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THE ROLE OF DIAGONALIZATION WITHIN A DIAGONALIZATION/MONTE CARLO SCHEME

Dean Lee

We discuss a method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach which combines both diagonalization and Monte Carlo techniques.

arXiv:hep-lat/0010095International Journal of Modern Physics ASep 1, 20011 citation

Quasi-sparse eigenvector diagonalization and stochastic error correction

Dean Lee

We briefly review the diagonalization of quantum Hamiltonians using the quasisparse eigenvector (QSE) method. We also introduce the technique of stochastic error correction, which systematically removes the truncation error of the QSE result by stochastically sampling the contribution of the remaining basis states. 1 Quasi-sparse eigenvector method Quasi-sparse eigenvector (QSE) diagonalization is a new computational method which finds approximate low-lying eigenvalues and eigenvectors for a general quantum field Hamiltonian H [1]. It handles the exponential increase in the dimension of Fock space by exploiting the sparsity of the Hamiltonian. The method is most effective when the splitting between low-lying eigenvalues is not too small compared to the size of the Hamiltonian off-diagonal entries. In such cases the low-lying eigenvectors are quasi-sparse, meaning that the vector is dominated by a small fraction of its largest components. The QSE algorithm is applied using the following steps: (1) Select a subset of orthonormal basis vectors {ei1, · · · , ein} and call theWe briefly review the diagonalization of quantum Hamiltonians using the quasisparse eigenvector (QSE) method. We also introduce the technique of stochastic error correction, which systematically removes the truncation error of the QSE result by stochastically sampling the contribution of the remaining basis states. 1 Quasi-sparse eigenvector method Quasi-sparse eigenvector (QSE) diagonalization is a new computational method which finds approximate low-lying eigenvalues and eigenvectors for a general quantum field Hamiltonian H [1]. It handles the exponential increase in the dimension of Fock space by exploiting the sparsity of the Hamiltonian. The method is most effective when the splitting between low-lying eigenvalues is not too small compared to the size of the Hamiltonian off-diagonal entries. In such cases the low-lying eigenvectors are quasi-sparse, meaning that the vector is dominated by a small fraction of its largest components. The QSE algorithm is applied using the following steps: (1) Select a subset of orthonormal basis vectors {ei1, · · · , ein} and call the

Jan 1, 20001 citation

The Variational Quantum Eigensolver (VQE): A Quantum-Classical Hybrid Approach for Computational Physics and Quantum Simulation.

Anand Thipperudra, Randeep Singh, Shambhulinga Aralekallu

Jan 1, 20250 citations

Variational Quantum Eigensolver: How to Use Any Quantum Device in Your Lab to Perform Quantum Simulation

Jarrod R. McClean

Bulletin of the American Physical SocietyMar 4, 20152 citations

Non-perturbative Methods in Modal Field Theory

Nathan Salwen

Several issues in the modal approach to quantum field theory are discussed. Within the formalism of spherical field theory, differential renormalization is presented and shown to result in a finite number of renormalization parameters. Computations of the massless Thirring model in 1+1 dimensions are presented using this approach. Diagonalization techniques in periodic field theory are demonstrated. Issues of very large Hilbert spaces are considered and several approaches are presented. The quasi sparse eigenvector (QSE) approach takes advantage of the relatively small number of basis states that typically contribute significantly to any particular eigenvector. Stochastic correction methods use Monte Carlo calculations to calculate higher order corrections to the quasi sparse result. The quasi sparse eigenvector method and stochastic error correction are applied to the Hubbard model. With U/t=4, the shift in the ground energy below the U=0 value is found to within 1% for the 8x8 Hubbard model with 25/64 filling.Several issues in the modal approach to quantum field theory are discussed. Within the formalism of spherical field theory, differential renormalization is presented and shown to result in a finite number of renormalization parameters. Computations of the massless Thirring model in 1+1 dimensions are presented using this approach. Diagonalization techniques in periodic field theory are demonstrated. Issues of very large Hilbert spaces are considered and several approaches are presented. The quasi sparse eigenvector (QSE) approach takes advantage of the relatively small number of basis states that typically contribute significantly to any particular eigenvector. Stochastic correction methods use Monte Carlo calculations to calculate higher order corrections to the quasi sparse result. The quasi sparse eigenvector method and stochastic error correction are applied to the Hubbard model. With U/t=4, the shift in the ground energy below the U=0 value is found to within 1% for the 8x8 Hubbard model with 25/64 filling.

arXiv:hep-lat/0212035Dec 19, 20024 citations

The diagonalization of quantum field Hamiltonians

Dean Lee, Nathan Salwen, Daniel Lee

arXiv:hep-th/0002251Physics Letters BMar 1, 200145 citations

TUTORIAL ON QUASI-SPARSE EIGENVECTOR DIAGONALIZATION

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TUTORIAL ON QUASI-SPARSE EIGENVECTOR DIAGONALIZATION

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