems. With the analysis of the magnetization curves and correlation functions we determine the ground state phase diagram consisting of (a) a Tomonaga-Luttinger (TL) liquid characterized by spin-3 2 Heisenberg model, (b) 3-sublattice state named UUD with 1/3 magnetization and (c) TL-liquid of massless chirality with 1/3 magnetization plateau, (d) TL-liquid of massless spin mode with or without chirality quasi long-range order.

We present a relativistic formulation of the quantum Hall effect on Haldane sphere. An explicit form of the pseudopotential is derived for the relativistic quantum Hall effect with/without mass term. We clarify particular features of the relativistic quantum Hall states with the use of the exact diagonalization study of the pseudopotential Hamiltonian. Physical effects of the mass term to the relativistic quantum Hall states are investigated in detail. The mass term acts as an interpolating parameter between the relativistic and non-relativistic quantum Hall effects. It is pointed out that the mass term unevenly affects the many-body physics of the positive and negative Landau levels as a manifestation of the "parity anomaly". In particular, we explicitly demonstrate the instability of the Laughlin state of the positive first relativistic Landau level with the reduction of the charge gap.

We propose a new method for solving binary optimization problems under inequality constraints using a quantum annealer. To deal with inequality constraints, we often use slack variables, as in previous approaches. When we use slack variables, we usually conduct a binary expansion, which requires numerous physical qubits. Therefore, the problem of the current quantum annealer is limited to a small scale. In this study, we employ the alternating direction method of multipliers. This approach allows us to deal with various types using constraints in the current quantum annealer without slack variables. To test the performance of our algorithm, we use quadratic knapsack problems (QKPs). We compared the accuracy obtained by our method with a simulated annealer and the optimization and sampling mode of a D-Wave machine. As a result of our experiments, we found that the sampling mode shows the best accuracy. We also found that the computational time of our method is faster than that of the exact solver when we tackle various QKPs defined on dense graphs.

Steel manufacturing involves multiple processes, and optimizing the production schedule is essential for improving the quality, cost, and delivery of products. However, the corresponding optimization problems are usually NP-hard and generally intractable even for modern high-performance computers. Recent advances in quantum computers, such as those developed by D-Wave Systems, have opened new possibilities for handling this type of optimization problem. Nevertheless, a major obstacle to the application of currently available quantum computers is the relatively small number of quantum bits, which limits the feasible problem size. To overcome this obstacle, we propose an algorithm for solving optimization problems based on Lagrangian decomposition and coordination. D-Wave quantum computers can explore diverse solutions simultaneously, and we leverage this unique characteristic to obtain the upper bound for Lagrangian decomposition and coordination. We apply the proposed algorithm to a simplified problem of two-process production scheduling. By decomposing the problem into two stages for the processes, the number of steel products that can be handled increases from five to eight (60% increase). The optimal solutions are reached even for the extended case of the eight products. The proposed algorithm is a promising technique for enabling the application of quantum computers to real-world problems by thoroughly exploiting quantum bits. KEY WORDS: steel manufacturing; multiple-process scheduling; Lagrangian decomposition and coordination; quantum annealing, and quantum-digital hybrid algorithm. 300 tons, whereas the volume downstream when reaching surface treatment is approximately 10 tons. Therefore, a production unit in an upstream process includes different products or semi-products across downstream processes. Between all the process layers, the grouping of different sequence orders or semi-products should be jointly optimized. Such optimization problems must be solved under several constraints. For instance, two products or semiproducts requiring very different processes or different specifications, such as chemical components, widths, and thicknesses, cannot be grouped. On the other hand, grouping products obtained from similar processing conditions is favorable for quality, productivity, and cost. Nevertheless, excessive grouping may lead to stock accumulation and delay because grouping implies waiting for other products

Fractional quantum Hall (FQH) effects in graphene are studied because of their relativistic characteristics and the valley degree of freedom. Recently FQH effects have been observed at various filling factors with graphene on a hexagonal boron nitride (h-BN) substrate. However, it is known that h-BN creates the mass term in the Dirac Hamiltonian that acts as the effective model of graphene. To understand recent experiments, we shall investigate many-body effects in the massive Dirac electron system. In this paper, we study the mass-term effects on the FQH states of Dirac electrons by exact diagonalization. We examine the ground state at filling factor 1/3 in the n = ±1 Landau level. Without the mass term, the ground state in the Laughlin state featuring valley degeneracy and the lowest excitation is characterized by the valley unpolarized state (known as the valley skyrmion state). Conversely, we find that the mass-term lifts the valley degeneracy due to the breaking of the inversion symmetry. We also demonstrate that the valley unpolarized excitation is suppressed and that the fully or partially polarized state appears in the lowest excitation by increasing the mass term. Finally, we discuss the stability of FQH states in the massive Dirac Hamiltonian in experimental situations. We find that our numerical results are in agreement with previous experimental results.