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

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. 1. Introduction Calculation of eigenvalues and eigenvectors of the Hamiltonian operator is one of the most frequent problems of physics. Most of the classical algorithms require an exponential amount of time for this. However, not all eigenvalues are of physical interest, but only few lower lying ones. Abrams and Lloyd’s quantum algorithm [1], considered to be the most important [2] quantum algorithm known so far, computes an eigenvalue of a unitary operator, or specifically the time evolution operator , in polynomial time. The eigenvalues of unitary operator lie around the unit circle on complex plane. Abrams-Lloyd algorithm starts with an approximation to the eigenvector. Normally, it is the result of a classical calculation. The algorithm needs two quantum registers. The first, an index register, is prepared in a uniform superposition of all the computational basis states. The second, a target register, is prepared with approximate eigenvector. After the application of Abrams-Lloyd algorithm, the eigenvalue is calculated from the measurement on the index register. The corresponding eigenstate is in the target register but is not measured. There are three disadvantages with the approximation. First, if the component of an eigenstate