Iterative approach for the eigenvalue problems

Datta, Jayita; Bera, P.

doi:10.1007/s12043-011-0118-z

Cited by 2 publications

(1 citation statement)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It still does not solve the problem of repeated eigenvalues. An iterative approximation method using perturbation series for the eigenvectors and the eigenvalues is presented in [4], but it is not a quantum algorithm. A recursive quantum algorithm by Bang et al [5] finds the lowest eigenstate of a general Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm

Warrier

2013

Journal of Computational Methods in Physics

View full text Add to dashboard Cite

The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector . The eigenstate is a basis vector in the orthonormal eigenspace. Finding another eigenvalue, using a random approximate eigenvector, may require many trials as the trial may repeatedly result in the eigenvalue measured earlier. We present a method involving orthogonalization of the eigenstate obtained in a trial. It is used as the for the next trial. Because of the orthogonal construction, Abrams-Lloyd algorithm will not repeat the eigenvalue measured earlier. Thus, all the eigenvalues are obtained in sequence without repetitions. An operator that anticommutes with a unitary operator orthogonalizes the eigenvectors of the unitary. We implemented the method on the programming language model of quantum computation and tested it on a unitary matrix representing the time evolution operator of a small spin chain. All the eigenvalues of the operator were obtained sequentially. Another use of the first eigenvector from Abrams-Lloyd algorithm is preparing a state that is the uniform superposition of all the eigenvectors. This is possible by nonorthogonalizing the first eigenvector in all dimensions and then applying the Abrams-Lloyd algorithm steps stopping short of the last measurement.

show abstract

Section: Introductionmentioning

confidence: 99%

Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm

Warrier

2013

Journal of Computational Methods in Physics

View full text Add to dashboard Cite

show abstract

Supersymmetric Expansion Algorithm and Complete Analytical Solution for the Hulthén and Anharmonic Potentials

Napsuciale,

Rodríguez,

Kirchbach

2024

Progress of Theoretical and Experimental Physics

View full text Add to dashboard Cite

An algorithm for providing analytical solutions to Schrödinger’s equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the supersymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian H0 is obtained from the nodeless states of the Hamiltonian H0 and of a set of supersymmetric partners H1, H2, …, Hr. The nodeless states (dubbed “edge” states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite system of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as “Supersymmetric Expansion Algorithm” step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulthén–, and the one-dimensional anharmonic oscillator potentials.

show abstract

Iterative approach for the eigenvalue problems

Cited by 2 publications

References 23 publications

Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm

Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm

Supersymmetric Expansion Algorithm and Complete Analytical Solution for the Hulthén and Anharmonic Potentials

Contact Info

Product

Resources

About