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O(N) algorithm for linear response functions of non-interacting electrons

Toshiaki Iitaka, 敏晃 飯高

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

An O(N) algorithm is proposed for calculating linear response functions of non-interacting electrons. This program is simple and suitable to parallel- and vector- computation. Since it avoids O(N 3) computational effort of matrix diagonalization, it requires only O(N) computational efforts where N is the dimension of the statevector. The use of this O(N) algorithm is very effective since otherwise we have to calculate large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang (L.W. Wang, Phys. Rev. B 49, 10154 (1994);L.W. Wang, Phys. Rev. Lett. 73, 1039 (1994) ) is that our method can calculate linear response functions without any storage of huge statevectors on external storage while their method should store many statevectors. 1

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An O(N) algorithm is proposed for calculating linear response functions of non-interacting electrons. This program is simple and suitable to parallel- and vector- computation. Since it avoids O(N 3) computational effort of matrix diagonalization, it requires only O(N) computational efforts where N is the dimension of the statevector. The use of this O(N) algorithm is very effective since otherwise we have to calculate large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang (L.W. Wang, Phys. Rev. B 49, 10154 (1994);L.W. Wang, Phys. Rev. Lett. 73, 1039 (1994) ) is that our method can calculate linear response functions without any storage of huge statevectors on external storage while their method should store many statevectors. 1
An O(N)O(N)O(N) algorithm is proposed for calculating linear response functions of noninteracting electrons. This algorithm is simple and suitable to parallel and vector computation. Since it avoids O(N3){O(N}^{3})O(N3) computational effort of matrix diagonalization, it requires only O(N)O(N)O(N) computational efforts, where NNN is the dimension of the state vector. The use of this O(N)O(N)O(N) algorithm is very effective since, otherwise, we have to calculate a large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang and Zunger [L. W. Wang, Phys. Rev. B 49, 10 154 (1994); L. W. Wang and A. Zunger, Phys. Rev. Lett. 73, 1039 (1994)] is that our method can calculate linear response functions without any storage of huge state vectors on external storage.
We introduce an efficient and numerically stable method for calculating linear response functions chi(q,omega) of quantum systems at finite temperatures. The method is a combination of numerical solution of the time-dependent Schrödinger equation, random vector representation of trace, and Chebyshev polynomial expansion of Boltzmann operator. This method should be very useful for a wide range of strongly correlated quantum systems at finite temperatures. We present an application to the ESR spectrum of s=1 / 2 antiferromagnet Cu benzoate.
Accurate modeling of the response of molecular systems to an external electromagnetic field is challenging on classical computers, especially in the regime of strong electronic correlation. In this paper, we develop a quantum linear response (qLR) theory to calculate molecular response properties on near-term quantum computers. Inspired by the recently developed variants of the quantum counterpart of equation of motion (qEOM) theory, the qLR formalism employs "killer condition" satisfying excitation operator manifolds that offers a number of theoretical advantages along with reduced quantum resource requirements. We also used the qEOM framework in this work to calculate state-specific response properties. Further, through noise-less quantum simulations, we show that response properties calculated using the qLR approach are more accurate than the ones obtained from the classical coupled-cluster based linear response models due to the improved quality of the ground-state wavefunction obtained using the ADAPT-VQE algorithm.
Accurate modeling of the response of molecular systems to an external electromagnetic field is challenging on classical computers, especially in the regime of strong electronic correlation. In this article, we develop a quantum linear response (qLR) theory to calculate molecular response properties on near-term quantum computers. Inspired by the recently developed variants of the quantum counterpart of equation of motion (qEOM) theory, the qLR formalism employs "killer condition" satisfying excitation operator manifolds that offer a number of theoretical advantages along with reduced quantum resource requirements. We also used the qEOM framework in this work to calculate the state-specific response properties. Further, through noiseless quantum simulations, we show that response properties calculated using the qLR approach are more accurate than the ones obtained from the classical coupled-cluster-based linear response models due to the improved quality of the ground-state wave function obtained using the ADAPT-VQE algorithm.
Response functions are a key quantity to describe the near-equilibrium dynamics of strongly interacting many-body systems. Recent techniques that attempt to overcome the challenges of calculating these ab initio have employed expansions in terms of orthogonal polynomials. We employ a neural network prediction algorithm to reconstruct a response function $S(\ensuremath{\omega})$ defined over a range in frequencies $\ensuremath{\omega}$. We represent the calculated response function as a truncated Chebyshev series whose coefficients can be optimized to reduce the representation error. We compare the quality of response functions obtained using coefficients calculated using a neural network (NN) algorithm with those computed using the Gaussian integral transform (GIT) method. In the regime where only a small number of terms in the Chebyshev series are retained, we find that the NN scheme outperforms the GIT method.
We present and test a new algorithm for time-evolving quantum many-body systems initially proposed by Holzner et al. [Phys. Rev. B 83, 195115 (2011)]. The approach is based on merging the matrix product state (MPS) formalism with the method of expanding the time-evolution operator in Chebyshev polynomials. We calculate time-dependent observables of a system of hardcore bosons quenched under the Bose-Hubbard Hamiltonian on a one-dimensional lattice. We compare the new algorithm to more standard methods using the MPS architecture. We find that the Chebyshev method gives numerically exact results for small times. However, the reachable times are smaller than the ones obtained with the other state-of-the-art methods. We further extend the new method using a spectral-decomposition-based projective scheme that utilizes an effective bandwidth significantly smaller than the full bandwidth, leading to longer evolution times than the nonprojective method and more efficient information storage, data compression, and less computational effort.