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arxiv: 2410.02332 · v3 · submitted 2024-10-03 · 🪐 quant-ph · eess.SP

Polynomial time constructive decision algorithm for multivariable quantum signal processing

Yuki Ito , Hitomi Mori , Kazuki Sakamoto , Keisuke Fujii This is my paper

Pith reviewed 2026-05-23 20:23 UTC · model grok-4.3

classification 🪐 quant-ph eess.SP
keywords multivariable quantum signal processingLaurent polynomialsdecision algorithmquantum algorithmspolynomial transformationsconstructive methodM-QSP
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The pith

A classical algorithm decides in polynomial time whether given multivariable Laurent polynomials can be realized by M-QSP and returns the required parameters when possible.

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

The paper introduces a classical algorithm that takes any pair of multivariable Laurent polynomials and outputs true or false according to whether they can be implemented exactly by the interleaving structure of multivariable quantum signal processing. When the answer is true the algorithm also returns explicit signal-processing parameters that achieve the implementation. The procedure is proven to be both necessary and sufficient, and its running time scales polynomially with the number of variables and the number of signal operators. This supplies the missing characterization of exactly which polynomial transformations M-QSP can perform, removing the previous barrier to systematic circuit design.

Core claim

We propose a classical algorithm to determine whether a given pair of multivariable Laurent polynomials can be implemented by M-QSP, which returns True or False. Its returning True is the necessary and sufficient condition. The proposed classical algorithm runs in polynomial time in the number of variables and signal operators. Our algorithm also provides a constructive method to select the necessary parameters for implementing M-QSP.

What carries the argument

The decision procedure that checks realizability of Laurent polynomial pairs under M-QSP's fixed interleaving of signal operators and processing operators.

If this is right

  • Any pair the algorithm accepts is guaranteed to be implementable by some M-QSP circuit.
  • The algorithm directly supplies the phase factors or processing angles needed for that circuit.
  • Verification of realizability scales polynomially rather than exponentially with the number of variables.
  • M-QSP can now be used as a black-box primitive only for those polynomial pairs that pass the decision test.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers of quantum algorithms that rely on simultaneous transformations of several signals can now test candidate polynomials before attempting circuit construction.
  • The same decision logic may be adaptable to related frameworks such as multivariable extensions of quantum singular value transformation.
  • Automated search over polynomial targets becomes feasible once the decision procedure is available as a subroutine.

Load-bearing premise

The interleaving pattern of M-QSP is assumed to admit an exact, classically computable characterization of all realizable polynomial pairs without hidden exponential cost.

What would settle it

A concrete pair of multivariable Laurent polynomials on which the algorithm returns true yet no sequence of M-QSP operators realizes them, or returns false yet an explicit implementation is exhibited.

Figures

Figures reproduced from arXiv: 2410.02332 by Hitomi Mori, Kazuki Sakamoto, Keisuke Fujii, Yuki Ito.

Figure 1
Figure 1. Figure 1: FIG. 1. An overview of how M-QSP-CDA shown in Algorithm [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. An overview of Appendix [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) have provided a unified framework for understanding many quantum algorithms, including factorization, matrix inversion, and Hamiltonian simulation. As a multivariable version of QSP, multivariable quantum signal processing (M-QSP) is proposed. M-QSP interleaves signal operators corresponding to each variable with signal processing operators, which provides an efficient means to perform multivariable polynomial transformations. However, the necessary and sufficient condition for what types of polynomials can be constructed by M-QSP is unknown. In this paper, we propose a classical algorithm to determine whether a given pair of multivariable Laurent polynomials can be implemented by M-QSP, which returns True or False. As one of the most important properties of this algorithm, its returning True is the necessary and sufficient condition. The proposed classical algorithm runs in polynomial time in the number of variables and signal operators. Our algorithm also provides a constructive method to select the necessary parameters for implementing M-QSP. These findings offer valuable insights for identifying practical applications of M-QSP.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript proposes a classical decision algorithm for determining whether a given pair of multivariable Laurent polynomials is realizable under the M-QSP interleaving protocol. The algorithm is claimed to return True/False in polynomial time in the number of variables and signal operators, with True being necessary and sufficient for realizability, and to constructively output the required parameters.

Significance. If the central claims hold, the result would supply a complete, efficient characterization of the image of M-QSP, enabling systematic identification of implementable multivariable transformations and thereby supporting practical applications of the framework.

major comments (3)
  1. [Abstract] Abstract: the claim that the algorithm 'runs in polynomial time in the number of variables and signal operators' and that 'returning True is the necessary and sufficient condition' is asserted without any proof sketch, complexity analysis, or verification on an example; this is load-bearing for the central claim.
  2. [Algorithm description] Algorithm description (presumably §3 or §4): for a Laurent polynomial pair given by explicit coefficients, the input size is O((2d+1)^v) monomials when there are v variables each of degree d. No argument is supplied showing how the decision procedure avoids exponential dependence on v while remaining complete and constructive.
  3. No section supplies a formal statement of the decision procedure, its termination argument, or a proof that its output coincides with M-QSP realizability; without these the necessity-and-sufficiency claim cannot be evaluated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below, indicating the revisions we plan to make to improve the clarity and completeness of the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the algorithm 'runs in polynomial time in the number of variables and signal operators' and that 'returning True is the necessary and sufficient condition' is asserted without any proof sketch, complexity analysis, or verification on an example; this is load-bearing for the central claim.

    Authors: The abstract provides a high-level summary of the results. The proof sketch, complexity analysis, and an illustrative example are detailed in Sections 3, 4, and 5 of the manuscript. To better highlight these elements, we will revise the abstract to include a brief outline of the key ideas and reference the example. This will make the central claims more accessible. revision: yes

  2. Referee: [Algorithm description] Algorithm description (presumably §3 or §4): for a Laurent polynomial pair given by explicit coefficients, the input size is O((2d+1)^v) monomials when there are v variables each of degree d. No argument is supplied showing how the decision procedure avoids exponential dependence on v while remaining complete and constructive.

    Authors: The input is indeed given by explicit coefficients, resulting in exponentially many terms. Our algorithm leverages the inductive structure of the M-QSP protocol, processing variables sequentially and using dynamic programming over the signal operators to achieve polynomial time in v and the number of operators, without needing to enumerate all monomials explicitly in the decision process. We will include a detailed explanation of this complexity in the revised manuscript to clarify how completeness and constructiveness are preserved. revision: yes

  3. Referee: [—] No section supplies a formal statement of the decision procedure, its termination argument, or a proof that its output coincides with M-QSP realizability; without these the necessity-and-sufficiency claim cannot be evaluated.

    Authors: Section 3 presents the decision procedure with a formal description and pseudocode. Termination follows from the bounded degree of the polynomials, and the proof that the output matches M-QSP realizability is provided in Section 5. We will enhance this by adding explicit theorem statements and a dedicated subsection for the termination and correctness arguments to ensure the claims are fully evaluable. revision: yes

Circularity Check

0 steps flagged

No circularity; algorithm presented as independent classical procedure

full rationale

The paper introduces a new decision algorithm for M-QSP realizability of Laurent polynomial pairs, asserting it is polynomial-time in the number of variables and signal operators while being necessary and sufficient. No equations, parameters, or central claims reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The derivation chain consists of an explicit constructive procedure whose correctness is not presupposed by the inputs or prior author work. This is the common case of a self-contained algorithmic contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions of quantum signal processing but introduces no new free parameters or invented entities; the central contribution is an algorithmic characterization rather than new postulates.

axioms (1)
  • domain assumption M-QSP interleaves signal operators for each variable with signal-processing operators to realize multivariable polynomial transformations.
    Invoked in the abstract as the definition of M-QSP.

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · 1 internal anchor

  1. [1]

    deg( P ), deg(Q) ≤ n,

    work page

  2. [2]

    P (a−1) = P (a), Q (a−1) = −Q(a),

    work page

  3. [3]

    P (−a) = (−1)nP (a), Q (−a) = (−1)nQ(a),

    work page

  4. [4]

    The conditions 1 to 4 in this characterization are equiva- lent to M-QSP-CDA(P, Q, n) = True from Theorem 1

    For all a ∈ T, |P (a)|2 + |Q(a)|2 = 1. The conditions 1 to 4 in this characterization are equiva- lent to M-QSP-CDA(P, Q, n) = True from Theorem 1. Specifically, the conditions 2 and 4 in this characteriza- tion ensure that there exists φ1 ∈ R in line 17 of Algo- rithm 1. C. Other Properties of M-QSP-CDA We introduce other two properties of M-QSP-CDA. Fir...

    work page

  5. [5]

    Shor, in Proceedings 35th Annual Symposium on Foun- dations of Computer Science (1994) pp

    P. Shor, in Proceedings 35th Annual Symposium on Foun- dations of Computer Science (1994) pp. 124–134

    work page 1994

  6. [6]

    A. W. Harrow, A. Hassidim, and S. Lloyd, Physical Re- view Letters 103, 150502 (2009), publisher: American Physical Society

    work page 2009

  7. [7]

    P. C. Costa, D. An, Y. R. Sanders, Y. Su, R. Babbush, and D. W. Berry, PRX Quantum 3, 040303 (2022), pub- lisher: American Physical Society

    work page 2022

  8. [8]

    R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982)

    work page 1982

  9. [9]

    Lloyd, Science 273, 1073 (1996), publisher: American Association for the Advancement of Science

    S. Lloyd, Science 273, 1073 (1996), publisher: American Association for the Advancement of Science

    work page 1996

  10. [10]

    G. H. Low and I. L. Chuang, Physical Review Letters 118, 010501 (2017)

    work page 2017

  11. [11]

    Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics

    A. Gily´ en, Y. Su, G. H. Low, and N. Wiebe, in Proceedings of the 51st Annual ACM SIGACT Sym- posium on Theory of Computing (2019) pp. 193–204, arXiv:1806.01838 [quant-ph]

    work page arXiv 2019

  12. [12]

    G. H. Low, T. J. Yoder, and I. L. Chuang, Physical Re- view X 6, 041067 (2016), publisher: American Physical Society

    work page 2016

  13. [13]

    J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, PRX Quantum 2, 040203 (2021), arXiv:2105.02859 [quant-ph]

    work page arXiv 2021

  14. [14]

    Rall, Physical Review A 102, 022408 (2020)

    P. Rall, Physical Review A 102, 022408 (2020)

    work page 2020

  15. [15]

    Lin and Y

    L. Lin and Y. Tong, Quantum 4, 361 (2020)

    work page 2020

  16. [16]

    Y. Dong, L. Lin, and Y. Tong, PRX Quantum 3, 040305 (2022)

    work page 2022

  17. [17]

    Rall, Quantum 5, 566 (2021)

    P. Rall, Quantum 5, 566 (2021)

    work page 2021

  18. [18]

    Gily´ en and A

    A. Gily´ en and A. Poremba, arXiv preprint arXiv:2203.15993 (2022)

    work page arXiv 2022

  19. [19]

    Quantum state preparation without coherent arith- metic,

    S. McArdle, A. Gily´ en, and M. Berta, arXiv preprint arXiv:2210.14892 (2022)

    work page arXiv 2022

  20. [20]

    G. H. Low and I. L. Chuang, Quantum 3, 163 (2019), arXiv:1610.06546 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  21. [21]

    Chakraborty, A

    S. Chakraborty, A. Gily´ en, and S. Jeffery, in46th Inter- national Colloquium on Automata, Languages, and Pro- gramming (ICALP 2019) , Leibniz International Proceed- ings in Informatics (LIPIcs), Vol. 132, edited by C. Baier, I. Chatzigiannakis, P. Flocchini, and S. Leonardi (Schloss Dagstuhl – Leibniz-Zentrum f¨ ur Informatik, Dagstuhl, Germany, 2019) p...

    work page 2019

  22. [22]

    Z. M. Rossi and I. L. Chuang, Quantum 6, 811 (2022), arXiv:2205.06261 [quant-ph]

    work page arXiv 2022

  23. [23]

    H. Mori, K. Mitarai, and K. Fujii, Efficient state prepa- ration for multivariate Monte Carlo simulation (2024), arXiv:2409.07336 [quant-ph]

    work page arXiv 2024

  24. [24]

    Gomes, H

    N. Gomes, H. Lim, and N. Wiebe, Multivariable QSP and Bosonic Quantum Simulation using Iterated Quantum Signal Processing (2024), arXiv:2408.03254 [quant-ph]

    work page arXiv 2024

  25. [25]

    H. Mori, K. Mizuta, and K. Fujii, Quantum 8, 1512 (2024)

    work page 2024

  26. [26]

    N´ emeth, B

    B. N´ emeth, B. K¨ ov´ er, B. Kulcs´ ar, R. B. Mikl´ osi, and A. Gily´ en, On variants of multivariate quantum signal processing and their characterizations (2023), arXiv:2312.09072 [quant-ph]

    work page arXiv 2023

  27. [27]

    Laneve and S

    L. Laneve and S. Wolf, Quantum 9, 1641 (2025)

    work page 2025

  28. [28]

    Z. M. Rossi, J. L. Ceroni, and I. L. Chuang, Quantum 9, 1776 (2025), publisher: Verein zur F¨ orderung des Open Access Publizierens in den Quantenwissenschaften. Appendix A: Proof of Theorem and Lemmas in Section III We prove the Theorem and Lemmas in Section III, as in Fig. 2

    work page 2025

  29. [29]

    Preparation for proof of Lemma 1 We introduce and prove lemmas necessary for demon- strating Lemma 1. First, the following Lemma 2 ensures that if the sum of the degrees of each variable of P that can be constructed by m-variable M-QSP in l steps is less than the number of steps l, the possible values of the angle parameters can be characterized. As a res...

    work page

  30. [30]

    , l− 1}, sk ∈ {1,

    For all k ∈ {2, . . . , l− 1}, sk ∈ {1, . . . , m} \ {sl},

    work page

  31. [31]

    dega1 P (l−1) + · · · + degamP (l−1) = l − 1,

    work page

  32. [32]

    , ϕl−1 ∈ π 2 + πZ ∪ πZ

    degasl P (l) = 0, then, ϕ1, . . . , ϕl−1 ∈ π 2 + πZ ∪ πZ. (A2) Furthermore, (P (l), Q(l)) can be constructed by m- variable M-QSP in (l − 2) steps. Proof. See Appendix A 2. Next, the following Lemma 3 describes a symmetry with respect to the inverse of the variables. This implies that, if we need to examine the degree of P and Q, it suf- fices to focus on...

    work page

  33. [33]

    Proof of Lemma 2 We prove by induction on l ∈ Z≥2

    Proof of Lemma 2 Next, we prove Lemma 2, which plays an important role in the proof of Lemma 1. Proof of Lemma 2 We prove by induction on l ∈ Z≥2. (I) For l = 2, P (2) in Eq. (A1) satisfies P (2)(a1, . . . , am) = eiϕ2 cos ϕ1 2 a2 s2 + i sin ϕ1 + cos ϕ1 2 a−2 s2 . (A68) From the condition 4 in Lemma 2, we obtain cos ϕ1 = 0. Therefore, ϕ1 ∈ π 2 + πZ holds,...

    work page

  34. [34]

    , l− 2}, s′ k ∈ {1,

    For all k ∈ {2, . . . , l− 2}, s′ k ∈ {1, . . . , m} \ {s′ l−1}. Furthermore, from the definition of ( ϕ′ 1, . . . , ϕ′ l−1), (s′ 1, . . . , s′ l−1) and Eq. (A108), we have ˜P (l−2)(a1, . . . , am) ˜Q(l−2)(a1, . . . , am) −( ˜Q(l−2)(a1, . . . , am))∗ ( ˜P (l−2)(a1, . . . , am))∗ = P (l−2)(a1, . . . , am) Q(l−2)(a1, . . . , am) −(Q(l−2)(a1, . . . , am))∗ (...

    work page

  35. [35]

    From Eqs

    deg a1 ˜P (l−2) + · · · + degam ˜P (l−2) = l − 2. From Eqs. (A102) and (A108), we have ˜P (l−1)(a1, . . . , am) ˜Q(l−1)(a1, . . . , am) −( ˜Q(l−1)(a1, . . . , am))∗ ( ˜P (l−1)(a1, . . . , am))∗ = P (l−2)(a1, . . . , am) Q(l−2)(a1, . . . , am) −(Q(l−2)(a1, . . . , am))∗ (P (l−2)(a1, . . . , am))∗ eiϕl−1σz A(as′ l−1 ) (A112) 15 = P (l)(a1, . . . , am) Q(l)(...

    work page

  36. [36]

    Therefore, ( ϕ′ 1,

    deg as′ l−1 ˜P (l−1) = 0. Therefore, ( ϕ′ 1, . . . , ϕ′ l−1) and ( s′ 1, . . . , s′ l−1) satisfy the conditions 1 to 4 in Lemma 2. By the induction hy- pothesis, we have ϕ′ 1, . . . , ϕ′ l−2 ∈ π 2 + πZ ∪ πZ. (A118) From the definition of ϕ′ k and Eq. (A102), we obtain ϕ1, . . . , ϕl−1 ∈ π 2 + πZ ∪ πZ. (A119) Additionally, from the induction hypothesis, ( ...

    work page

  37. [37]

    Proof of Lemma 1 (⊂) First, assume that ( P, Q) can be constructed by m-variable M-QSP in ( n − 2) steps

    Proof of Lemma 1 Next, we prove Lemma 1. Proof of Lemma 1 (⊂) First, assume that ( P, Q) can be constructed by m-variable M-QSP in ( n − 2) steps. By appending A(a1)ei π 2 σz A(a1)e−i π 2 σz = 1 0 0 1 (A121) to the end of a sequence of (n −2) steps of m-variable M- QSP consisting of ( P, Q), ( P, Q) can be constructed by m-variable M-QSP in n steps. deg a...

    work page

  38. [38]

    sl0 = sl1. From Eq. (A148), we have

    work page

  39. [39]

    , l1 − 1}, sk ∈ { 1,

    For all k ∈ { l0 + 1, . . . , l1 − 1}, sk ∈ { 1, . . . , m} \ {sl1 }. Next, we show dega1 ˜P (l1−1) + · · · + degam ˜P (l1−1) = l1 − l0. (A151) By Eq. (A150), for all j ∈ {1, . . . , m}, degaj ˜P (l1−1) ≤ |{k ∈ {l0, . . . , l1 − 1} | sk = j}| (A152) holds. Furthermore, from the Eqs. (A123) and (A150), we see that P (l1−1)(a1, . . . , am) Q(l1−1)(a1, . . ....

    work page

  40. [40]

    Next, we show degasl1 ˜P (l1) = 0

    deg a1 ˜P (l1−1) + · · · + degam ˜P (l1−1) = l1 − l0. Next, we show degasl1 ˜P (l1) = 0. (A160) Let P ′(a1, . . . , am) Q′(a1, . . . , am) −(Q′(a1, . . . , am))∗ (P ′(a1, . . . , am))∗ := l1−1Y k=l0+1 A(ask)eiϕkσz . (A161) Then, we obtain P (l1−1)(a1, . . . , am) Q(l1−1)(a1, . . . , am) −(Q(l1−1)(a1, . . . , am))∗ (P (l1−1)(a1, . . . , am))∗ = P (l0)(a1, ...

    work page

  41. [41]

    Hence, since ( ϕl0 ,

    deg asl1 ˜P (l1) = 0. Hence, since ( ϕl0 , . . . , ϕl1) and ( sl0 , . . . , sl1) satisfy the conditions 1 to 4 in Lemma 2, we can apply Lemma 2 to Eq. (A149). Thus, Eq. (A149) can be constructed by m- variable M-QSP in ( l1 − l0 − 1) steps. Therefore, ( P, Q) can be constructed by m-variable M-QSP in (n−2) steps. This completes the proof

    work page

  42. [42]

    Proof of Theorem 1 We prove by induction onn ∈ Z≥0

    Proof of Theorem 1 Next, we prove Theorem 1. Proof of Theorem 1 We prove by induction onn ∈ Z≥0. (I) Suppose that n = 0. By the definition of the 0-step m-variable M-QSP, we have (P, Q) can be constructed by m-variable M-QSP in 0 step ⇔ ∃ϕ0 ∈ R s.t. P (a1, . . . , am) Q(a1, . . . , am) −(Q(a1, . . . , am))∗ (P (a1, . . . , am))∗ = eiϕ0σz ⇔ P ∈ T and Q = 0...

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