Phase estimation using an approximate eigenstate

Abstract

A basic building block of many quantum algorithms is the Phase Estimation algorithm (PEA). It estimates an eigenphase $ϕ$ of a unitary operator $U$ using a copy of the corresponding eigenstate $|ϕ\rangle$. Suppose, in place of $|ϕ\rangle$, we have a copy of an approximate eigenstate $|ψ\rangle$ whose overlap magnitude with $|ϕ\rangle$ is at least $\sqrt{2/3}$. Then PEA fails with a constant probability. However, using multiple copies of $|ψ\rangle$, 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 $|ψ\rangle$, a single copy is sufficient to estimate $ϕ$. 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 $Θ(\ln n)$ copies of the eigenstate. Using our algorithm, BKS algorithm can work using just a single copy of the eigenstate.