REVIEW 4 major objections 5 minor 47 references
Classical post-processing approach for quantum amplitude estimation
T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (4)
- [Section II B, Eq. (14)] 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 II A, Eq. (9) and Appendix A, Eq. (A4)] 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 IV B, Eq. (26)] 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 III, Figs. 3–8] 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.
minor comments (5)
- [Section IV B, Fig. 7] 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.
- [Appendix B, Eq. (B3)] 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 IV A, Eq. (20)] 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 II B] 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.
- [Throughout] 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}.
Circularity Check
The prescribed initial state and the variance-minimizing magnification factor are both defined through the unknown angle θ, so the numerical peak "predictions" reduce by construction; the central Fourier extraction itself retains independent mathematical content.
-
self definitional
[Section II.A, Eqs. (9)-(10); Appendix A, Eq. (A4)]
"Given an initial state |ψ0⟩ = 1/√2(|y+⟩ +|y−⟩), where |y±⟩, corresponding to eigenvalues ±1, represent the eigenstates of Pauli operator σy in the subspace H ... Am|ψ0⟩ = 1/√2(exp(i2mθ)|y+⟩ +exp(−i2mθ)|y−⟩)."
Lemma 1 fixes the subspace basis by |ψ⟩=|1⟩ and |φ⟩=cosθ|1⟩+sinθ|0⟩, so |y±⟩=(|0⟩±i|1⟩)/√2 and hence |ψ0⟩=(|y+⟩+|y−⟩)/√2=|0⟩=(|φ⟩−cosθ|ψ⟩)/sinθ. Preparing |ψ0⟩ therefore requires knowing θ, the quantity the algorithm is supposed to estimate. The signal whose peaks are read off as ±2mθ in Eq. (14) can only be generated if θ is already known; the numerical demonstrations choose this initial state using the known θ, so the observed peak agreement with 2mθ is enforced by construction rather than by an estimation procedure.
-
self definitional
[Section IV.B, Eq. (26)]
"we can conclude that the variance of the overlap ⟨y−| Amt|y−⟩ becomes minimal when the magnification factor satisfies m = Nπ/(2tθ), where N = 1, 2, 3,... . This is a desirable property in QAE."
The recommended value of m is an explicit function of the unknown angle θ. To apply this variance-minimization recipe one must already know the answer the algorithm is designed to find; without a preliminary estimate of θ, m cannot be set, and with such an estimate the prescription merely restates the target. This is a parameter choice defined in terms of the sought quantity, so the claimed noise-reduction advantage is circular as stated.
full rationale
The core Fourier post-processing step is not circular: for any state with equal overlap on the two σy eigenstates, ⟨ψ0|A^{mt}|ψ0⟩=cos(2mtθ), and the Gaussian-weighted discrete-time Fourier transform yields peaks at ±2mθ (modulo 2π). That analytic relation has independent content and is not obtained by fitting. However, the paper's prescribed probe |ψ0⟩=(|y+⟩+|y−⟩)/√2 is the state |0⟩ of the H-basis introduced in Lemma 1, where |0⟩=(|φ⟩−cosθ|ψ⟩)/sinθ; preparing it requires θ. The numerical simulations accordingly select this state using the known θ, so their peak agreement with 2mθ is by construction. Likewise, the shot-noise minimization recipe m=Nπ/(2tθ) requires the unknown θ as an input, so the recommended parameter choice presupposes the answer. A minor self-citation for the Gaussian window (Ref. [38], with overlapping first author) is not load-bearing because any decaying symmetric window would preserve the peak locations. A separate correctness flaw—Eq. (14) omits the 2π-periodic replicas of the discrete-time Fourier transform, making peak-based θ extraction ambiguous without an unwrapping rule—is noted but is not counted as circularity under the definitional-equivalence criterion. Overall, two load-bearing steps reduce by construction, giving partial circularity.
Assumptions & free parameters
free parameters (4)
- Gaussian smoothing parameter a =
1/(20*sqrt(2)) and 1/(10*sqrt(2)) in examples
- Summation cutoff T =
40 and 60 in examples
- Magnification factor m =
varies from 1 to 22 in figures
- Classical x-grid resolution =
not specified
assumptions (5)
- domain assumption The Gaussian window p(t) = e^{-a^2 t^2} is an appropriate cooling function and gives the best performance.
- ad hoc to paper The probe state |psi0> is an equal superposition of the eigenstates |y+> and |y-> of the amplification operator A.
- domain assumption Finite truncation of the summation does not shift the peak position of S(x).
- ad hoc to paper The variance of the overlap estimator is minimized when alpha = cos(2mt theta) = +/-1, and beta can then be inferred from alpha.
- standard math The two-dimensional subspace reduction with |psi> = |1> and |phi> = cos(theta)|1> + sin(theta)|0> captures the full n-qubit problem.
Cite this review
Pith. "Pith review of Classical post-processing approach for quantum amplitude estimation." pith.science (2026). https://pith.science/paper/DR55RDFF
@misc{pith2026250205617,
author = {Pith},
title = {Pith review of: Classical post-processing approach for quantum amplitude estimation},
year = {2026},
howpublished = {\url{https://pith.science/paper/DR55RDFF}},
note = {Machine review of arXiv:2502.05617}
}
read the original abstract
We propose an approach for quantum amplitude estimation (QAE) designed to enhance computational efficiency while minimizing the reliance on quantum resources. Our method leverages quantum computers to generate a sequence of signals, from which the quantum amplitude is inferred through classical post-processing techniques. Unlike traditional methods that use quantum phase estimation (QPE), which requires numerous controlled unitary operations and the quantum Fourier transform, our method avoids these complex and resource-demanding steps. By integrating quantum computing with classical post-processing techniques, our method significantly reduces the need for quantum gates and qubits, thus optimizing the utilization of quantum hardware. We present numerical simulations to validate the effectiveness of our method and provide a comprehensive analysis of its computational complexity and error. This hybrid strategy not only improves the practicality of QAE but also broadens its applicability in quantum computing.
Figures
Figures from the paper (5 more)
Reference graph
Works this paper leans on
- [40]
-
[1]
Select an initial state |ψ0⟩ which has a reasonable overlap with the eigenstates |y±⟩ in the subspace H
-
[2]
(11) and set a specific magnification factor, i.e., m =M
Choose an appropriate summation range [−T,T ] as defined in Eq. (11) and set a specific magnification factor, i.e., m =M
-
[3]
Evolve the selected initial state|ψ0⟩ with amplifica- tion operator Amt in a quantum computer, where t must be a integer and range from −T to T
-
[4]
Perform measurements to obtain ⟨ψ0| Amt|ψ0⟩ or |⟨ψ0| Amt|ψ0⟩| 2, and multiply it by eixt for the parameter x
-
[5]
2 4 6 8 10 12 14 16 0.0 0.2 0.4 0.6 0.8 1.0 x S(x) m=2 m=4 m=6 m=8 m=10 m=12 FIG
Compute the summation S(x) = ∑T t=−Tf(t)eixt by a classical computer. 2 4 6 8 10 12 14 16 0.0 0.2 0.4 0.6 0.8 1.0 x S(x) m=2 m=4 m=6 m=8 m=10 m=12 FIG. 3. The quantum amplitude of two 4-qubit state |ψ⟩ and |ϕ⟩. In the figure, the red vertical line indicates the exact value of the 2mθ, while the light blue line represents S(x) as evaluated with our method....
-
[6]
We use a cutoff of the sum range from [−∞,∞] to [−60, 60] when evaluatingS(x) in Eq. (14). We set the exact value of the angle as θ = 0.6 and change the magnification factor m from 2 to 12. As shown in Fig. 3, the location of peaks matches well with the exact value of angle 2mθ. To further demonstrate the robustness of our method, we randomly select six p...
-
[7]
We apply a cutoff in the summation range from [−∞,∞] to [−60, 60] when evaluatingS(x) in Eq.(14). As shown in the figure, the location of the peaks in S(x) closely matches the exact value of 2 mθ, demon- strating the accuracy and reliability of our method in the context of randomly selected quantum states. B. Estimate the observable Our method can also be...
Show all 47 references
-
[8]
P. W. Shor, in Proc. 35th Annual Symp. on Foundations of Computer Science (IEEE, 1994) pp. 124–134
1994
-
[9]
M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, Vol. 2 (Cambridge University Press Cambridge, 2001)
2001
-
[10]
Rebentrost, B
P. Rebentrost, B. Gupt, and T. R. Bromley, Physical Review A 98, 022321 (2018)
2018
-
[11]
Woerner and D
S. Woerner and D. J. Egger, npj Quantum Information 5, 15 (2019)
2019
-
[12]
D. J. Egger, R. G. Guti´ errez, J. C. Mestre, and S. Wo- erner, IEEE Transactions on Computers70, 2136 (2020)
2020
-
[13]
Stamatopoulos, D
N. Stamatopoulos, D. J. Egger, Y. Sun, C. Zoufal, R. Iten, N. Shen, and S. Woerner, Quantum 4, 291 (2020)
2020
-
[14]
Wiebe, A
N. Wiebe, A. Kapoor, and K. M. Svore, Quantum In- formation & Computation 15, 316 (2015)
2015
-
[15]
Wiebe, A
N. Wiebe, A. Kapoor, and K. M. Svore, Quantum In- formation & Computation 16, 541 (2016)
2016
-
[16]
Kerenidis, J
I. Kerenidis, J. Landman, A. Luongo, and A. Prakash, Advances in neural information processing systems 32 (2019)
2019
-
[17]
Kapoor, N
A. Kapoor, N. Wiebe, and K. Svore, Advances in neural information processing systems 29 (2016)
2016
-
[18]
T. Li, S. Chakrabarti, and X. Wu, in International Con- ference on Machine Learning (PMLR, 2019) pp. 3815– 3824
2019
-
[19]
Miyahara, K
H. Miyahara, K. Aihara, and W. Lechner, Physical Re- view A 101, 012326 (2020)
2020
-
[20]
Knill, G
E. Knill, G. Ortiz, and R. D. Somma, Physical Review A—Atomic, Molecular, and Optical Physics 75, 012328 (2007)
2007
-
[21]
Kassal, S
I. Kassal, S. P. Jordan, P. J. Love, M. Mohseni, and A. Aspuru-Guzik, Proceedings of the National Academy of Sciences 105, 18681 (2008)
2008
-
[22]
Gacon, C
J. Gacon, C. Zoufal, and S. Woerner, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, 2020) pp. 47–55
2020
-
[23]
Montanaro, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, 20150301 (2015)
A. Montanaro, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, 20150301 (2015)
2015
-
[24]
Miyamoto and K
K. Miyamoto and K. Shiohara, Physical Review A 102, 022424 (2020)
2020
-
[25]
Brassard, P
G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Con- temporary Mathematics 305, 53 (2002)
2002
-
[26]
A. Y. Kitaev, arXiv preprint quant-ph/9511026 (1995)
1995 arXiv
-
[27]
Preskill, Quantum 2, 79 (2018)
J. Preskill, Quantum 2, 79 (2018)
2018
-
[28]
Lin and Y
L. Lin and Y. Tong, PRX Quantum 3, 010318 (2022)
2022
-
[29]
R. D. Somma, New Journal of Physics 21, 123025 (2019)
2019
-
[30]
Clinton, J
L. Clinton, J. Bausch, J. Klassen, and T. Cubitt, New Journal of Physics 25, 033027 (2023)
2023
-
[31]
T. E. O’Brien, B. Tarasinski, and B. M. Terhal, New Journal of Physics 21, 023022 (2019)
2019
-
[32]
A. E. Russo, K. M. Rudinger, B. C. Morrison, and A. D. Baczewski, Physical review letters 126, 210501 (2021)
2021
-
[33]
Zhang, G
R. Zhang, G. Wang, and P. Johnson, Quantum 6, 761 (2022)
2022
-
[34]
Suzuki, S
Y. Suzuki, S. Uno, R. Raymond, T. Tanaka, T. Onodera, and N. Yamamoto, Quantum Information Processing 19, 1 (2020)
2020
-
[35]
Grinko, J
D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, npj Quantum Information 7, 52 (2021)
2021
-
[36]
Aaronson and P
S. Aaronson and P. Rall, in Symposium on simplicity in algorithms (SIAM, 2020) pp. 24–32
2020
-
[37]
Giurgica-Tiron, I
T. Giurgica-Tiron, I. Kerenidis, F. Labib, A. Prakash, and W. Zeng, Quantum 6, 745 (2022)
2022
- [38]
-
[39]
Rall and B
P. Rall and B. Fuller, Quantum 7, 937 (2023)
2023
-
[41]
Plekhanov, M
K. Plekhanov, M. Rosenkranz, M. Fiorentini, and M. Lubasch, Quantum 6, 670 (2022)
2022
-
[42]
Bravyi, D
S. Bravyi, D. P. DiVincenzo, and D. Loss, Annals of physics 326, 2793 (2011)
2011
-
[43]
A. N. Chowdhury, Y. Subasi, and R. D. Somma, arXiv preprint arXiv:1803.02466 (2018)
2018 arXiv
-
[44]
P. Zeng, J. Sun, and X. Yuan, arXiv preprint arXiv:2109.15304 (2021)
2021 arXiv
-
[45]
Y. Yang, Y. Li, X. Xu, and X. Yuan, Physical Review A 109, 052416 (2024)
2024
-
[46]
Jones and S
T. Jones and S. Benjamin, Quantum Science and Tech- nology 5, 034012 (2020)
2020
-
[47]
H. H. S. Chan, R. Meister, M. L. Goh, and B. Koczor, PRX Quantum 6, 010352 (2025). 9 Appendix A: Derivation Process of Eq. (14) This section details the derivation process of Eq.(14). S(x) = ∞∑ t=−∞ e−a2t2 Tr[Amt|ψ0⟩⟨ψ0|]eixt = Tr[ ∞∑ t=−∞ Amt|ψ0⟩⟨ψ0|e−a2t2 eixt]. (A1) Here, w...
2025
Reviewed August 8, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.