  This paper presents a hybrid variational quantum algorithm that finds a
random eigenvector of a unitary matrix with a known quantum circuit. The
algorithm is based on the SWAP test on trial states generated by a parametrized
quantum circuit. The eigenvector is described by a compact set of classical
parameters that can be used to reproduce the found approximation to the
eigenstate on demand. This variational eigenvector finder can be adapted to
solve the generalized eigenvalue problem, to find the eigenvectors of normal
matrices and to perform quantum principal component analysis (QPCA) on unknown
input mixed states. These algorithms can all be run with low depth quantum
circuits, suitable for an efficient implementation on noisy intermediate state
quantum computers (NISQC) and, with some restrictions, on linear optical
systems. Limitations and potential applications are discussed.
