We are now approaching the “post-Moore era” because of the limitation of microfabrication technology for semiconductor devices. Quantum computers are expected to play an important role in next generation computer science. Quantum annealer is a kind of quantum computers, which is specialized to combinatorial optimization problems, and is now available to use. However, there are still various technical issues that have to be resolved. One of the technical issues is how to deal with the energy gap that becomes smaller with increasing the number of qubits. The energy gap between the ground state and the first excited state has to be reasonably large to perform stabilized quantum annealing. To overcome these technical issues, we have been developing a quantum annealing simulator “QUARTS” by the density matrix renormalization group method. QUARTS offers features that are useful to not only researchers but also hardware developers to develop quantum devices. QUARTS calculates various physical quantities on an arbitrary qubit system, and can perform a simulation of systems with a scale of 10,000 qubits on PCs by realistic memory usage and elapsed time. In addition, QUARTS can also perform efficient calculations of much larger systems using large-scale supercomputers such as supercomputer “Fugaku”. This project was supported by the IPA MITOU target program (quantum annealing division) in FY2018 and FY2019. “QUARTZ” is opened to the public.

For efficient development of quantum algorithms that can take advantage of a power of quantum computers, it is essential to develop a fast quantum circuit simulator that can evaluate the performance of quantum circuits on classical computers. In this study, we developed a quantum circuit simulator specialized for the evaluation of tensor network states constructed by a product of arbitrary two-qubit unitary gates. The simulator is implemented in Fortran 90 and runs in a hybrid-parallel computation with the OpenMP and MPI. For the gate operations in each MPI process, we refer to the 2-qubit gate operations in the OpenMP-adapted Qulacs. For the address management of the state vectors distributed among the processes, we refer to a reported study using the K computer. In our benchmark calculations for a system of 28 qubits on the RIKEN supercomputer "HOKUSAI GreatWave," we confirm that the simulation speed becomes up to 36 times faster than that with the single process. This research was supported by the IPA MITOU Target program (Gate-type Quantum Computer Division) in FY2018 and FY2019.

The density matrix renormalization group (DMRG) method is known as one of the most efficient numerical methods for one-dimensional quantum many-body systems. Although the degree of freedom of quantum many-body systems increases exponentially with the system size, the DMRG method can accurately describe the quantum states by an arbitrary number of bases and can treat larger systems that exact numerical methods cannot afford. On the other hand, the DMRG method for higher-dimensional systems requires quite large computational costs to keep its numerical accuracy because of the rapid increase of quantum entanglement. Nonetheless, recently developed large-scale supercomputers enable us to perform accurate DMRG calculations of two-dimensional systems that other numerical methods such as quantum Monte Carlo methods cannot treat because of the notorious negative sign problem. Thus, it is important to develop an efficient massively parallel DMRG algorithm. We have developed massively parallel DMRG codes that have achieved more than 70% (7.8 PFLOPS) for the theoretical performance of the K computer using all nodes (82,944 nodes). Our large-scale parallel DMRG codes are open to the public: "Dynamical DMRG" for quantum dynamics, "2-D DMRG" for higher dimensional systems, and "paraDMRG" for ab-initio calculations of strongly correlated molecules.

A quantum Monte Carlo (QMC) method is one of the most reliable and efficient techniques for strongly-interacting electrons. We have been engaged in developing a highly efficient QMC code based on the auxiliary field scheme for lattice fermion systems. Since numerical calculations involved in this formulation are mostly linear algebraic procedures such as matrix-matrix product and numerical orthogonalization, we can take full advantage of numerical libraries highly optimized for supercomputers such as K and Fugaku to calculate physical observables with a high degree of accuracy on unprecedentedly large systems. Typical target systems we aim at are of the order of 10,000 electrons and more in two or three dimensions unless the notorious negative-sign problem occurs. Our simulation code has achieved up to 80% of the theoretical performance per node of the K computer, in addition to the inherent suitability of the QMC method for parallel computation. For example, we have successfully performed QMC simulations for the Hubbard model on a honeycomb lattice, a canonical model for graphene, with up to 10,952 sites at half filling to reveal quantum criticality of a metal-insulator transition.

The search for a spin liquid phase, a novel state of matter that exhibits no long-range order despite having mutual interactions, is one of the most challenging subjects in modern condensed-matter physics. Conventionally, it is believed that not only the strong quantum effects but also geometrical frustration is necessary to realize a spin liquid phase. In 2010, a surprising and even controversial proposal was reported; it claims that the spin-liquid phase emerges in the Hubbard model on the non-frustrating honeycomb lattice as an intermediate phase between a semimetal (SM) and an antiferromagnetic Mott insulator (AFMI). We have examined this interesting possibility by performing large-scale QMC simulations and eventually concluded that the spin-liquid phase reported previously is unlikely.

Interacting Dirac electrons constitute an ideal platform to study interaction-driven metal-insulator transitions since it does not involve the nesting instability or the divergence of the density of states, both of which bring about a trivial phase transition even in a weak-coupling limit. We have investigated the quantum criticality of the phase transitions between a semimetal and an antiferromagnetic insulator in the Hubbard model on the honeycomb lattice and on the square lattice with π-flux. By performing large-scale QMC simulations, we have determined critical exponents for the transitions with a high degree of accuracy and found that these two models share the same critical exponents, which is strong numerical evidence for the existence of a universality class. This universality class is known as the chiral-Heisenberg universality class in terms of the Gross-Neveu model, the celebrated effective model in elementary particle physics.