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arXiv:physics/9802021

Computing the real-time Green's Functions of large Hamiltonian matrices

T. Iitaka

Published Feb 11, 1998• 3 citations
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Abstract

A numerical method is developed for calculating the real time Green's functions of very large sparse Hamiltonian matrices, which exploits the numerical solution of the inhomogeneous time-dependent Schroedinger equation. The method has a clear-cut structure reflecting the most naive definition of the Green's functions, and is very suitable to parallel and vector supercomputers. The effectiveness of the method is illustrated by applying it to simple lattice models. An application of this method to condensed matter physics will be found in H. Tanaka, Phys. PRB 57, 2168 (1998).

DOI
10.48550/arxiv.physics/9802021

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Computation of the Green's function is crucial to study the properties of quantum many-body systems such as strongly correlated systems. Although the high-precision calculation of the Green's function is a notoriously challenging task on classical computers, the development of quantum computers may enable us to compute the Green's function with high accuracy even for classically-intractable large-scale systems. Here, we propose an efficient method to compute the real-time Green's function based on the local variational quantum compilation (LVQC) algorithm, which simulates the time evolution of a large-scale quantum system using a low-depth quantum circuit constructed through optimization on a smaller-size subsystem. Our method requires shallow quantum circuits to calculate the Green's function and can be utilized on both near-term noisy intermediate-scale and long-term fault-tolerant quantum computers depending on the computational resources we have. We perform a numerical simulation of the Green's function for the one- and two-dimensional Fermi-Hubbard model up to $4\ifmmode\times\else\texttimes\fi{}4$ sites lattice (32 qubits) and demonstrate the validity of our protocol compared to a standard method based on the Trotter decomposition. We finally present a detailed estimation of the gate count for the large-scale Fermi-Hubbard model, which also illustrates the advantage of our method over the Trotter decomposition.
The finite-element discrete variable representation is proposed to express the nonequilibrium Green's function for strongly inhomogeneous quantum systems. This method is highly favorable against a general basis approach with regard to numerical complexity, memory resources, and computation time. Its flexibility also allows for an accurate representation of spatially extended Hamiltonians and thus opens the way toward a direct solution of the two-time Schwinger-Keldysh-Kadanoff-Baym equations on spatial grids, including, for example, the description of highly excited states in atoms. As benchmarks, we compute and characterize, in Hartree-Fock and second Born approximations, the ground states of the He atom, the H2{\mathrm{H}}_{2}H2​ molecule, and the LiH molecule in one spatial dimension. Thereby, the ground-state and binding energies, densities, and bond lengths are compared with the direct solution of the time-dependent Schr\"odinger equation.
Part I: Green's Functions in Mathematical Physics - Time-Independent Green's Functions / Time-dependent Green's Functions / Part II: One-Body Green's Functions - Physical Significance of One-Body Green's Functions / Green's Functions and Perturbation Theory / Green's Functions for Tight-Binding Hamiltonians / Single Impurity Scattering / Extension Theory for Lattice Green's Functions / Part III: Many-Body Green's Functions - Field Operators and Three Pictures / Definition and Properties of Many-Body Green's Functions / The Diagram Technique for Zero-Temperature Green's Functions / Definition and Properties of Matsubara Green's Functions / Diagram Technique for the Matsubara Green's Functions / Three Approximation Schemes of the Diagram Technique / Linear Response Theory / The Equation of Motion Technique for the Green's Functions / Magnetic Systems Described by Heisenberg Model / The Green's Functions for Boson Systems with Condensation / Superconductors with Weak Interaction Between Electrons / Nonequilibrium Green's Functions / Electronic Transport through a Mesoscopic Structure / Appendixes / Index.
A fast and stable numerical method is formulated to compute the time evolution of a wave function in a magnetic field by solving the time-dependent Schroedinger equation. This computational method is based on the finite element method in real space to improved accuracy without any increase of computational cost. This method is also based on Suzuki's exponential product theory to afford an efficient way to manage the TD-Schroedinger equation with a vector potential. Applying this method to some simple electron dynamics, we have confirmed its efficiency and accuracy.