  Quantum phase estimation 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.
  In this paper, we show that the principal eigenvalue of an irreducible
non-negative operator can be determined by using an equal superposition initial
state in the phase estimation algorithm. This removes the necessity of the
existence of an initial good approximate eigenvector. Moreover, we show that
the success probability of the algorithm is related to the closeness of the
operator to a stochastic matrix. Therefore, we draw an estimate for the success
probability by using the variance of the column sums of the operator. This
provides a priori information which can be used to know the success probability
of the algorithm beforehand for the non-negative matrices and apply the
algorithm only in cases when the estimated probability reasonably high.
Finally, we discuss the possible applications and show the results for random
symmetric matrices and 3-local Hamiltonians with non-negative off-diagonal
elements.
