Quantum Eigenvalue Estimation for Irreducible Non-negative Matrices

Quantum phase estimation algorithm finds the ground state energy, the lowest eigenvalue, of a quantum Hamiltonian more efficiently than its classical counterparts. Furthermore, with different settings, the algorithm has been successfully adapted as a sub frame of many other algorithms applied to a wide variety of applications in different fields. However, the requirement of a good approximate eigenvector given as an input to the algorithm hinders the application of the algorithm to the problems where we do not have any prior knowledge about the eigenvector. This paper presents a modification to the phase estimation algorithm for the positive operators to determine the eigenvalue corresponding to the positive eigenvector without the necessity of the existence of an initial approximate eigenvector. Moreover, by this modification, we show that the success probability of the algorithm becomes to depend on the normalized absolute sum of the matrix elements of the unitary operator whose eigenvalue is being estimated. This provides a priori information to know the success probability of the algorithm beforehand and makes the algorithm output the right solution with high probability in many cases.