Optimal Coherent Quantum Phase Estimation via Tapering

Quantum phase estimation is one of the fundamental primitives that underpins many quantum algorithms, including Shor's algorithm for efficiently factoring large numbers. Due to its significance as a subroutine, in this work, we consider 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. Most existing phase estimation algorithms involve intermediary measurements that disrupt coherence. Only a couple of algorithms, including the standard quantum phase estimation algorithm, consider this coherent setting. However, the standard algorithm only succeeds with a constant probability. To boost this success probability, it employs the coherent median technique, resulting in an algorithm with optimal query complexity (the total number of calls to U and controlled-U). However, this coherent median technique requires a large number of ancilla qubits and a computationally expensive quantum sorting network. To address this, in this work, we propose an improved version of this standard algorithm called the tapered quantum phase estimation algorithm. It leverages tapering/window functions commonly used in signal processing. Our algorithm achieves the optimal query complexity without requiring the expensive coherent median technique to boost success probability. 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 propose an efficient algorithm to prepare the quantum state corresponding to the optimal tapering function.
Comment: 24 pages, 6 figures; Changes in version 2: Improved presentation