Optimal Coherent Quantum Phase Estimation via Tapering

Andrew T. Sornborger; Dhrumil Patel; Shi Jie Samuel Tan; Yigit Subasi

arxiv: 2403.18927 · v3 · submitted 2024-03-27 · 🪐 quant-ph · math-ph· math.MP

Optimal Coherent Quantum Phase Estimation via Tapering

Dhrumil Patel , Shi Jie Samuel Tan , Yigit Subasi , Andrew T. Sornborger This is my paper

Pith reviewed 2026-05-24 02:14 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords quantum phase estimationcoherent QPEtapering functionsquery complexityancilla preparationquantum algorithms

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The pith

Tapered quantum phase estimation reaches asymptotically optimal query complexity without coherent median.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes replacing the coherent median technique in quantum phase estimation with tapering functions drawn from classical signal processing. This produces an algorithm whose total number of calls to the unitary and controlled unitary matches the best known asymptotic scaling while avoiding a large ancilla register and quantum sorting network. The authors identify the single taper that is optimal not only in scaling but in exact finite performance and supply an efficiently preparable approximation to it whose error probability is at most twice that of the ideal taper.

Core claim

The tapered quantum phase estimation algorithm, by shaping the phase-estimation amplitudes with an optimal taper function, attains the asymptotically optimal query complexity for coherent phase estimation while using only an efficiently preparable ancilla state that approximates the ideal taper and raises the error probability by at most a factor of two.

What carries the argument

The tapering (window) function applied to the amplitudes of the phase-estimation register, which concentrates the output distribution to raise success probability without intermediary measurements.

If this is right

Asymptotically optimal query complexity is obtained without the ancilla overhead and sorting cost of coherent median.
The identified taper is exactly optimal among all possible windows, not merely asymptotically.
The approximate taper state can be prepared by an explicit circuit given in the appendices.
An error bound holds when the phase estimate is subsequently used as a control and uncomputed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same window-shaping idea may reduce ancilla cost in other coherent quantum routines that currently rely on median amplification.
Finite-size performance gains from the exact optimal taper could be measured directly on small quantum devices.
Because the optimal taper is derived from a classical signal-processing criterion, other classical windows may yield further practical improvements.

Load-bearing premise

An efficiently preparable ancilla state approximating the optimal taper exists and raises error probability by at most a factor of two while preserving the stated query complexity and error bounds when the phase is later used as a control.

What would settle it

Numerical evaluation, for concrete numbers of qubits and a fixed phase, of whether the success probability of the proposed taper equals or exceeds that of every other window function and stays within a factor of two of the ideal median-boosted algorithm.

read the original abstract

Due to its significance as a subroutine, in this work, we consider the coherent version of the quantum 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 (tQPE) 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 appendices, we give an explicit construction of the taper state preparation circuit. Finally, we derive an error bound for coherent QPE when the phase estimate is used as a control and subsequently uncomputed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Tapering for coherent QPE looks promising on paper but can't be checked from the abstract alone.

read the letter

The main point here is that the authors apply tapering functions from signal processing to coherent quantum phase estimation and claim this reaches asymptotically optimal query complexity without the ancilla overhead and sorting cost of the coherent median technique. They also identify what they call the absolutely optimal taper and give an efficiently preparable approximation that increases error probability by at most a factor of two. An explicit circuit for the taper state is promised in the appendices, along with an error bound when the phase estimate controls another operation and is later uncomputed. That is the actual new content beyond the standard algorithm and the median boost. The approach makes sense as a way to reduce practical costs in a subroutine used across quantum algorithms. The soft spot is straightforward: only the abstract is available. No derivations, no circuit diagrams, no numerical checks, and no proof of the optimality or the factor-of-two bound can be examined. The key assumption that an approximating ancilla state can be prepared without spoiling the query complexity or error guarantees is stated but not demonstrated here. This leaves the central claims untested. The work is aimed at researchers who need coherent phase estimation with lower resource overhead on limited hardware. If the full paper contains the promised constructions and bounds, it is worth sending to referees so the details can be checked. Until then the idea is interesting but remains provisional.

Referee Report

1 major / 0 minor

Summary. The manuscript presents the tapered quantum phase estimation (tQPE) algorithm for the coherent version of quantum phase estimation. Using tapering (window) functions from classical signal processing, it claims to achieve asymptotically optimal query complexity to the unitaries U and controlled-U without the coherent median technique, identifies the absolutely optimal taper for exact performance, supplies an efficiently preparable ancilla state based on an approximation of that taper (incurring at most a factor-of-two error-probability increase), provides an explicit taper-state-preparation circuit in the appendices, and derives an error bound for the case in which the phase estimate is subsequently used as a control and uncomputed.

Significance. If the optimality claims, the factor-of-two error bound, and the controlled-use error analysis hold, the result would be significant for coherent quantum algorithms: it removes the need for large ancilla registers and quantum sorting networks while retaining near-optimal scaling and practical performance. The explicit circuit construction and the focus on exact (not merely asymptotic) taper performance would be concrete strengths for reproducibility and implementation.

major comments (1)

[Abstract] Abstract: the central claim that the algorithm achieves asymptotically optimal query complexity without the coherent median technique, together with the factor-of-two error bound on the ancilla approximation, cannot be verified because the promised derivations, comparisons, and appendices are not available in the provided text.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their review of our manuscript. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the algorithm achieves asymptotically optimal query complexity without the coherent median technique, together with the factor-of-two error bound on the ancilla approximation, cannot be verified because the promised derivations, comparisons, and appendices are not available in the provided text.

Authors: We acknowledge that the text provided for review contains only the abstract. As a result, the detailed derivations, comparisons to the coherent median technique, analysis of the factor-of-two error bound, and the appendices with the explicit taper-state-preparation circuit are not present. The complete manuscript on arXiv contains these sections, but they cannot be reproduced or verified from the material supplied here. revision: no

standing simulated objections not resolved

The derivations, comparisons, and appendices supporting the central claims are not available in the provided text, preventing us from supplying them in this response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract contains no equations, no derivation steps, and no self-citations. The algorithm is described as drawing on external classical signal-processing tapering/window functions and standard quantum oracles, with the central claims (asymptotically optimal query complexity without coherent median, efficient ancilla preparation) presented as new constructions rather than reductions to prior self-authored results. No load-bearing step reduces by construction to fitted inputs or self-citations; the text is self-contained against external benchmarks and yields a normal non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no explicit free parameters, invented entities, or non-standard axioms identified beyond routine quantum computing assumptions.

axioms (1)

domain assumption Black-box access to unitary U and controlled-U is available.
Stated directly in the abstract as the problem setting.

pith-pipeline@v0.9.0 · 5819 in / 1163 out tokens · 41349 ms · 2026-05-24T02:14:42.917635+00:00 · methodology

discussion (0)

Forward citations

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