In this work we consider practical implementations of Kitaevs algorithm for quantum phase estimation. We analyze the use of phase shifts that simplify the estimation of successive bits in the estimation of unknown phase $varphi$. By using increasingly accurate shifts we reduce the number of measurements to the point where only a single measurements in needed for each additional bit. This results in an algorithm that can estimate $varphi$ to an accuracy of $2^{-(m+2)}$ with probability at least $1-epsilon$ using $N_{epsilon} + m$ measurements, where $N_{epsilon}$ is a constant that depends only on $epsilon$ and the particular sampling algorithm. We present different sampling algorithms and study the exact number of measurements needed through careful numerical evaluation, and provide theoretical bounds and numerical values for $N_{epsilon}$.
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