High-Performance Quantum Simulation for Coupled Josephson Junctions on the Earth Simulator: a Challenge To the Schrödinger Equation On 256 4 Grids

Abstract

In order to explore quantum dynamics of coupled Josephson junctions, we develop a program solving directly the time-dependent Schrödinger equation by diagonalizing the Hamiltonian matrix and obtaining its ground and multiple low-lying excitation states. The Schrödinger equation is defined on mⁿ grids, in which m is the number of grid points discretized on a characteristic phase space of each junction and n is the number of coupled junctions. In this paper, the calculated maximum system is that m = 256 and n = 4, i.e. the number of degrees of freedom reaches 256⁴ (=4,294,967,296). We examine possible effective numerical schemes and make a parallel tuning to optimize the communication on the Earth Simulator. We sustain floating-point operation performance exceeding 20% of the peak on 512 nodes (4,096 PEs). From systematic calculations, we find a new concept that “quantum-assisted synchronization” occurs with downsizing the junction plane. This is a discovery adding a quantum flavor to the classical concept “synchronization”.

Keywords

coupled Josephson junctions eigenvalue problem the Earth Simulator conjugate gradient method pre-conditioner convergence property

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