Risk-minimizing states for the quantum-phase-estimation algorithm

Crispin H. W. Barnes; David R. M. Arvidsson-Shukur; Joseph G. Smith

REVIEW 3 major objections 5 minor 33 references

Risk-minimizing states for the quantum-phase-estimation algorithm

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper derives the input state that minimizes Bayesian risk for quantum phase estimation: the smallest-eigenvalue eigenvector of a Toeplitz matrix built from the loss function's Fourier coefficients.

desk verdict The Toeplitz-eigenvector idea is neat and the cosine approximation works well, but the central optimality claim rests on an assumption about the Bayes estimator that is false for simple valid input states, so the main theorem is unproven. read the letter →

arxiv 2505.00764 v1 pith:GWD442YU submitted 2025-05-01 quant-ph

Joseph G. Smith , Crispin H. W. Barnes , David R. M. Arvidsson-Shukur This is my paper · ORCID

classification quant-ph

keywords quantumphaseestimationBayesianriskToeplitzmatrixcosinestateHeisenberglimitHolevovariancedepolarizingnoiselossfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks which input state to the quantum-phase-estimation algorithm (QPEA) gives the lowest Bayesian risk for a given loss function. It shows that the risk is a quadratic form in the register amplitudes, $R(\Psi)=L_0+(1-\lambda)^N c^\dagger R c$, where $R$ is a Toeplitz matrix whose entries are the Fourier coefficients of the loss. The risk-minimizing state is therefore the eigenvector of $R$ with the smallest eigenvalue, and it is independent of the noise strength $\lambda$. Because that eigenvector has no closed form in general, the paper proves that a cosine-shaped state with one tuned frequency approximates it well; for absolute, squared, and Holevo losses, the cosine state reaches the same large-$N$ risk scaling as Heisenberg-limited measurements. It also proves the traditional uniform superposition is suboptimal for every nonconstant loss except the one-qubit case, and gives a multi-measurement strategy that lowers risk under depolarizing noise.

What carries the argument

The central object is the $(N+1)\times(N+1)$ symmetric Toeplitz matrix $R$ whose constant diagonals are the Fourier coefficients $\{L_k\}$ of the loss function, with zero on the diagonal. It carries the argument because the risk $R(\Psi)=L_0+(1-\lambda)^N c^\dagger R c$ is a quadratic form in the register amplitudes, so minimizing risk over normalized states is exactly the problem of finding the minimum-eigenvalue eigenvector of $R$. Around this matrix the paper builds the cosine-state family $c_i(\omega)\propto\cos((N/2-i)\omega)$, which supplies an analytic near-optimal substitute when the true eigenvector lacks a closed form, and it uses the $M$-fold product likelihood to analyze repeated measurements under depolarizing noise.

What would settle it

For a small register (say $m=3$) and the absolute or squared loss, prepare the cosine state with the paper's optimal frequency, compute the true posterior $p(\theta|y)$ for each outcome $y$, and numerically minimize $\int p(\theta|y)L(\hat\theta-\theta)d\theta$ over $\hat\theta$ without fixing $\hat\theta=2\pi y/2^m$. If any outcome's minimizing estimate differs from the bin center, the state-independent risk formula is not the actual Bayes risk and the Toeplitz eigenvector need not be risk-minimizing.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that for any even, nondecreasing, $2\pi$-periodic loss function $L(\theta-\hat\theta)$, the Bayes risk of QPEA with first-register state $|\Psi\rangle=\sum_j c_j|j\rangle$ is $R(\Psi)=L_0+(1-\lambda)^N c^\dagger R c$, with $L_0$ the zeroth Fourier coefficient of $L$ and $R$ the Toeplitz matrix whose entries are the Fourier coefficients $L_{j-k}$ with zero on the diagonal. The risk-minimizing input is thus the normalized eigenvector of $R$ with smallest eigenvalue, and this holds for any loss in that class and any depolarizing strength $\lambda$; the optimal state itself does not depend on $\lambda$. For the Holevo loss $L(\delta)=4\sin^2(\delta/2)$, this eigenvector is exactly a cosine state with frequency $\omega=\pi/(N+2)$, reproducing a previously known optimum. For general losses the authors introduce cosine states with a tunable frequency, optimize the frequency, and show numerically that the resulting risk nearly matches the true optimum; in the noiseless limit these cosine states attain the Heisenberg-limited asymptotic scalings for absolute, squared, and Holevo losses, while the uniform state does not. A further theorem states that the uniform state is strictly suboptimal for any nonconstant loss except in the one-qubit register, and the paper shows that repeating measurements of a fixed-size circuit can reduce risk below the shot-noise limit when depolarizing noise is present.

Load-bearing premise

The derivation rests on assuming that the estimate minimizing expected posterior loss is always the bin center $\hat\theta=2\pi y/2^m$ for every input state; if any state has a posterior whose minimum lies elsewhere, the risk formula overestimates the true minimal risk and the derived optimal state may not be Bayes-optimal.

Editorial extensions

If this is right

For any even, nondecreasing, periodic loss, designing the optimal QPEA input becomes a standard linear-algebra computation: diagonalize the $2^m\times 2^m$ Toeplitz matrix and take the bottom eigenvector.
The cosine-state family with optimized frequency gives a practical, preparable input that achieves Heisenberg-limited risk for absolute, squared, and Holevo losses in noiseless settings, so the traditional uniform initialization is provably suboptimal for registers with $m>1$.
For the Holevo loss the optimum is known exactly: the cosine frequency is $\omega=\pi/(N+2)$, giving a closed-form risk $R=2-2(1-\lambda)^N\cos(\pi/(N+2))$.
Under depolarizing noise, increasing register size alone drives the risk to the constant $L_0$; taking $M$ repeated measurements of a fixed-size circuit can push the risk below the shot-noise limit, with $M\ge 3$ needed for the 1-0 loss in noiseless circuits.
The optimal state's independence from $\lambda$ means the same input remains risk-minimizing regardless of the depolarizing error rate, even though the achievable risk floor worsens as noise increases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper: the same Toeplitz construction should apply to any algorithm whose final measurement is a discrete Fourier transform of controlled phase shifts, so the optimal-input calculus could transfer to Fourier-transform-based metrology and spectral estimation.
Going beyond the paper: because the risk formula is linear in the loss's Fourier coefficients, the optimal state for a mixture of losses is the minimum eigenvector of the corresponding averaged Toeplitz matrix; this offers a route to inputs that perform well under several loss functions at once, which the paper does not explore.
Going beyond the paper: the paper's own numerics show the cosine approximation degrades for the 1-0 loss at $m\ge 7$, suggesting that a different shaped envelope, such as a Gaussian or prolate-spheroidal profile, might close the gap to the true optimum in the regime where the eigenvector concentrates away from the edges.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely useful before it overreaches. It observes that if you fix the estimator to be the standard QPEA grid point, then the risk of any input state is a quadratic form whose matrix is Toeplitz built from the loss's Fourier coefficients. Minimizing that form over states is an eigenvector problem, and the eigenvector generalizes the known Holevo-optimal state. That is a clean piece of linear algebra, and the cosine-state approximation to that eigenvector is an attractive, practical result. The scaling plots are credible, and the proof that the uniform state is suboptimal for fixed grid-point estimation is a nice observation. So there is real content here.

But the paper claims to minimize the Bayes risk, i.e., risk after choosing the optimal estimator for each outcome. The proof rests on Eq. (13), which asserts that the grid point always minimizes the posterior expected loss. That is not true. For m=2, take the valid normalized input state c=(1/√2,0,1/√2,0). For squared loss, the expected loss is π²/3 + (1/4)cos(2δ), which is minimized at δ=±π/2, not δ=0. So the grid point is not the Bayes estimator. The risk R(Ψ) defined in Eq. (9) is then strictly less than Eq. (14) for this state, and minimizing Eq. (14) over c gives an upper bound on the true Bayes risk, not the minimum. The main theorem is therefore unproven: the eigenvector of R may be optimal for the fixed grid-point estimator, but not for the Bayes risk as defined. This is a load-bearing flaw, not a typo. The paper needs to either restrict to states for which the grid point is actually the Bayes estimator (e.g., unimodal posteriors centered at the grid point) or reframe the contribution as risk minimization for a fixed estimator. The latter is still worth publishing, but the abstract and conclusions currently overclaim.

There are smaller issues too: Eq. (14) is missing a factor of 2 that appears in Eq. (15), and the abstract's statement that the uniform state is suboptimal for any reasonable loss is false for m=1, a point the text itself concedes. The conclusion already acknowledges that adaptive methods outperform the proposed repeated-measurement scheme, which is honest and useful.

If the authors can fix or correctly scope the Bayes-estimator step, the Toeplitz characterization and cosine states will be a solid contribution. As it stands, I would not rely on the optimality theorem, but the paper deserves a careful referee rather than a desk reject. The core idea is salvageable and the numerical work is likely reproducible.

Referee Report

3 major / 5 minor

Summary. The paper proposes a method to find the Bayes-risk-minimizing input state for the quantum-phase-estimation algorithm. It claims that, for any loss function, the posterior expected loss is minimized by the grid-point estimate 2πy/2^m, so the risk becomes a state-independent quadratic form (Eq. 14). The optimal state is then identified with the minimum-eigenvalue eigenvector of a Toeplitz matrix whose entries are the Fourier coefficients of the loss (Eq. 15). The paper introduces cosine states as approximate optima, reports Heisenberg-limited scaling for absolute, squared, and Holevo losses, proves that the uniform state is suboptimal for non-constant losses, and studies depolarizing noise and repeated measurements.

Significance. If the central theorem held, the paper would provide an elegant reduction of QPEA input optimization to a linear-algebra problem for arbitrary loss functions, together with explicit near-optimal states and scaling claims. The agreement with the known Holevo-optimal state of van Dam et al. is a useful consistency check, and the argument that the uniform state is suboptimal is conceptually clean. However, the central derivation rests on Eq. (13), which is false for general input states. The main optimality theorem is therefore not established as stated, and the cosine-state scaling results are conditional on an unproven property of the posterior. The paper has merit and may be salvageable by restricting the claims and supplying the missing proofs, but the present version is not sound.

major comments (3)

[Section II, Eqs. (12)-(15)] Eq. (13) is false for general input states. For m=2 and the valid normalized state |Ψ⟩=(|0⟩+|2⟩)/√2 with squared loss, only the lag-2 autocorrelation is nonzero, A2=1/2. With the missing factor of 2 restored, Eq. (12) gives L(y,θ̂′)=π²/3+(1/2)cos(2ϑ), which is minimized at ϑ=±π/2, not at ϑ=0. The posterior is proportional to cos²(θ-2πy/4), a bimodal distribution, so the grid point is not the Bayes estimator. Consequently Eq. (14) is an upper bound on the Bayes risk rather than the risk itself, and the Toeplitz-eigenvector optimality claim in Eq. (15) does not follow. The authors must either prove that the Bayes estimator is the grid point for the class of states they actually optimize over, or reformulate the optimization in terms of the true Bayes risk.
[Section III, Eq. (20) and Table II] The cosine-state risk formula and the scaling claims inherit the same problem: Eq. (20) evaluates the expected loss at the fixed grid estimator. For states with bimodal posteriors this is not the Bayes risk, so the quoted R(ω′) may overstate the true risk. Even if the final numerical states are positive and monotone, the paper does not prove that their posteriors are unimodal with mode at the grid point, nor that the minimum eigenvector of R lies in that class. Without such a proof the Heisenberg-scaling claims in Table II are not established.
[Section III, paragraph before Eq. (19)] The assertion that the minimum eigenvector of R satisfies ci=c_{N-i} and c0<c1<...<c_{(N-1)/2} is not a general property of eigenvectors of symmetric banded Toeplitz matrices and is not proved for the loss functions considered. This monotonicity restriction is important because it may be exactly what is needed to make the posterior unimodal, and hence to make Eq. (13) valid on the candidate optimal states. The authors should provide a proof of these structural properties or state them as assumptions and show that the numerical optima satisfy them.

minor comments (5)

[Section II, Eqs. (12)-(14)] Eqs. (12)-(14) are missing a factor of 2 that appears in Eqs. (15) and (20); the inconsistency should be corrected throughout the derivation.
[Section III, Eqs. (19) and (21)] In the plain-text rendering the normalizing denominator appears as 2m rather than 2^m; as printed, the normalization of the cosine state is incorrect.
[Table II, 1-0 loss row] Using Eq. (22) with σ²=1/N and σ²=1/N², the 1-0 risk exponents should be O(e^{-Nϵ²/2}) and O(e^{-N²ϵ²/2}), not O(e^{-Nϵ/2}) and O(e^{-N²ϵ/2}); please correct the exponents.
[Section III, Eq. (24)] The four expressions for R(u) are not labeled; please indicate which expression corresponds to absolute, squared, Holevo, and 1-0 loss.
[Section IV, numerical calculations] The paper should state how the numerical risks for M measurements were evaluated (for example, the discretization of the phase interval and the handling of the 2^{mM} outcome sum) so that the results are reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Toeplitz-eigenvector state is a direct Rayleigh-quotient output, checked against the independent Holevo result of Ref. [29]; the flagged Eq. (13) issue is a correctness gap, not circularity.

full rationale

The derivation of the risk-minimizing register state is self-contained and is not circular. Eqs. (12)-(14) compute the expected loss from the Fourier coefficients of L; Eq. (15) rewrites the risk as L0+(1-lambda)^N c^dagger R c, and minimizing over normalized c is exactly the minimum-eigenvalue problem for R. The loss function enters only through the L_k defining R, and the eigenvector is an output of the minimization, not a re-labeled input. The cosine family in Eq. (19) is introduced explicitly as a variational ansatz, with omega' = argmin R(omega); the claimed scalings of R(omega') are computed from Eq. (20), not inferred from a fit and then presented as independent predictions. The Holevo case reproduces the independent external state of Ref. [29], providing a non-circular benchmark. The uniform-state suboptimality proof is a Fourier-coefficient argument and does not depend on the authors' prior work. The authors' self-citations ([14], [17], and co-authored refs [3], [5]) are used for context, comparison, and algorithmic background, not as the load-bearing justification of the Toeplitz theorem. The only significant flagged weakness is mathematical rather than circular: Eq. (13) asserts without proof that 'This loss is minimised by the estimate theta-hat = 2*pi*y/2^m'; for some losses, such as squared loss with the alternating L_k of Table I, the expected loss in Eq. (12) can be minimized at theta-hat' unequal to 2*pi*y/2^m, so Eq. (14) would not be the true Bayes risk. This would be a correctness gap in the derivation of the claimed optimality, not a reduction of the derived state to fitted parameters or to a self-citation chain. It therefore does not raise the circularity score.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The free parameters are the numerically optimized cosine frequencies for three loss functions; the Holevo case is derived exactly. The key axioms are the uniform prior, the depolarizing noise model, and the restriction to even, non-decreasing periodic loss functions.

free parameters (3)

cosine frequency omega' for absolute loss = pi/(1.04N+22.27) asymptotically
Numerically optimized to minimize the absolute risk R(omega); no closed form is derived.
cosine frequency omega' for squared loss = pi/(N+1.72) asymptotically
Numerically optimized to minimize the squared risk R(omega); no closed form is derived.
cosine frequency omega' for 1-0 loss = pi/(N+1.00) asymptotically
Numerically optimized to minimize the 1-0 risk R(omega); no closed form is derived.

assumptions (6)

domain assumption Uniform prior pi(theta)=1/(2*pi) over the phase interval Theta=[theta0-pi,theta0+pi]
Used in Eqs. (5)-(6) to compute posterior and risk; represents complete prior ignorance.
domain assumption Depolarizing noise model: after each controlled-U gate, the first register is subject to a depolarizing channel of strength lambda, yielding Eq. (2)
A tractable but specific noise model; real device noise (e.g., dephasing) would change the results.
domain assumption Loss functions are even, non-decreasing in |delta|, and periodic with period 2*pi
Restricts the analysis to symmetric error measures; the 1-0 loss with threshold epsilon is included.
domain assumption The optimal input state can be taken to have real amplitudes c_j in R
The symmetry of the loss and the Toeplitz matrix allow a real eigenvector; the paper does not justify complex amplitudes.
standard math For general banded Toeplitz matrices, no closed-form eigenvector exists (Refs [31,32])
Used to justify numerical methods for the optimal state for non-Holevo losses.
standard math Shot-noise (sigma^2=1/N) and Heisenberg (sigma^2=1/N^2) limits from Fisher-information-optimal measurements (Refs [12,13]) serve as benchmarks
RC and RQ in Table II are computed from these external limits via Eq. (22).

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Cite this review

Pith. "Pith review of Risk-minimizing states for the quantum-phase-estimation algorithm." pith.science (2026). https://pith.science/paper/GWD442YU

@misc{pith2026250500764,
  author       = {Pith},
  title        = {Pith review of: Risk-minimizing states for the quantum-phase-estimation algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWD442YU}},
  note         = {Machine review of arXiv:2505.00764}
}

read the original abstract

The quantum-phase-estimation algorithm (QPEA) is widely used to find estimates of unknown phases. The original algorithm relied on an input state in a uniform superposition of all possible bit strings. However, it is known that other input states can reduce certain Bayesian risks of the final estimate. Here, we derive a method to find the risk-minimizing input state for any risk. These states are represented by an eigenvector of a Toeplitz matrix with elements given by the Fourier coefficients of the loss function of interest. We show that, while the true optimal state does not have a closed form for a general loss function, it is well approximated by a state with a cosine form. When the cosine frequency is chosen appropriately, these states outperform the original QPEA and achieve the optimal theoretical quantum-advantage scaling for three common risks. Furthermore, we prove that the uniform input state is suboptimal for any reasonable loss function. Finally, we design methods to mitigate the impact of depolarizing noise on the performance of QPEA.

Figures

Figures reproduced from arXiv: 2505.00764 by the authors.

**Figure 2.** FIG. 2. Numerically calculated [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. The four common risks from measurements of a noise [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. The four common risks from measurements of a [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: plots the Holevo and 1-0 risks as a function of the resources used for 1 ≤ M ≤ 5 when the QPEA register is prepared in the optimal cosine state. For noiseless circuits (top row), the Holevo risk scales as RQ for any fixed M. The minimum risk per resource occurs when M…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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