Phase estimation using an approximate eigenstate
  A basic building block of many quantum algorithms is the Phase Estimation algorithm (PEA). It estimates an eigenphase $\phi$ of a unitary operator $U$ using a copy of the corresponding eigenstate $|\phi\rangle$. Suppose, in place of $|\phi\rangle$, we have a copy of an approximate eigenstate $|\psi\rangle$ whose overlap magnitude with $|\phi\rangle$ is at least $\sqrt{2/3}$. Then PEA fails with a constant probability. However, using multiple copies of $|\psi\rangle$, the failure probaility can be made to decrease exponentially with the number of copies. In this paper, we show that as long as we can perform a selective inversion of $|\psi\rangle$, a single copy is sufficient to estimate $\phi$. An important application is to improve the spatial complexity of eigenpath traversal algorithm, a "digital" analogue of quantum adiabatic evolution, having applications ranging from quantum physics simulation to optimization. Here the goal is to travel a path of eigenstates of $n$ different unitary operators satisfying some conditions. The fastest algorithm is due to Boixo, Knill and Somma (BKS) which needs $\Theta(\ln n)$ copies of the eigenstate. Using our algorithm, BKS algorithm can work using just a single copy of the eigenstate. 
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