In this work we develop EPOCH (equilibrium propagator by orthogonal polynomial chain), a computationally efficient method to calculate the time-dependent equilibrium Green's functions, including the anomalous Green's functions of superconductors, to capture the time evolution in large inhomogeneous systems. The EPOCH method generalizes the Chebyshev wave-packet propagation method from quantum chemistry and efficiently incorporates the Fermi-Dirac statistics that is needed for equilibrium quantum condensed matter systems. The computational cost of EPOCH scales only linearly in the system degrees of freedom, generating an extremely efficient algorithm also for very large systems. We demonstrate the power of the EPOCH method by calculating the time evolution of an excitation near a superconductor--normal metal interface in two and three dimensions, capturing transmission as well as normal and Andreev reflections.
We propose a highly efficient numerical method to describe inhomogeneous superconductivity by using the kernel polynomial method in order to calculate the Green's functions of a superconductor. Broken translational invariance of any type (impurities, surfaces, or magnetic fields) can be easily incorporated. We show that limitations due to system size can be easily circumvented and therefore this method opens the way for the study of scenarios and/or geometries that were unaccessible before. The proposed method is highly efficient and amenable to large scale parallel computation. Although we only use it in the context of superconductivity, it is applicable to other inhomogeneous mean-field theories.
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.
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).
In the non-equilibrium Green function calculations, the use of the generalized Kadanoff–Baym ansatz (GKBA) allows for a simple approximate reconstruction of the two-time Green function from its time-diagonal value. With this, a drastic reduction of the computational needs is achieved in time-dependent calculations, making longer time propagation possible and more complex systems accessible. This paper gives credit to the GKBA that was introduced 25 years ago. After a detailed derivation of the GKBA, we recall its application to homogeneous systems and show how to extend it to strongly correlated, inhomogeneous systems. As a proof of concept, we present the results for a two-electron quantum well, where the correct treatment of the correlated electron dynamics is crucial for a correct description of the equilibrium and dynamic properties.
We propose the ultra-fast numerical approach to large-scale inhomogeneous superconductors, which we call the Localized Krylov Bogoliubov-de Gennes method (LK-BdG). In the LK-BdG method, the computational complexity of the local Green's function, which is used to calculate the local density of states and the mean-fields, does not depend on the system size
N. The calculation cost of self-consistent calculations is
O(N), which enables us to open a new avenue for treating extremely large systems with millions of lattice sites. To show the power of the LK-BdG method, we demonstrate a self-consistent calculation on the 143806-site Penrose quasicrystal lattice with a vortex and a calculation on 1016064-site two-dimensional nearest-neighbor square-lattice tight-binding model with many vortices. We also demonstrate that it takes less than 30 seconds with one CPU core to calculate the local density of states with whole energy range in 100-millions-site tight-binding model.
A majority of measurable dynamic quantities of a physical system is described as an equilibrium correlation function of the form
GAB(t−t′)=⟨A(t)B(t′)⟩, where
t′ represents the impact time and
t represents the response time. Conventional quantum algorithms to compute such quantities via Hamiltonian simulation such as the Hadamard test, variational methods, or linear-response-based algorithms share one feature in common: each time point
t−t′ is calculated by a different quantum circuit, leading to at least
O(Nt) total shots, and at least
O(Nt2) total quantum gates. Here, we provide a different approach: utilizing
O(logNt) ancilla qubits to store time as a quantum variable, and effectively parallelize the computation of an equilibrium correlation function by requiring only a single quantum circuit for all time points. We show that this circuit needs to be run only
O(logNt) times, which in total requires
O(NtlogNt) quantum resources in the large system size limits. We prove that this scaling is optimal, and demonstrate our algorithm calculating the Green's function of a one-dimensional spinless Hubbard model.