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REVIEW 4 major objections 5 minor 19 references

Application of an upsampling algorithm to quantum state preparation of continuous and discrete probability distributions

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read An upsampling construction prepares quantum states encoding sampled probability distributions, with quantum forking reducing gate depth to polylogarithmic.

desk verdict The continuous half is a sound but non-novel re-derivation of Mottonen-style state prep; the discrete half rests on a false premise and a normalization error, so the paper's main extension claim fails. read the letter →

arxiv 2504.14349 v2 pith:LPNM22UY submitted 2025-04-19 quant-ph

R. P. Erickson This is my paper · ORCID
classification quant-ph
keywords quantumstatepreparationupsamplingalgorithmprobabilitydistributionspolylogarithmicgatedepthforkingGaussiandistributionLaplacebinomial
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 derives a quantum state-preparation routine that loads a sampled probability distribution into the amplitudes of an $n$-qubit state. The construction treats the distribution as a periodic upsampled function, factors the amplitudes into products of sines and cosines, and arranges those factors as a divide-and-conquer circuit of controlled $y$-rotations. Because that circuit has exponential gate count, the paper applies quantum forking to produce an equivalent circuit with polylogarithmic gate depth. The method is worked out for continuous distributions on the whole real line, demonstrated on Gaussian and Laplace examples, and extended to arbitrary discrete distributions such as the binomial. If the derivation holds, it offers a general, cheap state-loading primitive for quantum algorithms that need repeated reloading, including financial and scientific applications.

What carries the argument

The load-bearing object is the upsampling envelope $\xi_n(x)$ with its period-splitting identity: splitting the period $w$ into halves turns the envelope into products of $\cos(\theta_{n,m}(i)/2)$ and $\sin(\theta_{n,m}(i)/2)$ factors. That factorization is what converts a smooth target distribution into a concrete circuit of controlled $y$-rotations. The second piece of machinery is quantum forking, which reuses the controlled rotations across a shared control qubit and ancillas to compress the circuit's time complexity from exponential to polylogarithmic gate depth. For the discrete extension, the same envelope is formed from Dirac-delta densities placed at the bitstring labels, and the reduction to $\theta_{n,1}=2\arccos\sqrt{P(i)}$, $\theta_{n,m}=\pi/2$ for $m>1$ is what turns the general circuit into a discrete state-preparation routine.

What would settle it

Calculate the squared norm of the discrete state in Eq. (33) for $n>1$: with the prefactor $1/\sqrt{2^n-1}$ it equals $2^{n-1}/(2^n-1)$, not $1$, so a simulation of the $n=3$ binomial circuit would reveal measured probabilities that miss the target distribution by that factor. For the continuous branch, measure bitstring frequencies from the prepared Gaussian state and compare them with $P(x_n)\delta x_n$; any systematic deviation beyond sampling error would refute the preparation claim.

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Extended reading notes

Core claim

The paper's central claim is that a target probability density $P(x)$ can be captured by a periodic envelope $\xi_n(x) = \sqrt{\delta x_n \sum_{j=-\infty}^{\infty} P(x+jw)}$, whose period $w$ contains the $2^n$ sample points of an $n$-qubit register. Repeatedly applying the identity $\sum_j P(x+jy) = \sum_j P(x+2jy) + \sum_j P(x+(2j+1)y)$ splits the envelope into a product of cosine and sine factors, so each qubit is prepared by controlled $y$-rotations with angles $\theta_{n,m}(i)$. The same formula yields the discrete branch: for a probability distribution $P(i)$ over bitstrings, the paper introduces a complementary-probability auxiliary state and finds that the rotation angles reduce to $2\arccos\sqrt{P(i)}$ on the leading qubit and $\pi/2$ on all others, leaving a circuit whose Hadamard gates and controlled rotations prepare the desired amplitudes. Applied to Gaussian, Laplace, and binomial examples, this is the construction the paper claims achieves polylogarithmic depth after quantum forking.

Load-bearing premise

The discrete branch rests on the premise that the divide-and-conquer circuit of Fig. 2 cannot prepare arbitrary real-amplitude states directly, which is why a complementary-probability auxiliary state with a particular normalization factor is introduced; if that premise fails, the discrete extension loses its justification.

Editorial extensions

If this is right

  • For any univariate distribution supported on the whole real line, the same angle formulas prepare the corresponding state; the paper names Cauchy and Student's $t$ as immediate follow-ons to Gaussian and Laplace.
  • Distributions on the positive half-line, such as lognormal, are covered after a logarithmic change of variables that maps their support onto $\mathbb{R}$.
  • Arbitrary discrete distributions over $2^n$ outcomes can be loaded with the discrete circuit, demonstrated on the binomial distribution.
  • Because quantum states cannot be cloned, a calculation that consumes the state must reload it repeatedly; cutting preparation depth to polylogarithmic makes that reloading affordable in practice.
  • The univariate construction is stated to extend to multivariate probability distributions.

Reading between the lines

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

  • If the period-splitting identity is read in reverse, the same circuit gives a natural tomography dual: measured amplitudes could be inverted to reconstruct a sampled PDF, a use the paper does not pursue.
  • The $\zeta$ shift in the sampling window could support an ensemble strategy — prepare several states at different $\zeta$ values and average measurement statistics — to suppress aliasing artifacts beyond the single-window construction the paper presents.
  • The normalization factor in the discrete auxiliary state is one of the first things to check in an independent re-derivation: requiring unit norm fixes the prefactor at $2^{-(n-1)}$ rather than $1/(2^n-1)$, and the downstream angle formulas would remain unchanged.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a quantum state-preparation method based on time-series upsampling. For a continuous probability density P(x) with support on the real line, the authors define a sampling lattice over a window w, derive a normalization factor δx_n, and express the target state as a product of cosine rotations parameterized by angles θ_{n,m}; the resulting divide-and-conquer circuit has exponential gate count, which they then convert to polylogarithmic depth using quantum forking. The same formalism is extended to discrete probability distributions by preparing a two-branch auxiliary state |ψ_n> = (|0>|φ_{n-1}> + |1>|φ*_{n-1}>)/√(2^n−1). Examples include Gaussian, Laplace, and binomial distributions.

Significance. The continuous part is a coherent and explicit construction: Eq. (16) correctly solves the normalization condition, Eqs. (8)-(11) give correct limiting behaviors, and the angle prescription (26) is parameter-free in the sense that all angles are computed directly from the target PDF. The paper also provides explicit small circuits, which is helpful. However, the advertised extension to 'arbitrary' discrete distributions is the central novelty of the abstract, and that extension is invalid: the proposed state is not normalized, Eq. (38) is inconsistent with Eq. (33), the premise that Fig. 2 cannot prepare arbitrary real-amplitude states is false, and the construction has only half the required degrees of freedom. Because these defects are in the load-bearing discrete section, the paper's main claim is not established.

major comments (4)
  1. [Section II.B, Eqs. (33)-(34) and Appendix A, Eq. (A7)] The state |ψ_n> defined in Eq. (33) is not normalized for n>1: because |φ_{n-1}> has squared norm Σ P(i_{n-1}) and |φ*_{n-1}> has squared norm 2^{n-1}−Σ P(i_{n-1}), the squared norm of (33) is 2^{n-1}/(2^n−1). Correspondingly, the PDF in Eq. (34) integrates to 2^{n-1}/(2^n−1), not to unity. The correct prefactor would be 2^{−(n−1)/2}, and the same erroneous 1/(2^n−1) factor appears in Appendix A, Eq. (A7).
  2. [Section II.B, preamble to Eq. (33)] The claim that 'It is not possible to prepare (28) from the set of unitary gate operations depicted in Fig. 2, except when n=1' is incorrect. The circuit of Fig. 2 contains 2^n−1 controlled Ry rotations, which is the standard parameter count for arbitrary real-amplitude state preparation; the Mottonen et al. construction cited as [9] uses exactly such uniformly controlled Ry gates. Thus the stated motivation for the auxiliary-state construction (33) is unsupported, and the apparent need for the two-branch state disappears.
  3. [Section II.B, Eq. (38)] Equation (38) does not follow from substituting Eq. (37) into Eq. (25). For m>1, each factor in the product in (25) evaluates to cos(π/4)=1/√2 independent of the bit, so the product contributes an overall factor 2^{−(n−1)/2}; Eq. (38) instead has prefactor 1/√(2^n−1). Consequently (38) is not equal to (33), and even after correcting the prefactor the state would have amplitudes √P(i_{n−1})/2^{(n−1)/2} in the first half and √(1−P(i_{n−1}))/2^{(n−1)/2} in the second half, which is not the arbitrary distribution |φ_n> of Eq. (28). The construction contains only 2^{n−1} free parameters P(i_{n−1}), not the 2^n−1 parameters needed for arbitrary discrete probabilities.
  4. [Section III.B.1, Fig. 6] The binomial demonstration reports only the projection onto the |0> state of the most significant qubit. This is not the full n-qubit state promised by the abstract; it discards half of the amplitudes, and a projection of a normalized state need not match the target distribution. The example therefore does not verify preparation of the 2^n-point binomial distribution.
minor comments (5)
  1. [Abstract and Section I] The word 'polylogarthmic' is misspelled; it should be 'polylogarithmic'.
  2. [Section II.B, text before Eq. (29)] The phrase 'the compliment of P(0)' should read 'the complement of P(0)'.
  3. [Section II.C] The number of ancilla qubits is given as 'd = 2n− 1'; from context and Refs. [10,11] this should be d = 2^n−1.
  4. [Fig. 1(a) caption and Eq. (3)] The caption says the sampling window is centered about x = x̄ + wζ/2, while Eq. (3) gives x_0 = x̄ + w(ζ−1)/2; these statements are inconsistent.
  5. [Eq. (25)] The notation i_{n−m} is not defined explicitly; it would help to state that it denotes the integer formed by the lower n−m bits.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the state-preparation angles are computed from the target probability distributions by explicit closed-form formulas, and no fitted parameter is relabeled as a prediction.

full rationale

The paper's continuous derivation is self-contained. Equation (26) gives theta_{n,m}(i_{n-1}) directly as 2 arccos of a ratio of sums of the target PDF P(x), and the normalization factor delta x_n is fixed by the normalization condition (7), not by matching a subset of outputs. The Gaussian and Laplace examples choose standard textbook parameters (mu, sigma, b), so there are no fitted inputs. The discrete extension likewise determines its rotation angles by the explicit formula theta_{n,1}(i_{n-1}) = 2 arccos sqrt(P(i_{n-1})) in (B2)/(37); the target probabilities themselves enter as the input to the construction, which is the normal state-preparation task rather than a hidden circularity. The conversion to polylogarithmic depth relies on the external quantum-forking results of Araujo et al. and Park et al., with public source code, and is not a self-citation chain. Although the discrete-probability construction contains apparent mathematical inconsistencies (e.g., the normalization of (33) and the equality of (33) with (38) are questionable), those are correctness concerns, not circularity. No load-bearing premise is justified by citing the author's own prior work, and no quantity fitted from data is relabeled as a prediction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The continuous derivation uses a Poisson-summation-like identity and a large-window approximation; the discrete claim relies on a false impossibility premise and carries a normalization error. No data fitting occurs; w and zeta are chosen by hand. No new physical entities are introduced.

free parameters (2)
  • window width w
    Chosen 'sufficiently large' so that (9) holds; no principled selection rule is given, and the examples do not report the values used.
  • random shift zeta
    Chosen at random in [0, 1/2^{n-1}) to reduce aliasing; its value affects the prepared amplitudes but is not determined by the distribution.
assumptions (5)
  • standard math The summation identity (12) holds for all x,y and for the PDFs considered.
    Poisson summation-like identity used repeatedly in (14)-(16); valid for sufficiently regular integrable functions.
  • domain assumption The window w is chosen large enough that sum_j P(x_n + jw) is approximately P(x_n) for all sampled points (Eq. 9).
    This approximation is load-bearing for the interpretation of the prepared state's probabilities in (10)-(11). If w is not large relative to the distribution's tails, the periodic copies alias into the window.
  • domain assumption Quantum forking as described in [10,11,14] transforms the divide-and-conquer circuit to depth O(log^2 d) with the stated ancilla overhead.
    The paper relies on the external forking results without derivation; this is acceptable as prior work but not checked here.
  • domain assumption For the discrete case, Dirac deltas can be replaced by narrow Gaussians and the limit epsilon to 0 commuted with the sums (Appendices A and B).
    This is a standard distributional limiting argument, though the paper does not justify the interchange of limits and sums.
  • ad hoc to paper The circuit of Fig. 2 cannot prepare a state with arbitrary discrete amplitudes except for n=1.
    This assertion in Section II.B is load-bearing for the discrete construction, but it is false: the cited Mottonen et al. construction uses uniformly controlled rotations to prepare arbitrary real-amplitude states.

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Cite this review

Pith. "Pith review of Application of an upsampling algorithm to quantum state preparation of continuous and discrete probability distributions." pith.science (2026). https://pith.science/paper/LPNM22UY

@misc{pith2026250414349,
  author       = {Pith},
  title        = {Pith review of: Application of an upsampling algorithm to quantum state preparation of continuous and discrete probability distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPNM22UY}},
  note         = {Machine review of arXiv:2504.14349}
}
read the original abstract

Upsampling of time series data is often associated with classical fast Fourier transform analysis, but it also can be employed in the preparation of a quantum state vector. The quantum circuit of preparation derived from functional data sampled in this way exhibits exponential gate depth, like other divide-and-conquer algorithms, but is transformable to an equivalent circuit of polylogarithmic gate depth. In this study, we derive a polylogarthmic upsampling algorithm for construction of a state vector with amplitudes that are square roots of probabilities sampled from a continuous probability distribution having support over the entire real line. As examples, we prepare state vectors associated with Gaussian and Laplace distributions; state vectors for other continuous probability distributions such as Cauchy and Student's t follow in a similar manner. We also extend the algorithm to include preparation of a state vector whose amplitudes are square roots of an arbitrary distribution of discrete probabilities. Our analyses focus on univariate distributions, but can be extended to multivariate forms. The polylogarithmic upsampling algorithm has financial and scientific application.

Figures

Figures reproduced from arXiv: 2504.14349 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Upsampling capture of PDF, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Circuit diagram for the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. shows the circuit diagram for preparing the state vector of (38); gates corresponding to π/2 rotation angle reduce to Hadamard gates, H, and even multiples of these gates reduce further to identity operations. It is easy to see that (33) and (38) are both expressions of the same state vector, |ψn⟩. For example, comparing the two expressions, we have |ϕn⟩ = X in cos  θ(in) 2  |in⟩ = X in p P(in)|in⟩. (40) C. The Qu… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Gaussian distributed probabilities corresponding [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Plot of the binomial distribution obtained from [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Reference graph

Works this paper leans on

19 extracted references · 15 canonical work pages

  1. [9]

    Grover and T

    L. Grover and T. Rudolph, Creating superpositions that correspond to efficiently integrable probability dis- tributions (2002), arXiv:quant-ph/0208112 [quant-ph]; A. Carrera Vazquez and S. Woerner, Phys. Rev. Appl. 15, 034027 (2021)

    arXiv 2002

  2. [1]

    (43) Figure 4(a) shows the sampling measurement of a state vector prepared using the quantum-forking circuit of Fig

    The Gaussian Distribution The univariate normal, or Gaussian, probability den- sity can be expressed as P (x|µ,σ ) = 1 σ √ 2π exp " −(x−µ)2 2σ2 # , (41) where the upsampling center of capture on the line of support is ¯x =µ, with µ = Z ∞ −∞ xP (x|µ,σ )dx, (42) σ2 = Z ∞ −∞ (x−µ)2P (x|µ,σ )dx. (43) Figure 4(a) shows the sampling measurement of a state vecto...

    work page

  3. [2]

    (46) Figure 5(a) shows the sampling measurement of a state vector prepared using the quantum-forking circuit of Fig

    The Laplace Distribution The Laplace probability distribution can be expressed as P (x|µ,b ) = 1 2b exp −|x−µ| b , (44) where the upsampling center of capture on the line of support is ¯x =µ, with µ = Z ∞ −∞ xP (x|µ,b )dx, (45) b2 = 1 2 Z ∞ −∞ (x−µ)2P (x|µ,b )dx. (46) Figure 5(a) shows the sampling measurement of a state vector prepared using the quantum-...

    work page

  4. [3]

    The Lognormal Distribution Also, the upsampling algorithm can be applied to con- tinuous probability distributions that can be mapped to FIG. 5. (a) Laplace distributed probabilities corresponding to a state vector prepared via the quantum-forking circuit of panel (b). The original upsampling circuit, prior to conversion to the quantum-forking circuit, co...

    work page

  5. [4]

    Zoufal, A

    C. Zoufal, A. Lucchi, and S. Woerner, npj Quantum Inf. 5, 103 (2019); E. Y. Zhu, S. Johri, D. Bacon, M. Esen- can, J. Kim, M. Muir, N. Murgai, J. Nguyen, N. Pisenti, A. Schouela, K. Sosnova, and K. Wright, Phys. Rev. Res. 4, 043092 (2022)

    work page 2019

  6. [5]

    P(kn−1) ∞X j=−∞ e−(w2/8ϵ2)[2j+(in−kn−1)/2n−1] 2 + (1− P(kn−1)) ∞X j=−∞ e−(w2/8ϵ2)[2j−1+(in−kn−1)/2n−1] 2 #)−1 . (A3) Substituting (A3) into (A1) gives ξn(in) = X kn−1

    The Binomial Distribution As an example of a discrete case we consider the bino- mial distribution. Here, we define binomial probabilities in the form P(k,l ) = l k pkql−k ; k∈{ 0, 1, 2,...,l }, (48) where the support is the set of integers k, such that 0≤ k≤ l, and we define probabilities 0 ≤ p,q ≤ 1, where p +q = 1. 7 FIG. 6. (a) Plot of the binomial di...

    work page

  7. [6]

    M. A. Nielsen and I. L. Chuang, Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion (Cambridge University Press, 2010) pp. 24–25

    work page 2010

  8. [7]

    Aaronson, Nat

    S. Aaronson, Nat. Phys. 11, 291 (2015); J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature 549, 195 (2017)

    work page 2015

Show all 19 references
  1. [8]

    composed of controlled y-rotation gates [9], with ex- ponential gate depth, but it can be transformed to an equivalent circuit of polylogarthmic gate depth [10] us- ing a method of quantum forking [11]. In our analysis we consider the specific univariate examples of Gauss and ...

    2025 arXiv

  2. [10]

    Y. R. Sanders, G. H. Low, A. Scherer, and D. W. Berry, Phys. Rev. Lett. 122, 020502 (2019); J. Bausch, Quan- tum 6, 773 (2022)

    2019

  3. [11]

    Lubasch, J

    M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Phys. Rev. A 101, 010301 (2020); K. Plekhanov, M. Rosenkranz, M. Fiorentini, and M. Lubasch, Quantum 6, 670 (2022); I. Y. Akhal- waya, A. Connolly, R. Guichard, S. Herbert, C. Kargi, A. Krajenbrink, M. Lubasch, C. M. K...

    2020 arXiv

  4. [12]

    X. Sun, G. Tian, S. Yang, P. Yuan, and S. Zhang, IEEE Transactions on Computer-Aided Design of Inte- grated Circuits and Systems 42, 3301 (2023); X.-M. Zhang, T. Li, and X. Yuan, Phys. Rev. Lett. 129, 230504 (2022); A. G. Rattew and B. Koczor, Preparing ar- bitrary continuous ...

    2023 arXiv

  5. [13]

    T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to algorithms , 4th ed. (The MIT Press,

  6. [14]

    M¨ ott¨ onen, J

    M. M¨ ott¨ onen, J. J. Vartiainen, V. Bergholm, and M. M. Salomaa, Quantum Info. Comput. 5, 467–473 (2005)

    2005

  7. [15]

    I. F. Araujo, D. K. Park, F. Petruccione, and A. J. da Silva, Sci. Rep. 11, 6329 (2021)

    2021

  8. [16]

    D. K. Park, I. Sinayskiy, M. Fingerhuth, F. Petruccione, and J.-K. K. Rhee, New J. of Phys. 21, 083024 (2019)

    2019

  9. [17]

    Kitaev and W

    A. Kitaev and W. A. Webb, Wavefunction prepara- tion and resampling using a quantum computer (2009), arXiv:0801.0342 [quant-ph]

    2009 arXiv

  10. [18]

    If ζ were varied continuously from 0 to 1 /2n−1 then er- godic coverage of the entire PDF over the breadth of the sample window would be achieved

  11. [19]

    A. J. da Silva/Centro de Inform´ atica UFPE, Github— adjs/dcsp, https://github.com/adjs/dcsp (July 2020), accessed: November 2024

    2020

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