Quantum principal component analysis Seth Lloyd12*,Masoud Mohseni3 and Patrick Rebentrost2 The usual way to reveal properties of an unknown quantum state,given many copies of a system in that state,is to perform measurements of different observables and to analyse the results statistically12.For non-sparse but low-rank quantum states,revealing eigenvectors and corresponding eigenvalues in classical form scales super-linearly with the system dimension3-6.Here we show that multiple copies of a quantum system with density matrix o can be used to construct the unitary transformation e-it.As a result,one can perform quantum principal component analysis of an unknown low-rank density matrix,revealing in quantum form the eigenvectors corresponding to the large eigenvalues in time exponentially faster than any existing algorithm.We discuss applications to data analysis,process tomography and state discrimination