Elementary particles have no internal structures. They are not composed of other particles. At present six flavors of quarks and six leptons are known as the elementary particles in the universe. They interact via four fundamental interactions or forces: gravitation, electromagnetism, the weak interaction and the strong interaction. Among them the strong interaction, which acts only between the quarks through exchanges of the gluons, shows peculiar features depending on the energy scale. In the high energy region, where the quarks come close to each other, the interaction strength becomes getting weaker (asymptotic freedom) so that the quarks are allowed to move more freely. On the other hand, the quarks are "confined" in the hadrons at the low energy scale. They are never observed individually. Once the energy scale becomes even lower, the protons and the neutrons, which are members of the hadrons, constitute the nuclei in the atoms. The purpose of the lattice QCD calculation is to prove that QCD is the fundamental theory of the strong interaction and investigate its dynamics nonperturbatively based on the first principles.

Large scale lattice QCD simulations with the Monte Carlo method enable us to make quantitative predictions for various physical quantities after controlling both the statistical and the systematic errors. The former is monotonically diminished by increasing the statistics. Troublesome is the latter, which is never removed without simulating the dynamical up (u), down (d), strange (s) quarks at their physical values on sufficiently large spatial volume. This is a very demanding calculation against which the lattice community has struggled long time. The aim of our project is to perform the calculation satisfying the above condition.

The lattice QCD simulations generate a set of the gluon configurations as the vacuum of nature, with which we can investigate various physical quantities. Our first target is to establish QCD as the fundamental theory of the strong interaction by reproducing basic physical quantities, e.g., the hadron spectrum. A second step is to investigate the nuclear force acting between the nucleons, which has awaited a proper treatment based on the first principles since Yukawa proposed the meson theory in 1934. Although the lattice QCD calculations have concentrated on the one-body properties of the hadrons partly due to the computational cost, the direct construction of the nuclei on the lattice based on the dynamics of the quarks and the gluons allow us to embark on a new era of multiscale physics with lattice QCD. We know that it is the multiscale physics where the computational science should explore and show its potential ability. Another important application of lattice QCD is the contributions to a new physics search and the examination of the standard model. Typical examples are evaluation of the hadron matrix elements required for the prediction of the proton lifetime and the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements in the Weinberg-Salam theory.

One of the biggest issues in the lattice QCD simulations is the computational cost which quickly increases as the quark mass decreases. Over last 30 years this problem has prevented us from a direct simulation at the physical u, d, s quark masses: We have been forced to make simulations at unphysical u, d quark masses which are degenerate and typically ten times as heavy as the physical values so that the simulation results are extrapolated to the physical point. With the use of the PACS-CS (Parallel Array Computer System for Computational Sciences) computer with a peak speed of 14.3 Tflops developed at University of Tsukuba, we have succeeded in reducing the ud quark mass to the physical point by the algorithmic improvements and shown how important it is from a view point of physics.

The right figure illustrate how the computational cost is reduced by the algorithmic improvements in the last decade. Black line expresses an expected computational cost of the plain HMC (Hybrid Monte Carlo) algorithm which was widely used as of 2001. The pi-to-rho meson mass ratio in the horizontal axis is defined from zero to one as a strictly increasing function of the ud quark mass. The cost seems to almost diverge as the ratio approaches the physical point. This expectation tells us that we definitely need not only the increase of the computational power but also the algorithmic improvements. Years later the difficulty is overcome by the DDHMC (Domain-Decomposed Hybrid Monte Carlo) algorithm. Blue circles denote the measured computational cost in our simulations, which clearly shows that the direct simulation at the physical point is possible with the current computational resources.

Although the physical point simulation is an inevitable ingredient under the name of the first principle calculation with lattice QCD, it is also important to understand why it is necessary in a practical sense. The lower left figure plots the pion mass squared divided by the degenerate ud quark mass. The PACS-CS results (red circles) show a clear curvature near the physical point, while the previous CP-PACS results look almost linear in the heavier quark mass region. This implies a possibility that the extrapolation method may give the erroneous estimations at the physical point. On the theoretical side the chiral perturbation theory predicts the logarithmic quark mass dependence near the chiral limit, which is consistent with the curvature observed in the PACS-CS results. However, the point is that it is practically very difficult to follow the logarithmic quark mass dependence precisely so that we estimate the "correct" value at the physical point by the extrapolation. We can avoid any possible systematic errors associated with the chiral extrapolations by the physical point simulations. The lower right compares our results for the hadron spectrum with the experimental values. We find that most of them are consistent within the error bars, though some cases show 2 - 3% deviations at most.

Once the physical point simulations are realized, the first task should be the determination of the quark masses which are free parameters in QCD. The gauge configurations become available for measurement of various physical quantities. There are a couple of physical quantities which have special interest and importance. η' meson mass and ρ-ππ decay are representative cases.　These are completely governed by the dynamics of the strong interaction and await the nonperturbative investigation with lattice QCD. The next steps should be directed to the precision measurement and the multiscale physics. The former demand us to incorporate the isospin breaking effects and the electromagnetic interactions. The precise determination of the quark masses is never achieved without taking account of these effects. The latter means the investigation of the interactions between the nuclei based on the dynamics of the quarks and the gluons using lattice QCD. Note that the precision measurement and the multiscale physics are not independent issues. For example, the deuteron binding energy is about 2 MeV, which is roughly comparable to the charged-neutral pion and the proton-neutron mass differences. A quantitative calculation of the macroscopic physical quantities necessarily requires to incorporate the isospin breaking effects and the electromagnetic interactions.