The dynamics of Glauber-Fock lattice of size $N$ is given through exact diagonalization of the corresponding Hamiltonian; the spectra ${{\ensuremath{\lambda}}_{k}}$ is given as the roots of the $N$ th Hermite polynomial, ${H}_{N}({\ensuremath{\lambda}}_{k}/\sqrt{2})=0$, and the eigenstates are given in terms of Hermite polynomials evaluated at these roots. The exact dynamics is used to study coherent phenomena in discrete lattices. Due to the symmetry and spacing of the eigenvalues ${{\ensuremath{\lambda}}_{k}}$, oscillatory behavior is predicted with highly localized spectra, that is, near complete revivals of the photon number and partial recovery of the initial state at given waveguides.The dynamics of Glauber-Fock lattice of size $N$ is given through exact diagonalization of the corresponding Hamiltonian; the spectra ${{\ensuremath{\lambda}}_{k}}$ is given as the roots of the $N$th Hermite polynomial, ${H}_{N}({\ensuremath{\lambda}}_{k}/\sqrt{2})=0$, and the eigenstates are given in terms of Hermite polynomials evaluated at these roots. The exact dynamics is used to study coherent phenomena in discrete lattices. Due to the symmetry and spacing of the eigenvalues ${{\ensuremath{\lambda}}_{k}}$, oscillatory behavior is predicted with highly localized spectra, that is, near complete revivals of the photon number and partial recovery of the initial state at given waveguides.

arXiv:1108.3004Physical Review ANov 23, 201131 citations

Exact Diagonalization of Small Systems

X. Zotos

NATO ASI series. Series B : PhysicsJan 1, 19900 citations

Comparative study of matrix product state/quantized tensor-train algorithms for solving time-independent partial differential equations

Paula García-Molina, Luca Tagliacozzo, Juan José García‐Ripoll

This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to $2^{30}$ points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism (QTT). This study focuses on Hamiltonian equations, for which five algorithms are introduced: explicit imaginary-time evolution methods, steepest gradient descent in conventional and optimized forms, a power method, and an explicitly restarted Arnoldi method. The first five methods are engineered using a framework of limited-precision linear algebra, in which operators -- i.e., the equation itself -- and vectors are represented using matrix product operator (MPO) and matrix product state (MPS) formalisms, and where operator-vector multiplication and vector addition are approximated with limited resources. All methods are benchmarked using an exactly solvable PDE for a quantum harmonic oscillator in one and two dimensions over a regular grid with up to $2^{30}$ points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.

arXiv:2303.09430Mar 16, 20231 citation

On-the-fly exact diagonalization solver for quantum electronic models

Кашин Илья Владимирович, Vladimir V. Mazurenko

We propose a distinct numerical approach to effectively solve the problem of partial diagonalization of the super-large-scale quantum electronic Hamiltonian matrices. The key ingredients of our scheme are the new method for arranging the basis vectors in the computer's RAM and the algorithm allowing not to store a matrix in RAM, but to regenerate it on-the-fly during diagonalization procedure. This scheme was implemented in the program, solving the Anderson impurity model in the framework of dynamical mean-field theory (DMFT). The DMFT equations for electronic Hamiltonian with 18 effective orbitals that corresponds to the matrix with the dimension of 2.4 * 10^9 were solved on the distributed memory computational cluster.We propose a distinct numerical approach to effectively solve the problem of partial diagonalization of the super-large-scale quantum electronic Hamiltonian matrices. The key ingredients of our scheme are the new method for arranging the basis vectors in the computer's RAM and the algorithm allowing not to store a matrix in RAM, but to regenerate it on-the-fly during diagonalization procedure. This scheme was implemented in the program, solving the Anderson impurity model in the framework of dynamical mean-field theory (DMFT). The DMFT equations for electronic Hamiltonian with 18 effective orbitals that corresponds to the matrix with the dimension of 2.4 * 10^9 were solved on the distributed memory computational cluster.

arXiv:1508.04895Aug 20, 20150 citations

Diagonalization with matrix-product states

Chen Liu, Anders W. Sandvik

Bulletin of the American Physical SocietyMar 23, 20110 citations

The dynamics of Glauber-Fock lattice of size

N

is given through exact diagonalization of the corresponding Hamiltonian; the spectra ${{\ensuremath{\lambda}}_{k}}$ is given as the roots of the

N

th Hermite polynomial, ${H}_{N}({\ensuremath{\lambda}}_{k}/\sqrt{2})=0$, and the eigenstates are given in terms of Hermite polynomials evaluated at these roots. The exact dynamics is used to study coherent phenomena in discrete lattices. Due to the symmetry and spacing of the eigenvalues ${{\ensuremath{\lambda}}_{k}}$, oscillatory behavior is predicted with highly localized spectra, that is, near complete revivals of the photon number and partial recovery of the initial state at given waveguides.

2^{30}

points and compared with the density matrix renormalization group (DMRG) method. Our study reveals that all MPS-based techniques exponentially outperform exact diagonalization techniques based on vectors regarding memory usage. Imaginary-time algorithms are shown to underperform any gradient descent in terms of calibration needs and costs. Finally, MPS DMRG and interpolated Arnoldi-like asymptotically outperform all other methods, including state-of-the-art vector-based exact diagonalization, with significant advantages in time and memory use.