  Quantum phase estimation is one of the fundamental primitives that underpins
many quantum algorithms, including quantum amplitude estimation, the HHL
algorithm for solving linear systems of equations, and quantum principal
component analysis. Due to its significance as a subroutine, in this work, we
study the coherent version of the phase estimation problem, where given an
arbitrary input state and black-box access to unitaries $U$ and controlled-$U$,
the goal is to estimate the phases of $U$ in superposition. Unlike most
existing phase estimation algorithms, which employ intermediary measurements
steps that inevitably destroy coherence, only a couple of algorithms, including
the well-known standard quantum phase estimation algorithm, consider this
coherent setting. In this work, we propose an improved version of this standard
algorithm that utilizes tapering/window functions. Our algorithm, which we call
tapered quantum phase estimation algorithm, achieves the optimal query
complexity (total number of calls to $U$ and controlled-$U$) without requiring
the use of a computationally expensive quantum sorting network for median
computation, which the standard algorithm uses to boost the success probability
arbitrarily close to one. We also show that the tapering functions that we use
are optimal by formulating optimization problems with different optimization
criteria. Beyond the asymptotic regime, we also provide non-asymptotic query
complexity of our algorithm, as it is crucial for practical implementation.
Finally, we also propose an efficient algorithm that prepares the quantum state
corresponding to the optimal tapering function.
