{"id":"07c033a5-29c6-4855-bbe1-3fe45ea63266","arxiv_id":"2502.05617","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A hybrid quantum-classical algorithm estimates quantum amplitudes from the Fourier peaks of Gaussian-filtered overlap measurements, without the quantum Fourier transform.","lead":"This paper proposes estimating quantum amplitudes by measuring a series of overlap signals and processing them with a classical Fourier transform, avoiding quantum phase estimation. The method is numerically demonstrated on 4-qubit amplitudes and 6-qubit observables, but key resource claims and error analysis have gaps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The peak-location step in Eq. (14) determines 2mθ only modulo 2π; without a phase-unwrapping procedure, a single-m scan cannot recover θ whenever 2mθ exceeds 2π, so the central extraction claim is ambiguous as stated.","rationale":"The most load-bearing condition for the central claim is not the specific form of |ψ0⟩ in Eq. (9), but the uniqueness of the peak location. The reader's weakest assumption concerns preparation of (|y+⟩+|y−⟩)/√2. That issue is real but not decisive: choosing |ψ0⟩=|ψ⟩ gives identical squared overlaps with |y±⟩, so Eq. (14) and the coefficient structure in Eq. (A4) are preserved; the probe can be trivially fixed. The aliasing problem, by contrast, affects every probe and every implementation of Section II.B or II.C. It strikes at the extraction step itself: because the Fourier sum over integer t is periodic, S(x) has equal-height peaks at all points ±2mθ+2πk, so the 'location' of the peak is only defined modulo 2π. Without an explicit unwrapping rule, a user who follows the paper and scans a reasonable range of x can infer a θ that differs from the true value by π/m. The simulations sidestep this by plotting near the known θ, which is not available in the intended estimation task. This is an internal correctness gap, not an outside-consensus disagreement. It is fixable by standard phase-unwrapping or multi-m consistency, and the underlying Gaussian-derivation idea is sound, so the appropriate verdict remains conditional rather than reject. I do not see the reader's probe-state concern as the same load-bearing issue, hence 'disagree' on agreement; nevertheless, the recommended verdict category is unchanged from CONDITIONAL because the paper needs revision in either case.","tokens_in":11267,"tokens_out":27244,"duration_ms":279531,"concrete_test":"Reproduce Eq. (14) numerically for θ=0.6, m=6, a=1/(20√2), T=60, computing S(x) on a fine grid over x∈[0,2π). Locate the argmax. If, as predicted, the peak lies at about 0.917 rad (not 7.2 rad), then θ_est=argmax/(2m)≈0.076, disagreeing with the true θ. Repeat for m=2,...,12 and verify that the argmax equals (2mθ mod 2π) whenever 2mθ>2π; this confirms the aliasing flaw. A satisfactory fix should also be tested: e.g., use a coarse m=1 estimate to select the correct copy for larger m, or solve a multi-m consistency condition, and then check that the reported θ estimates are stable over at least two independent m choices.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Because t is an integer, the exact discrete-time Fourier transform in Eq. (11) is 2π-periodic in x. The evaluation in Eq. (14) should read (√π/2a) Σ_k [exp(-(x+2mθ-2πk)²/4a²) + exp(-(x-2mθ-2πk)²/4a²)], not a single Gaussian pair. The paper drops the k-sum. Consequently, the 'peak at x=±2mθ' is accompanied by equal-height peaks at every ±2mθ+2πk. In any fundamental interval [0,2π), the observable peak is at (2mθ mod 2π) and (-2mθ mod 2π). For the paper's own parameters (θ=0.6, m=6), 2mθ=7.2 rad, whose periodic copy is 0.917 rad; a scan of x∈[0,2π) would show its maximum near 0.917 and infer θ≈0.076, not 0.6. The numerical demonstrations avoid this only by plotting a window around the known exact value. Switching the probe to |ψ⟩ does not help: ⟨ψ|A^{mt}|ψ⟩=cos(2mθt) retains the same periodicity, so the ambiguity is independent of the probe-state issue. The paper assumes 0≤θ≤π/2 but does not use that prior to select among periodic replicas, and it gives no multi-m consistency rule for unwrapping. Thus Eq. (14) is presented as a unique-peak extraction when in fact the peak location is defined only up to an integer multiple of 2π (equivalently, θ up to π/m).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quantum amplitude estimation (QAE) scheme that avoids full quantum phase estimation. The authors construct an amplitude amplification operator A = exp(i2θσ_y) in the two-dimensional subspace spanned by the states |ψ⟩ and |φ⟩, prepare an initial state |ψ0⟩ = (|y+⟩ + |y−⟩)/√2, and compute the classical Fourier sum S(x) = Σ_t p(t)⟨ψ0|A^{mt}|ψ0⟩e^{ixt}. They claim that S(x) has peaks at x = ±2mθ (Hadamard-test version, Eq. (14)) or at x = ±4mθ and x = 0 (no-Hadamard version using |⟨ψ0|A^{mt}|ψ0⟩|², Eq. (16)), so that θ can be read off from the peak positions. The paper also extends the method to estimating expectation values of Pauli strings, analyzes three error sources (summation cutoff, shot noise, and circuit noise), and presents noiseless and noisy numerical simulations for 4-qubit and 6-qubit examples.","tokens_in":11635,"tokens_out":11450,"duration_ms":121692,"significance":"If the proposed extraction were correct and implementable, the contribution would be a simple classical-post-processing alternative to QPE-based QAE, closely related to Fourier-based phase estimation methods such as algorithmic shadow spectroscopy. The algebraic derivation of Eq. (14) is largely correct, and the no-Hadamard version in Eq. (16) is a useful variant. However, the central peak-extraction claim is affected by a 2π periodicity ambiguity that is not addressed, and the prescribed initial state is defined in a basis that depends on the unknown angle θ. The numerical simulations plot S(x) in windows around the known exact value and therefore do not demonstrate an end-to-end estimation procedure. These are load-bearing issues that require substantial revision.","major_comments":[{"comment":"Because t is an integer, S(x) defined in Eq. (11) is 2π-periodic in x. The exact evaluation of Eq. (14) contains not a single Gaussian pair but a Poisson sum over k of pairs centered at x = ±2mθ + 2πk. Consequently, a scan over any fundamental interval only determines 2mθ modulo 2π. For example, with the paper's own parameters θ = 0.6 and m = 6, the peak at 2mθ = 7.2 has a periodic copy at 7.2 − 2π ≈ 0.917 rad, so a scan of x ∈ [0, 2π) would place its maximum near 0.917 and infer θ ≈ 0.076, not 0.6. The assumption 0 ≤ θ ≤ π/2 is not used to select among replicas, and the paper gives no multi-m consistency rule or phase-unwrapping procedure. The numerical demonstrations in Figs. 3 and 5 circumvent this by plotting x-windows that contain the known exact value. The protocol must specify how θ is extracted uniquely, e.g., via multiple magnification factors or an explicit restriction on m, and this issue also affects Eq. (16).","section":"Section II B, Eq. (14)"},{"comment":"The initial state |ψ0⟩ = (|y+⟩ + |y−⟩)/√2 is defined in terms of the eigenstates |y±⟩ of σ_y in the two-dimensional subspace H spanned by |ψ⟩ and |φ⟩. This basis is fixed by the unknown angle θ: in the basis where |ψ⟩ = |1⟩ and |φ⟩ = cos θ|1⟩ + sin θ|0⟩, the state |ψ0⟩ equals |0⟩, whose preparation requires knowing θ or otherwise constructing the component of |φ⟩ orthogonal to |ψ⟩. The manuscript provides no circuit for preparing |ψ0⟩ from the assumed state-preparation unitaries Uψ and Uφ. The derivation of Eq. (14) relies on Eq. (A4), which is only valid for this specific state. If the accessible probe is instead |ψ⟩ or |φ⟩, a separate calculation is needed; although those states happen to give the same peak positions, the paper should either provide a preparation procedure for |ψ0⟩ or reformulate the protocol with an explicitly accessible initial state and verify Eqs. (14) and (16) for it.","section":"Section II A, Eq. (9) and Appendix A, Eq. (A4)"},{"comment":"The variance-minimization rule m = Nπ/(2tθ) is circular: it requires knowledge of θ, which is exactly the quantity the algorithm is designed to estimate. The paper does not explain how a user can choose m without already having an estimate of θ, nor does it state which value of t should be used in Eq. (26) when S(x) involves a sum over t. In addition, m must be an integer number of applications of A, while Eq. (26) gives a real number that requires rounding or an explicit integer constraint. As written, the claim that one can reduce shot noise by adjusting m is not an implementable prescription.","section":"Section IV B, Eq. (26)"},{"comment":"The numerical simulations do not implement the full estimation task: they compute S(x) from the exact known overlaps and compare the resulting curves with vertical lines at the known values of 2mθ and 4mθ, over x-ranges chosen to contain those values. This does not test the extraction step, because the periodic-replica ambiguity and the finite-sampling estimation of the peak position are not addressed. To validate the method, the authors should simulate the complete procedure: sample the overlaps with finite shots, scan a fundamental interval, apply the proposed unwrapping or multi-m rule, and report the resulting θ estimates and their errors.","section":"Section III, Figs. 3–8"}],"minor_comments":[{"comment":"In the shot-noise demonstration the paper sets θ = 3/2, which lies outside the assumed domain 0 ≤ θ ≤ π/2 for amplitude estimation stated after Eq. (7); please clarify whether this example concerns observable estimation (where 0 ≤ θ ≤ π is allowed) or correct the value.","section":"Section IV B, Fig. 7"},{"comment":"The inequality Σ_t e^{-a²t² + i(x±2mθ)t} ≤ Σ_t e^{-a²t²} is not a valid inequality for complex quantities; it should be written with absolute values and justified by the triangle inequality.","section":"Appendix B, Eq. (B3)"},{"comment":"The statement that truncation 'does not change the correct position of the peak' should be qualified: for a finite T, the peak is at x = 2mθ modulo 2π, so the same periodicity ambiguity as in Eq. (14) remains; the truncation only affects the width and height of the peak.","section":"Section IV A, Eq. (20)"},{"comment":"The sentence 'we will demonstrate that our algorithm does not even require the Hadamard test' appears before the Hadamard-test version is introduced; please rephrase to avoid confusion about which version is being discussed.","section":"Section II B"},{"comment":"There are several minor presentation issues: 'trunction' should be 'truncation'; the parameter values written as '1/20√2' and '1/10√2' are ambiguous and should be typeset as 1/(20√2) and 1/(10√2); and negative magnification factors such as m = −1 in Fig. 5 require a definition of A^{−m}.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The core idea is a straightforward Fourier-spectroscopy approach to amplitude estimation, and the algebraic derivation is mostly sound. The main obstacles are the unaddressed 2π periodicity ambiguity, the unspecified preparation of the initial state, and the circular variance-optimization rule. These are fixable in principle, but the revision will need new material: a phase-unwrapping or multi-m estimation procedure, a clear statement of which initial states are actually preparable, and end-to-end simulations that do not use the known θ when locating peaks. I would also encourage the authors to compare the measurement complexity with existing methods such as IQAE and robust amplitude estimation, since the current text does not quantify the advantage claimed in the abstract."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on arXiv:2502.05617. The core idea — extract the amplitude angle θ from the Fourier peaks of a time series of overlap measurements — is sound in the ideal case, and the algebra in Lemma 1 and Eq. (14) is essentially correct up to global phase. The no-Hadamard variant and the Gaussian-windowed summation are a reasonable incremental extension of Algorithmic Shadow Spectroscopy, and the noiseless simulations show clean peaks at the expected positions. So there is a real contribution here: a concrete circuit-light recipe for QAE that is worth thinking about.\n\nBut there are two load-bearing gaps that as written make the central extraction claim ambiguous. First, the initial state |ψ0⟩=(|y+⟩+|y−⟩)/√2 of Eq. (9) is defined in the y-basis of the unknown subspace H, which is set by the very angle θ you are trying to estimate. The paper later switches to |y−⟩ in the error analysis without reconciling this, and the derivation in Appendix A relies on the superposition. If you probe with |ψ⟩ instead, the coefficients change and the protocol as written does not apply. This is fixable — using |ψ⟩ as the probe gives ⟨ψ|A^{mt}|ψ⟩=cos(2mθt), which also yields peaks — but the paper never states this.\n\nSecond, and more serious, the discrete-time Fourier transform S(x) is 2π-periodic in x, because t is an integer. The exact sum is a train of Gaussian peaks at ±2mθ+2πk, not the single Gaussian pair in Eq. (14). The paper drops the k-sum. For its own example (θ=0.6, m=6), 2mθ=7.2 rad, whose periodic copy is 0.917 rad inside [0,2π); a scan over one period would infer θ≈0.076, not 0.6. The numerical demonstrations avoid this by plotting a window around the known exact value, which is not a legitimate protocol. The assumption 0≤θ≤π/2 does not disambiguate the periodic copies. You need a multi-m consistency rule or a constraint on mθ<π to recover θ unambiguously. The paper provides neither.\n\nThe shot-noise analysis is also incomplete: Eq. (26) tells you to choose m to minimize variance, but that requires knowing θ to compute m, and the variance of the final θ estimate is never propagated. The cutoff error bound is fine. The circuit-noise section is qualitative and consistent with Ref. [40], but does not test the full estimator.\n\nOverall: the paper is a genuine, if incomplete, attempt. The flaws are fixable in principle, but as written the protocol is not a well-defined algorithm. If the authors address the probe-state and phase-unwrapping issues and provide an end-to-end statistical guarantee, it could be a solid incremental paper. As it stands, I'd send it back for major revision rather than accept.\n\nI'd bring it to a reading group as an example of how Fourier-based post-processing can look attractive in noiseless simulation and still be ambiguous in practice. I wouldn't cite it yet.","headline":"A Fourier-extraction QAE paper with a sound core idea but two load-bearing gaps — state preparation and phase-wrap ambiguity — that make the central claim unproven as written.","tokens_in":12152,"tokens_out":7511,"would_cite":false,"duration_ms":71319,"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 claims that quantum amplitude estimation reduces to locating peaks in a Gaussian-weighted Fourier transform of measured overlap signals, bypassing quantum phase estimation.","keywords":["quantum amplitude estimation","classical post-processing","Fourier transform","amplitude amplification","quantum phase estimation","Hadamard test","observable estimation","shot noise"],"falsifier":"For a known amplitude, say $\\theta=0.6$, run the protocol starting from the accessible probe state $|\\psi\\rangle$ rather than the $y$-basis superposition $|\\psi_0\\rangle=(|y_+\\rangle+|y_-\\rangle)/\\sqrt{2}$; the derivation of Eq. (14) uses $\\mathrm{Tr}[|y_\\pm\\rangle\\langle y_\\pm|\\psi_0\\rangle\\langle\\psi_0|]=1/2$, so with $|\\psi\\rangle$ the coefficients are unequal and the peak at $x=\\pm2m\\theta$ should shift or disappear. A numerical simulation of Eq. (11) for this initial state would settle whether the central claim depends on the $y$-basis preparation.","tokens_in":10987,"feed_emoji":"📈","tokens_out":16357,"duration_ms":138770,"temperature":0.7,"pith_summary":"This paper claims that estimating the overlap $|\\langle\\psi|\\phi\\rangle|^2=\\cos^2\\theta$ does not require the quantum phase estimation (QPE) circuit. A quantum computer prepares an initial state, applies powers of an amplification operator $A=e^{i2\\theta\\sigma_y}$, and outputs overlap signals; a classical computer then Fourier-transforms the Gaussian-weighted signal $S(x)=\\sum_t e^{-a^2t^2}\\langle\\psi_0|A^{mt}|\\psi_0\\rangle e^{ixt}$ and reads $\\theta$ from the peak positions $x=\\pm 2m\\theta$, or from $x=\\pm 4m\\theta$ when only absolute-squared overlaps are available. This removes controlled unitaries, the quantum Fourier transform, and, in the absolute-square variant, even the Hadamard test. Numerical simulations on 4-qubit amplitudes and 6-qubit observable estimation support the claim, and the error analysis covers truncation, shot noise, and two-qubit depolarizing noise. If correct, the method offers a hybrid route to amplitude estimation on shallow circuits.","feed_headline":"Quantum amplitudes read off Fourier peaks, no QPE circuit","feed_subtitle":"Peak of a Gaussian-weighted overlap signal recovers the amplitude classically, with fewer qubits and gates.","key_machinery":"The central object is the Gaussian-windowed Fourier signal $S(x)$ together with the $y$-basis decomposition of the amplification operator. Because $A^{mt}=e^{2imt\\theta}|y_+\\rangle\\langle y_+|+e^{-2imt\\theta}|y_-\\rangle\\langle y_-|$, and the initial state $|\\psi_0\\rangle=(|y_+\\rangle+|y_-\\rangle)/\\sqrt{2}$ weights the two branches equally, the trace $\\langle\\psi_0|A^{mt}|\\psi_0\\rangle$ becomes $\\cos(2mt\\theta)$. The summation over $t$ against the Gaussian window then evaluates in closed form to two Gaussians centered at $\\pm2m\\theta$ (Eq. 14), and the absolute-square version centers at $\\pm4m\\theta$ and $0$ (Eq. 16). The paper also notes that choosing $|y_-\\rangle$ as the initial state produces a cosine series whose peak remains at $2m\\theta$, so the peak location is stable under the choice of $y$-basis state.","core_discovery":"On the paper's own terms, the discovery is that the angle $\\theta$ defining the quantum amplitude is printed directly into the Fourier profile of a sequence of overlap measurements. In the two-dimensional subspace spanned by $|\\psi\\rangle$ and $|\\phi\\rangle$, the amplification operator is $A=\\exp(i2\\theta\\sigma_y)$. If the initial state is the equally weighted superposition of the $\\sigma_y$ eigenstates, $|\\psi_0\\rangle=(|y_+\\rangle+|y_-\\rangle)/\\sqrt{2}$, then the trace $\\langle\\psi_0|A^{mt}|\\psi_0\\rangle=\\cos(2mt\\theta)$ is a clean harmonic signal. After multiplying by the Gaussian window $e^{-a^2t^2}$ and summing over $t$, the identity $$S(x)=\\sum_{t=-\\infty}^{\\infty} $e^{{-a^2t^2}}$\\langle\\psi_0|$A^{{mt}}$|\\psi_0\\rangle $e^{{ixt}}$=\\frac{\\sqrt{\\pi}}{2a}\\big($e^{{-(x+2m\\theta)^2/4a^2}}$+$e^{{-(x-2m\\theta)^2/4a^2}}$\\big)$$ places Gaussian peaks at $x=\\pm2m\\theta$, so locating a peak gives $\\theta$ directly. Replacing the overlap by its absolute square yields the same extraction from peaks at $x=\\pm4m\\theta$ and $x=0$ without a Hadamard test. The central claim is that amplitude estimation reduces to peak-finding in a classically computed Fourier curve.","pith_inferences":["The paper leaves implicit that a practical run needs a recipe for preparing the $y$-basis superposition $|\\psi_0\\rangle$ from an unknown amplitude; a natural testable extension is to execute the protocol with the accessible probe states $|\\psi\\rangle$ or $|\\phi\\rangle$ and observe how the Fourier peak degrades.","In the absolute-square variant, the central peak at $x=0$ can mask the true peaks at $\\pm4m\\theta$ when the product $m\\theta$ is small; a practical extension would add a peak-discrimination rule or a two-step sweep in $m$ to separate the central and shifted peaks.","The error formulas suggest a design rule, namely choose $m$ so the peak separation $2m\\theta$ exceeds the Gaussian width set by $a$ and choose $T$ from $\\epsilon_c \\le \\frac{2}{a}e^{-a^2T^2}$; turning this into an explicit resource-optimization step is left for future work.","The setup could in principle estimate several amplitudes in one pass by using several initial-state preparations or magnification factors and separating their Fourier peaks, but the paper does not discuss this multiplexing."],"forward_implications":["Amplitude estimation can be run with shorter circuits: the quantum part applies powers of the amplification operator and measures an overlap, while the Fourier analysis is classical.","The same peak-location procedure estimates expectation values of Pauli strings by writing $P_i|\\psi\\rangle=\\cos\\theta|1\\rangle+\\sin\\theta|0\\rangle$.","Truncating the $t$-sum is not fatal: the cutoff error is bounded by $\\frac{2}{a}e^{-a^2T^2}$, and even the minimal range $[-1,1]$ leaves the peak at $2m\\theta$.","Shot noise can be controlled by choosing the magnification factor $m$ so that $\\cos(2mt\\theta)=\\pm1$, which minimizes the variance of the real part of the overlap estimator.","Moderate two-qubit depolarizing noise lowers the peak height but does not move the peak position."],"supporting_citations":[{"why":"Sets up the standard quantum amplitude estimation based on quantum phase estimation that this work aims to avoid.","marker":"[18]"},{"why":"Provides the reflection-operator construction used to define the rotation operator $U=\\exp(i\\theta\\sigma_y)$.","marker":"[35]"},{"why":"Supplies the cooling-function technique that lets the infinite Fourier sum be evaluated within a finite range.","marker":"[37]"},{"why":"Documents that the Gaussian window performs best among the tested cooling functions, justifying $p(t)=e^{-a^2t^2}$.","marker":"[38]"},{"why":"Establishes the classical-Fourier-extraction context and the gate-noise parameter $\\xi\\approx1$ used to interpret the circuit-noise simulations.","marker":"[40]"},{"why":"Provides the quantum-circuit simulator used to model two-qubit depolarizing noise in the numerical tests.","marker":"[39]"}],"fun_headline_variants":["Quantum amplitude from Fourier peaks, no QPE circuit","Fourier peak-finding replaces QPE for amplitude estimation","Classical post-processing reads quantum amplitude directly","No quantum Fourier transform: amplitude via classical peaks","Amplitude estimation turned into a peak-finding problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithm assumes that the initial state $|\\psi_0\\rangle$ can be prepared with known, balanced overlap with the two eigenstates of the amplification operator in the subspace spanned by the states whose overlap is being estimated, and the paper gives no procedure for this preparation.","fun_headline_variants_meta":{"raw":{"variants":["Quantum amplitude from Fourier peaks, no QPE circuit","Fourier peak-finding replaces QPE for amplitude estimation","Classical post-processing reads quantum amplitude directly","No quantum Fourier transform: amplitude via classical peaks","Amplitude estimation turned into a peak-finding problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00062,"raw_usage":{"total_tokens":2898,"prompt_tokens":989,"completion_tokens":1909,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":1833}},"tokens_in":605,"tokens_out":1909,"duration_ms":13161,"temperature":1.0,"reasoning_tokens":1833,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:37:02.904326+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a known amplitude, say $\\theta=0.6$, run the protocol starting from the accessible probe state $|\\psi\\rangle$ rather than the $y$-basis superposition $|\\psi_0\\rangle=(|y_+\\rangle+|y_-\\rangle)/\\sqrt{2}$; the derivation of Eq. (14) uses $\\mathrm{Tr}[|y_\\pm\\rangle\\langle y_\\pm|\\psi_0\\rangle\\langle\\psi_0|]=1/2$, so with $|\\psi\\rangle$ the coefficients are unequal and the peak at $x=\\pm2m\\theta$ should shift or disappear. A numerical simulation of Eq. (11) for this initial state would settle whether the central claim depends on the $y$-basis preparation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets up the standard quantum amplitude estimation based on quantum phase estimation that this work aims to avoid."},{"cited_title":"Grinko, J","cited_arxiv_id":null,"evidence_quote":"Provides the reflection-operator construction used to define the rotation operator $U=\\exp(i\\theta\\sigma_y)$."},{"cited_title":"Giurgica-Tiron, I","cited_arxiv_id":null,"evidence_quote":"Supplies the cooling-function technique that lets the infinite Fourier sum be evaluated within a finite range."},{"cited_title":"Rall and B","cited_arxiv_id":null,"evidence_quote":"Provides the quantum-circuit simulator used to model two-qubit depolarizing noise in the numerical tests."}],"review_version":1}