{"id":"af19dc31-087a-482e-a4f5-631a8b527a7b","arxiv_id":"2505.00764","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In quantum phase estimation, the risk-minimizing input state for any loss function is the minimum eigenvector of a Toeplitz matrix of the loss's Fourier coefficients, and cosine states approximate it well.","lead":"This paper derives the input state that minimizes any given Bayesian risk in quantum phase estimation: such states are eigenvectors of a Toeplitz matrix built from the loss function's Fourier coefficients. Cosine-shaped states approximate the optimum and beat the standard uniform input state for three common error measures.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (13) is false for squared loss: the Bayes estimator is not always the grid point, so the Toeplitz-eigenvector optimality theorem is not proven for general input states.","rationale":"The reader's weakest assumption identified Eq. (13) as the central gap. The stress-test confirms this concern and makes it concrete: the squared-loss counterexample with c=(1/sqrt{2},0,1/sqrt{2},0) shows that the grid-point estimator is not Bayes for a valid input state, so the risk formula Eq. (14) is not the true Bayes risk for all states. Consequently, the proof that the minimum eigenvector of R minimizes the risk is incomplete. However, this does not force a rejection: the theorem could still be true for the states that actually minimize R, and the numerical results for cosine states may survive because those states have positive, symmetric amplitudes and likely unimodal posteriors. The factor-of-2 inconsistency between Eq. (14) and Eq. (15) is a related but secondary issue that does not change the minimizer. A conditional acceptance requiring the authors to either prove the Bayes-estimator claim for the relevant states, restrict the theorem to that class, or provide a counterexample-free derivation is appropriate. The paper's independent support, including agreement with the van Dam et al. Holevo optimum and the explicit finite-N numerical scaling, counts in its favor but does not repair the logical gap in the central derivation.","tokens_in":9753,"tokens_out":27057,"duration_ms":246228,"concrete_test":"For m=2, squared loss, input state c=(1/sqrt{2},0,1/sqrt{2},0) and outcome y=0, the posterior is p(theta|0)=(1+cos(2*theta))/(2*pi) on Theta=[-pi,pi]. Compute the expected periodic squared loss at theta_hat=0: pi^2/3 + 1/2 ~ 3.79, and at theta_hat=pi/2: pi^2/3 - 1/2 ~ 2.79. Since the latter is smaller, Eq. (13) is false. If the authors instead claim the theorem applies only to states whose posterior is unimodal at the grid point, verify that the minimum eigenvector of R for squared loss (m=2, c ~ (0.350,0.614,0.614,0.350)) has a positive second derivative at delta=0, and additionally search over all normalized c (e.g., by brute-force optimization on the 3-sphere) for a state with true Bayes risk lower than the eigenvector's risk.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The derivation of the risk-minimizing state (Section II) hinges on Eq. (13), which asserts that for every input state the Bayes estimator is the grid point theta_hat = 2*pi*y/2^m. This is not true for the squared loss (Table I, with periodicization per Eq. (10)). Take m=2 and the valid normalized state c=(1/sqrt{2},0,1/sqrt{2},0). Only the lag-2 autocorrelation is nonzero, S_2=1/2. For L(delta)=min_l(delta+2*pi*l)^2 the coefficients are L_0=pi^2/3 and L_2=1/2, so Eq. (12) gives L(y,theta_hat') = pi^2/3 + (1/2) cos(2*delta), with delta = theta_hat' - 2*pi*y/2^m. This is minimized at delta = +/- pi/2, not delta = 0; equivalently, the second derivative at delta = 0 is negative. Thus Eq. (14) is not the Bayes risk of this state; it is the risk of a suboptimal estimator. Since Eq. (15) is derived by minimising Eq. (14), the claim that the smallest-eigenvalue eigenvector of R is the risk-minimizing input state is unproven for general states and losses. The paper's numerical cosine states have positive amplitudes and may satisfy the needed condition, but that must be proved or the theorem restricted to states with a unimodal posterior centred at the grid point. (A separate typo: Eq. (14) is missing the factor 2 that appears in Eq. (15).)","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10064,"tokens_out":15703,"duration_ms":150924,"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":[{"comment":"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":"Section II, Eqs. (12)-(15)"},{"comment":"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":"Section III, Eq. (20) and Table II"},{"comment":"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.","section":"Section III, paragraph before Eq. (19)"}],"minor_comments":[{"comment":"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":"Section II, Eqs. (12)-(14)"},{"comment":"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.","section":"Section III, Eqs. (19) and (21)"},{"comment":"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":"Table II, 1-0 loss row"},{"comment":"The four expressions for R(u) are not labeled; please indicate which expression corresponds to absolute, squared, Holevo, and 1-0 loss.","section":"Section III, Eq. (24)"},{"comment":"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.","section":"Section IV, numerical calculations"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nBut 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.\n\nThere 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.\n\nIf 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.","headline":"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.","tokens_in":10662,"tokens_out":8323,"would_cite":false,"duration_ms":80414,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["quantum phase estimation","Bayesian risk","Toeplitz matrix","cosine state","Heisenberg limit","Holevo variance","depolarizing noise","loss function"],"falsifier":"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.","tokens_in":9504,"feed_emoji":"🎯","tokens_out":9835,"duration_ms":87673,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quantum phase estimation's best input is a Toeplitz eigenvector","feed_subtitle":"A tuned cosine state reaches Heisenberg-limited risk for three common losses, while the uniform input is provably suboptimal.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the previously derived optimal QPEA input for Holevo risk, which the paper recovers as the tridiagonal Toeplitz minimum eigenvector and as a cosine state with $\\omega=\\pi/(N+2)$.","marker":"[29]"},{"why":"Provides earlier Bayesian phase-estimation results showing the standard QPEA lacks Heisenberg-limited Holevo variance, motivating non-uniform input states.","marker":"[18]"},{"why":"Defines the shot-noise-limited and Heisenberg-limited measurement models whose risk scalings the paper uses as benchmarks and extends to repeated measurements.","marker":"[17]"},{"why":"Gives the quantum metrology variance bounds used to set the shot-noise and Heisenberg limits for comparison.","marker":"[12]"},{"why":"Provides the companion quantum metrology bound for the Heisenberg limit with unitary resources.","marker":"[13]"},{"why":"Is the textbook presentation of QPEA with a uniform input register, the baseline state the paper proves suboptimal except in trivial cases.","marker":"[22]"},{"why":"Introduces the quantum phase estimation algorithm itself, whose standard uniform initialization is the object of comparison.","marker":"[25]"},{"why":"Documents the absence of closed-form eigenvectors for general banded Toeplitz matrices, justifying the cosine-state approximation.","marker":"[31]"}],"fun_headline_variants":["Optimal phase estimation input is a Toeplitz eigenvector","Cosine state reaches Heisenberg limit in QPEA","Uniform state fails: Toeplitz eigenvector minimizes risk","Risk-minimizing QPEA states discovered via Toeplitz analysis","Tuned cosine beats uniform for quantum phase estimation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Optimal phase estimation input is a Toeplitz eigenvector","Cosine state reaches Heisenberg limit in QPEA","Uniform state fails: Toeplitz eigenvector minimizes risk","Risk-minimizing QPEA states discovered via Toeplitz analysis","Tuned cosine beats uniform for quantum phase estimation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000395,"raw_usage":{"total_tokens":2127,"prompt_tokens":1058,"completion_tokens":1069,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":984}},"tokens_in":674,"tokens_out":1069,"duration_ms":10821,"temperature":1.0,"reasoning_tokens":984,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:37:16.972380+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"van Dam, G","cited_arxiv_id":null,"evidence_quote":"Supplies the previously derived optimal QPEA input for Holevo risk, which the paper recovers as the tridiagonal Toeplitz minimum eigenvector and as a cosine state with $\\omega=\\pi/(N+2)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the shot-noise-limited and Heisenberg-limited measurement models whose risk scalings the paper uses as benchmarks and extends to repeated measurements."},{"cited_title":"Giovannetti, S","cited_arxiv_id":null,"evidence_quote":"Gives the quantum metrology variance bounds used to set the shot-noise and Heisenberg limits for comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the textbook presentation of QPEA with a uniform input register, the baseline state the paper proves suboptimal except in trivial cases."},{"cited_title":"Ekstr¨ om, C","cited_arxiv_id":null,"evidence_quote":"Documents the absence of closed-form eigenvectors for general banded Toeplitz matrices, justifying the cosine-state approximation."}],"review_version":1}