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  • Open Access

Optimal Coherent Quantum Phase Estimation Via Tapering

Dhrumil Patel1,4,*, Shi Jie Samuel Tan2,3,4,*,†, Yiğit Subaşı4, and Andrew T. Sornborger4

  • *The first two authors contributed equally to this work.
  • †Contact author: stan97@umd.edu
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PRX Quantum 7, 020302 – Published 2 April, 2026

DOI: https://doi.org/10.1103/l5y6-6zxv

Abstract

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, one can employ the coherent median technique, resulting in an algorithm with asymptotically 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 the standard algorithm called the tapered quantum phase estimation algorithm, which leverages tapering (or window) functions commonly used in classical signal processing. Our algorithm achieves the asymptotically optimal query complexity without requiring the expensive coherent median technique to boost success probability. Moreover, we find the absolutely optimal taper—not only in the asymptotic scaling but in terms of exact performance. We provide an efficiently preparable ancilla state based on an approximation of the optimal taper, which incurs at most a factor-of-two increase in the probability of error, thereby maintaining near-optimal performance in practice. In the Appendixes, we give an explicit construction of the taper state preparation circuit. Finally, we derive an error bound for coherent quantum phase estimation when the phase estimate is used as a control and subsequently uncomputed.

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Quantum phase estimation (QPE) is a core subroutine for major quantum algorithms, ranging from factoring large numbers to simulating quantum chemistry. As a building block for these larger algorithms, QPE must operate coherently, meaning it must process quantum superpositions without prematurely causing them to collapse through measurement. Currently, the standard coherent QPE algorithm has a baseline success rate of only about 81%. Boosting this to near-certainty traditionally requires running the algorithm multiple times in parallel and calculating the median. This approach is highly resource-intensive, demanding a large number of extra ancilla qubits and complex quantum sorting networks.

To overcome this bottleneck, we propose a streamlined approach called tapered quantum phase estimation (tQPE). Rather than relying on costly sorting networks to boost success rates after the fact, tQPE optimizes the starting conditions of the algorithm. Instead of initializing the ancilla qubits in a standard uniform superposition, we shape their initial state using a “taper” or “window” function adapted from classical signal processing. Specifically, we utilize a quantum state based on a discrete prolate spheroidal sequence (DPSS). Just as a DPSS taper maximally concentrates a classical signal into a narrow frequency band, our quantum DPSS taper maximally concentrates the probability of the algorithm outputting the correct phase.

By strategically reshaping this initial state, tQPE achieves mathematically optimal success rates using exponentially fewer extra qubits than the traditional median-based approach. To facilitate practical application, we also provide an efficient quantum circuit for preparing this optimal DPSS state. Ultimately, our approach significantly reduces the hardware overhead required for high-precision, coherent QPE, paving the way for its implementation on near-term quantum devices.

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PRX Quantum 7, 020302– Published 2 April, 2026
  • Received 13 June 2025
  • Accepted 13 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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