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

Abstract EN

The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector Va.

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 Va 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.