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REVIEW 4 major objections 6 minor 1 cited by

Generalized tensor transforms and their applications in classical and quantum computing

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that replacing a fixed tensor-product transform with a tensor product of a tunable unitary $W^{\otimes n}$ lets quantum states and functions be compressed to a few coefficients, with higher reconstruction fidelity than…

desk verdict Correct math, standard construction, overclaimed applications: the compression and encoding results are circular because W is fitted to the data. read the letter →

arxiv 2507.02420 v2 pith:U62OE6TH submitted 2025-07-03 quant-ph

Alok Shukla , Prakash Vedula This is my paper
classification quant-ph MSC 65T5042C1081P68 PACS 03.67.Ac
keywords generalizedtensortransformsWalsh-HadamardtransformquantumstatecompressionfunctionencodingdigitalsignalprocessingfastunitaryKroneckerproductadaptivesparsity
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

This paper introduces generalized tensor transforms (GTTs), built as the $n$-fold tensor product $W^{\otimes n}$ of an arbitrary $b\times b$ unitary $W$, and argues that tuning $W$ lets one match the transform basis to a specific quantum dataset. The authors prove that the resulting basis functions are orthonormal, give a fast classical algorithm with $O(N\log_b N)$ cost, and give a quantum circuit implementation whose gate count and depth are $O(\log_b N)$, quadratically better than the quantum Fourier transform on $n$ qudits. The paper's central claim is that GTT-based compression and function encoding outperform fixed transforms such as the Walsh-Hadamard and Fourier transforms, reaching higher reconstruction fidelity with fewer retained components. If true, this gives a practical way to tailor spectral representations to quantum data.

What carries the argument

The central object is the GTT matrix $G_N = W^{\otimes n}$, the $n$-fold tensor (Kronecker) product of a $b\times b$ unitary base matrix $W$, with $N=b^n$; its columns are the GTT basis functions, recursively defined by $f_j^{(b^n)}(x)=W_{\lfloor bx\rfloor,\lfloor j/b^{n-1}\rfloor}\,f_{j\bmod b^{n-1}}^{(b^{n-1})}(\{bx\})$. When $W$ is the $2\times2$ Hadamard matrix this reproduces Walsh functions in natural order, and when $W$ is the DFT matrix it gives the multidimensional Fourier transform. The tunable parameters of $W$ (for qubits, the three angles of a general single-qubit unitary) are what let the basis adapt to data. The fast algorithm exploits separability of the Kronecker product: reshape, apply $W$ along the last index, recurse on the columns, giving $O(N\log_b N)$ classical work; the same structure becomes an $O(\log_b N)$-depth quantum circuit in natural ordering.

What would settle it

Take a random $n$-qubit state that is not constructed from a GTT basis, optimize $W$ on a training batch of similar states, and test the optimized transform on a held-out state; if the retained-$k$ fidelity on held-out states is no better than Hadamard or QFT, the adaptive-sparsity claim fails. A simpler check is to verify whether the fully quantum compression protocol's success probability, which scales as the retained norm squared, remains acceptable when the dominant indices are not known in advance.

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Extended reading notes

Core claim

The central claim is that replacing a fixed tensor-product transform with a tensor product of a tunable unitary $W$ yields a family of orthonormal, piecewise-constant Walsh-type bases, and that choosing $W$ to fit the data concentrates information into fewer coefficients. The paper proves orthonormality of the basis functions, exhibits the matrix $G_N = W^{\otimes n}$, derives a butterfly algorithm costing $O(N\log_b N)$ classically, and shows that the quantum circuit in natural spectral ordering costs $O(\log_b N)$ gates and depth. Its numerical demonstrations compare retaining $k=2$ coefficients of an eight-dimensional state: GTT fidelities are $1.0000$, $0.9797$, and $0.9637$ for the three test states, against $0.5685$-$0.5815$ for Hadamard and $0.4845$-$0.5001$ for QFT/FFT. For a 16-point function encoding, GTT fidelities reach $0.9181$, $0.9848$, and $0.9964$ for $k=4,8,12$, versus $0.2377$, $0.5386$, and $0.8425$ for Hadamard. The authors state that the advantage follows from adaptive sparsity: the tunable basis is selected to make the state or function sparse, which fixed transforms cannot do.

Load-bearing premise

The applications assume that the input state or function is already sparse in some GTT basis $W^{\otimes n}$, and that a suitable $W$ can be found without knowing the full state; for arbitrary data no selection method is demonstrated.

Editorial extensions

If this is right

  • For states or functions that are sparse in some GTT basis, transmission and storage can be reduced from $n$ qudits to $\lceil\log_b k\rceil$ qudits plus classical side information about the $k$ retained indices.
  • The $O(\log_b N)$ gate count and circuit depth give GTT-based filtering a quadratic improvement over QFT-based quantum filtering, when input state preparation and measurement are not the bottleneck.
  • Classically, the fast GTT provides an $O(N\log_b N)$ transform for any unitary base $W$, matching FFT and FWHT complexity while adding tunable basis shapes.
  • The compression examples show that for the same retained-component budget, GTT can reconstruct with fidelity near unity where Hadamard and QFT retain under sixty percent of the state information.

Reading between the lines

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

  • The paper leaves open how to find a good $W$ for arbitrary data; a natural testable extension is to optimize $W$ on a training set of states and measure whether the resulting basis compresses unseen states from the same family.
  • Because $W^{\otimes n}$ has only $O(b^2)$ tunable parameters, not every sparse vector can be made sparse in a GTT basis; an interesting boundary question is which sparsity patterns are reachable by a single shared base matrix.
  • The same adaptive-sparsity idea might transfer to classical image and audio compression, where fixed wavelets or DCT bases currently dominate, though the paper only demonstrates quantum-state and function examples.
  • The hybrid and fully quantum protocols assume the dominant indices are known or cheaply discoverable; without that assumption, amplitude-estimation overhead can erase the communication advantage, as the paper itself acknowledges.
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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 / 6 minor

Summary. This paper introduces Generalized Tensor Transforms (GTTs), defined as the n-fold tensor product G_N = W^{⊗n} of an arbitrary b×b unitary W. It defines recursive GTT basis functions, states and proves their orthonormality, identifies G_N as the sampled basis matrix, and gives a fast butterfly algorithm with O(N log_b N) classical complexity and O(log_b N) quantum gate count. The remainder of the paper applies the GTT to quantum state compression (Sec. 4.1), function encoding (Sec. 4.2), and low/high-pass filtering in the natural ordering (Sec. 4.3), with numerical comparisons against Hadamard and QFT/FFT.

Significance. The mathematical identification of W^{⊗n} with a unitary transform is correct, and the fast classical algorithm is a standard tensor-product butterfly; these parts would be a useful reference. The claimed applications, however, are not established by the evidence presented. The compression example uses a state 'designed to be exactly sparse in the chosen GTT basis' (Sec. 4.1.5), the function-encoding parameters are fit on the test function itself (Sec. 4.2.2), and the filtering complexity bound omits the cost of the filtering step (Sec. 4.3.3). Because no method is given for selecting W or the sparse index set S_k for arbitrary data, the advertised advantage over fixed transforms is an artifact of the experimental construction rather than a demonstrated property of the framework.

major comments (4)
  1. [Sec. 2.4.1, Eq. (2.10)] The proof of Theorem 2.2 defines the 'standard L2 inner product' as ∫ f(x)g(x)dx without complex conjugation and then uses Σ_m W_{m,j}W_{m,k}=δ_{jk} as the unitarity condition. For a complex unitary W the correct column-orthonormality identity is Σ_m \overline{W_{m,j}}W_{m,k}=δ_{jk}; the displayed bilinear form is not an inner product, and the un-conjugated identity fails even for the unitary W used in the paper's own example in Sec. 2.5.1. The theorem's statement is true if the inner product is defined with a conjugate, but as written the proof is incorrect.
  2. [Secs. 4.1.5, 4.1.6, Algorithm 2] The compression advantage is circular. The text in Sec. 4.1.5 states that the initial state 'was designed to be exactly sparse in the chosen GTT basis,' so the reported GTT fidelity of 1.0 is true by construction, and the comparison against Hadamard/QFT only demonstrates that a transform chosen for one vector outperforms fixed transforms on that vector. No procedure is given to find a suitable W or dominant index set S_k for an arbitrary input state: Algorithm 2 takes S_k as known input, and Sec. 4.1.6 concedes that discovering S_k without a priori knowledge costs O(√N) amplitude-estimation queries. Moreover, the number of tunable parameters b^2−1 is independent of N, so it is not obvious that W^{⊗n} can sparsify arbitrary N-dimensional vectors; the paper provides no condition under which this is possible.
  3. [Sec. 4.2.2, Table 2] The function-encoding benchmark is overfitting. The tunable parameters (θ,φ,λ) are optimized by scipy on the same discretized function that is then used to compute reconstruction fidelity (Step 5 of Sec. 4.2 and Table 2), while the Hadamard transform has no fitted parameters. This is not a controlled comparison, and no train/test split or out-of-sample experiment is reported. Accordingly, the higher fidelities in Table 2 do not support the abstract's claim that GTTs generally improve function encoding over fixed transforms.
  4. [Sec. 4.3.3, Algorithm 3] The claimed O(log_2 N) gate complexity and depth for GTT-based filtering excludes the dominant filtering step. Algorithm 3 Step 4 implements the passband projection by controlling the ancilla on each basis state in the passband; for cutoff c this requires c (or N−c) multi-controlled X gates, each costing O(log N) gates in the cited decomposition. For c=O(N) the total is O(N log N) gates, not O(log N), and the depth is O(N) if the gates are applied sequentially. Thus the claimed quadratic improvement over QFT-based filtering in Sec. 4.3.3 is not supported by the algorithm as written.
minor comments (6)
  1. [Sec. 2.5.1] In the computation of C1, the first integral evaluates to i/2 but the text reports −i/2; this is presumably a typo, but it should be fixed for consistency with the conjugate inner product.
  2. [Sec. 4] The word 'applicaiton' in the first paragraph should be 'application', and the sentence claiming the approach 'demonstrates the superiority' of GTT over WHT overstates what the later fitted examples can show.
  3. [Sec. 4.2] The normalization constant in the amplitude-encoding formula is denoted N, the same symbol as the transform dimension N=b^n, which makes the equation ambiguous.
  4. [Fig. 2 caption] The caption lists three values of θ (π/2, π/4, π/8) for a single array of subfigures; it should specify which θ is used for which panel.
  5. [Table 2] The header 'Ncomponents' and the text 'k=4,8,12' use inconsistent notation for the number of retained components.
  6. [Introduction and Conclusion] The claimed 'exponential advantage' over classical FFT/FWHT compares quantum gate counts with classical arithmetic operations without stating that different computational models are being compared; this should be made explicit.

Circularity Check

3 steps flagged · score 6.0 of 10

The numerical advantage claims for GTT compression and function encoding reduce to in-sample construction: states are designed to be sparse in the chosen GTT basis and encoding parameters are optimized on the test function, while the fully quantum protocol assumes the sparse index set as an input.

  1. self definitional [Section 4.1.5 (Quantum state compression for n=3 qubits)]
    "A fidelity of 1.0 indicates perfect reconstruction in this noiseless simulation, achieved because the initial state was designed to be exactly sparse in the chosen GTT basis, and all dominant components were retained."

    The test state |ψ⟩ was constructed (with W=U3(θ=π/4, φ=π/3, λ=π/6)) so that its GTT transform has only two non-zero coefficients, at indices 0 and 3. Truncating to k=2 then retains the full state by construction, giving fidelity 1.0. The comparison with Hadamard (0.5685) and QFT (0.5001) is therefore not a test of the GTT's general ability to find sparsity; it is a comparison between a basis chosen to match the state and fixed bases. The paper's advertised conclusion that 'GTTs enable improved performance in quantum state compression...achieving higher fidelities with fewer retained components' is supported only by such states that are sparse in the chosen basis by design.

  2. fitted input called prediction [Section 4.2.2 (Computational example: Function encoding) and Section 4.2 step 5 (Optimization)]
    "The GTT optimization procedure is employed to discover the optimal GTT parameters (θ,φ,λ) that maximize the fidelity of reconstruction for a specified number of retained components (k)."

    The same function f(x) is used both to select (θ,φ,λ) and to compute the reported fidelity. The GTT parameters are optimized to maximize F on this exact function, so the resulting fidelity values (0.9181–0.9964) are in-sample training scores, not predictions for unseen data. The Hadamard transform, having no free parameters, is evaluated without any analogous tuning. A basis optimized on the evaluation function will outperform a fixed basis on that function by construction, so the comparison does not establish a general advantage. The paper also provides no method for choosing (θ,φ,λ) for a new function without first knowing its representation in the resulting basis.

1 more flagged steps
  1. other [Algorithm 2 (quantum sparse state compression) and Section 4.1.6 (Computational complexity)]
    "Data: Known index set S_k = {y_0, y_1, ..., y_{k−1}} of dominant GTT coefficients. ... If the k dominant indices are not known a priori, discovering them from the quantum state through comprehensive amplitude amplification and estimation ... would typically incur a quantum query complexity scaling as O(√N) to find the dominant indices, plus O(k/ε) to estimate their amplitudes."

    The fully quantum compression protocol claims gate complexity O(k log_b N) and communication advantage O(log_b k), but these claims require S_k as an input. The paper explicitly concedes that without a priori knowledge, finding S_k costs O(√N), which dominates the advertised complexity and negates the claimed savings. The numerical demonstrations supply S_k by construction, because the initial state was designed to be sparse in the chosen basis. Thus the central 'compression' claim reduces to an assumed input that the algorithm does not derive, making the protocol's stated advantages, as presented, dependent on exactly the information the method is supposed to discover.

full rationale

The mathematical framework itself — GTT as W^⊗n, the fast classical algorithm, the O(log_b N) gate count for the tensor-product unitary when W is known, and the orthonormality proofs — is self-contained and not circular. The complexity statements are straightforward properties of tensor-product unitaries. The circularity lies in the applications and numerical demonstrations that are used to claim 'improved performance' over fixed transforms. In Sec. 4.1.5, the initial state is explicitly 'designed to be exactly sparse in the chosen GTT basis', so the fidelity comparison (1.0 vs. 0.5685/0.5001) is a tautology: the chosen basis contains the state's support by construction. In Sec. 4.2.2, the GTT parameters are optimized on the same function that is later used to compute fidelity, so the reported fidelities are in-sample fits; the fixed Hadamard transform receives no such tuning, making the comparison unbalanced by construction. Algorithm 2 and its complexity analysis assume the dominant index set S_k is known a priori, and Sec. 4.1.6 concedes that finding it costs O(√N), which undercuts the advertised quantum advantage. These are not cases of malicious self-citation; they are instances where the 'predictions' are equivalent to the inputs. The core unitary-transform result is real, but the paper's abstract claim of 'improved performance in quantum state compression and function encoding' is not established by the provided examples. Score 6 reflects partial circularity: the central mathematical claims are independent, but the numerical 'advantages' that form the headline are forced by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central mathematical construction (G_N = W^{⊗n}) is a standard Kronecker product, so no new axioms are needed for the transform itself. The load-bearing assumptions are in the applications: sparsity in a GTT basis, orderability of natural indices, and cheap multi-controlled gates. The tunable W parameters are fitted to the data, so the numerical demonstrations do not provide independent evidence.

free parameters (4)
  • GTT base matrix W parameters (θ,φ,λ) = Compression example: θ=π/4, φ=π/3, λ=π/6; encoding example: θ ∈ {0.2072,0.3242,0.1560}, φ,λ optimized in [0,2π]
    W is chosen to make the data sparse or to maximize fidelity on the test function; this is parameter fitting, not an independent prediction.
  • Sparse index set S_k = e.g., {0,3} for state S1 in Sec. 4.1.5
    The dominant coefficients are selected from the known transformed state; the result depends on this data-dependent choice.
  • Number of retained components k = k=2 for compression; k=4,8,12 for encoding
    Fidelity increases with k; the choice is arbitrary and affects conclusions.
  • Filter cutoff c = 4 for N=16 in the example; also N/2, N/4, N/8 in circuits
    Hand-chosen cutoff; complexity scales with c, but the paper claims O(log N).
assumptions (4)
  • standard math W^{⊗n} is unitary when W is unitary
    Used throughout; proven in Theorem 2.3 via column orthonormality.
  • ad hoc to paper The input state or function is sparsely representable in a GTT basis
    Assumed for all applications; in examples it is enforced by construction or by optimizing W on the same data (Secs. 4.1.5, 4.2.2).
  • ad hoc to paper Natural ordering of GTT coefficients orders components by frequency or importance for filtering
    Algorithm 3 filters by retaining low natural indices; no argument is given that natural order corresponds to a physically meaningful frequency ordering.
  • domain assumption Multi-controlled X gates with n controls can be implemented with O(n) gates and constant depth
    Cited from [46] and used for complexity claims; even if true, the number of such gates in Algorithm 3 is O(c), making total cost O(N log N) for c=O(N).

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Cite this review

Pith. "Pith review of Generalized tensor transforms and their applications in classical and quantum computing." pith.science (2026). https://pith.science/paper/U62OE6TH

@misc{pith2026250702420,
  author       = {Pith},
  title        = {Pith review of: Generalized tensor transforms and their applications in classical and quantum computing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U62OE6TH}},
  note         = {Machine review of arXiv:2507.02420}
}
abstract

We introduce a novel framework for Generalized Tensor Transforms (GTTs), constructed through an $n$-fold tensor product of an arbitrary $b \times b$ unitary matrix $W$. This construction generalizes many established transforms, by providing a adaptable set of orthonormal basis functions. Our proposed fast classical algorithm for GTT achieves an exponentially lower complexity of $O(N \log_b N)$ in comparison to a naive classical implementation that has an associated computational cost of $O(N^2)$. For quantum applications, our GTT-based algorithm, implemented in the natural spectral ordering, achieves both gate complexity and circuit depth of $O(\log_b N)$, where $N = b^n$ denotes the length of the input vector. This represents a quadratic improvement over Quantum Fourier Transform (QFT), which requires $O((\log_b N)^2)$ gates and depth for $n$ qudits, and an exponential advantage over classical Fast Fourier Transform (FFT) based and Fast Walsh-Hadamard Transform (FWHT) based methods, which incur a computational cost of $O(N \log_b N)$. We explore diverse applications of GTTs in quantum computing, including quantum state compression and transmission, function encoding and quantum digital signal processing. The proposed framework provides fine-grained control of the transformation through the adjustable parameters of the base matrix $W$. This versatility allows precise shaping of each basis function while preserving their effective Walsh-type structure, thus tailoring basis representations to specific quantum data and computational tasks. Our numerical results demonstrate that GTTs enable improved performance in quantum state compression and function encoding compared to fixed transforms (such as FWHT or FFT), achieving higher fidelities with fewer retained components. We also provided novel classical and quantum digital signal filtering algorithms based on our GTT framework.

Figures

Figures reproduced from arXiv: 2507.02420 by the authors.

Figure 1
Figure 1. Walsh basis functions in natural order labeled from [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. GTT basis functions of order N=8 for θ = π/2, π/4, π/8 are shown in subfigures (a)–(h). Note that the θ = π/2 case corresponds to Walsh basis functions in natural order shown in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Comparison of original and reconstructed signals for di [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Reconstruction of the function f (x) for N = 16 discrete points, comparing GTT and Hadamard transforms for varying numbers of retained components (k). The black ‘x’ markers represent the normal￾ized discretized input points, which lie on the normalized plot of the func…
Figure 5
Figure 5. Figure 5: Quantum filter circuits illustrating GTT-based low-pass filtering (with [PITH_FULL_IMAGE:figures/full_fig_p035_5.png]
Figure 6
Figure 6. Figure 6: Stem plots illustrating the GTT-based natural spectral components of the original, low-pass fil [PITH_FULL_IMAGE:figures/full_fig_p038_6.png]

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Reference graph

Works this paper leans on

46 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    The inequality of Erd ˝os-Turan-Koksma: Walsh and Haar functions over finite groups

    Vassil S Grozdanov and Stanislava S Stoilova. The inequality of Erd ˝os-Turan-Koksma: Walsh and Haar functions over finite groups. Math. Balkanica (NS), 19:349–366, 2005

    work page 2005

  2. [2]

    A generalised hadamard transform

    Kathy J Horadam. A generalised hadamard transform. In Proceedings. International Symposium on Information Theory, 2005. ISIT 2005., pages 1006–1008. IEEE, 2005

    work page 2005

  3. [3]

    Improved Haar and Walsh functions over triangular domains

    Ren-hong Wang and Wei Dan. Improved Haar and Walsh functions over triangular domains. Journal of the Franklin Institute, 347(9):1782–1794, 2010

    work page 2010

  4. [4]

    On generalized Walsh fourier series

    Chinami Watari. On generalized Walsh fourier series. Tohoku Mathematical Journal, Second Series, 10(3):211–241, 1958

    work page 1958

  5. [5]

    Episkoposian

    Sergo A. Episkoposian. On greedy algorithms with respect to generalized Walsh system

    work page

  6. [6]

    A generalized Walsh system and its fast algorithm

    Xixi Yuan and Zhanchuan Cai. A generalized Walsh system and its fast algorithm. IEEE Transactions on Signal Processing, 69:5222–5233, 2021

    work page 2021

  7. [7]

    Walsh functions and their applications

    Kenneth George Beauchamp. Walsh functions and their applications. Academic Press, 1975

    1975

  8. [8]

    Walsh functions for simplifying computation of blood pressure wave power spectra

    JB Shapiro and T Reich. Walsh functions for simplifying computation of blood pressure wave power spectra. Annals of Biomedical Engineering, 2(3):265–273, 1974

    work page 1974

Show all 46 references
  1. [9]

    A quantum approach for digital signal processing

    Alok Shukla and Prakash Vedula. A quantum approach for digital signal processing. The European Physical Journal Plus, 138(12):1–24, 2023. 40

    2023

  2. [10]

    Analysis of periodic data using Walsh functions

    Hilary A Broadbent and York A Maksik. Analysis of periodic data using Walsh functions. Behavior Research Methods, Instruments, & Computers, 24(2):238–247, 1992

    1992

  3. [11]

    Walsh functions and their use in the assessment of surface texture

    EH Smith and WM Walmsley. Walsh functions and their use in the assessment of surface texture. Wear, 57(1):157–166, 1979

    1979

  4. [12]

    A hybrid classical-quantum algorithm for digital image processing

    Alok Shukla and Prakash Vedula. A hybrid classical-quantum algorithm for digital image processing. Quantum Information Processing, 22(1):3, 2022

    2022

  5. [13]

    Hybrid classical-quantum image processing via polar Walsh basis functions

    Mohit Rohida, Alok Shukla, and Prakash Vedula. Hybrid classical-quantum image processing via polar Walsh basis functions. Quantum Machine Intelligence, 6(2):72, 2024

    2024

  6. [14]

    Edge detection using Walsh functions

    Frank O’Gorman. Edge detection using Walsh functions. Artificial Intelligence, 10(2):215–223, 1978

    1978

  7. [15]

    A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations

    Alok Shukla and Prakash Vedula. A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations. Applied Mathematics and Computation, 442:127708, 2023

    2023

  8. [16]

    Minimum energy control of time-delay systems via walsh functions

    KR Palanisamy and Ganti Prasada Rao. Minimum energy control of time-delay systems via walsh functions. Optimal Control Applications and Methods, 4(3):213–226, 1983

    1983

  9. [17]

    Transfer function determination from impulse response via walsh functions

    PN Paraskevopoulos and SJ Varoufakis. Transfer function determination from impulse response via walsh functions. International Journal of Circuit Theory and Applications, 8(1):85–89, 1980

    1980

  10. [18]

    Genetic algorithms and Walsh functions: Part I, A gentle introduction

    David E Goldberg. Genetic algorithms and Walsh functions: Part I, A gentle introduction. Complex systems, 3:129–152, 1989

    1989

  11. [19]

    A fast quantum mechanical algorithm for database search

    Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-eighthAnnual ACM Symposium on Theory of Computing, pages 212–219. ACM, 1996

    1996

  12. [20]

    An efficient implementation of a quantum search algorithm for arbitrary N

    Alok Shukla and Prakash Vedula. An efficient implementation of a quantum search algorithm for arbitrary N. The European Physical Journal Plus, 140(6):1–10, 2025

    2025

  13. [21]

    An efficient quantum algorithm for preparation of uniform quan- tum superposition states

    Alok Shukla and Prakash Vedula. An efficient quantum algorithm for preparation of uniform quan- tum superposition states. Quantum Information Processing, 23(2):38, 2024

    2024

  14. [22]

    Quantum complexity theory

    Ethan Bernstein and Umesh Vazirani. Quantum complexity theory. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 11–20, 1993

    1993

  15. [23]

    A generalization of Bernstein–Vazirani algorithm with multiple secret keys and a probabilistic oracle

    Alok Shukla and Prakash Vedula. A generalization of Bernstein–Vazirani algorithm with multiple secret keys and a probabilistic oracle. Quantum Information Processing, 22(6):244, 2023

    2023

  16. [24]

    Algorithms for quantum computation: discrete logarithms and factoring

    Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124–134. Ieee, 1994

    1994

  17. [25]

    Rapid solution of problems by quantum computation

    David Deutsch and Richard Jozsa. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907):553–558, 1992

    1907

  18. [26]

    On the power of quantum computation

    Daniel R Simon. On the power of quantum computation. SIAM journal on computing, 26(5):1474– 1483, 1997. 41

    1997

  19. [27]

    Quantum amplitude amplification and estimation, 2002

    Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation, 2002

    2002

  20. [28]

    Amplitude estimation without phase estimation.Quantum Information Processing, 19(2):1– 17, 2020

    Yohichi Suzuki, Shumpei Uno, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, and Naoki Ya- mamoto. Amplitude estimation without phase estimation.Quantum Information Processing, 19(2):1– 17, 2020

    2020

  21. [29]

    Efficient quantum algorithm for weighted partial sums and numer- ical integration

    Alok Shukla and Prakash Vedula. Efficient quantum algorithm for weighted partial sums and numer- ical integration. Advanced Quantum Technologies, 2025

    2025

  22. [30]

    Quantum algorithm for linear systems of equa- tions

    Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equa- tions. Physical Review Letters, 103(15):150502, 2009

    2009

  23. [31]

    A closed set of normal orthogonal functions

    Joseph L Walsh. A closed set of normal orthogonal functions. American Journal of Mathematics, 45(1):5–24, 1923

    1923

  24. [32]

    Walsh transforms

    Tom Beer. Walsh transforms. American Journal of Physics, 49(5):466–472, 1981

    1981

  25. [33]

    Walsh functions and the solution of nonlinear differential equations

    Henry F Ahner. Walsh functions and the solution of nonlinear differential equations. American Journal of Physics, 56(7):628–633, 1988

    1988

  26. [34]

    Geadah and MJG Corinthios

    Youssef A. Geadah and MJG Corinthios. Natural, dyadic, and sequency order algorithms and proces- sors for the Walsh–Hadamard transform. IEEE Transactions on Computers, 26(05):435–442, 1977

    1977

  27. [35]

    On sequency-complete and sequency-ordered matrices

    Alok Shukla and Prakash Vedula. On sequency-complete and sequency-ordered matrices. arXiv preprint arXiv:2402.11003, 2024

    2024 arXiv

  28. [36]

    A quantum algorithm for counting zero-crossings

    Alok Shukla. A quantum algorithm for counting zero-crossings. arXiv preprint arXiv:2212.11814, 2022

    2022 arXiv

  29. [37]

    Quantum compression of tensor network states

    Ge Bai, Yuxiang Yang, and Giulio Chiribella. Quantum compression of tensor network states. New Journal of Physics, 22(4):043015, 2020

    2020

  30. [38]

    Quantum data compression of a qubit ensemble

    Lee A Rozema, Dylan H Mahler, Alex Hayat, Peter S Turner, and Aephraim M Steinberg. Quantum data compression of a qubit ensemble. Physical Review Letters, 113(16):160504, 2014

    2014

  31. [39]

    Quantum state compression shadow

    He-Liang Huang and Chen Ding. Quantum state compression shadow. The European Physical Journal Special Topics, pages 1–8, 2025

    2025

  32. [40]

    Compression of entanglement im- proves quantum communication

    Yu Guo, Hao Tang, Jef Pauwels, Emmanuel Zambrini Cruzeiro, Xiao-Min Hu, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, and Armin Tavakoli. Compression of entanglement im- proves quantum communication. Laser & Photonics Reviews, 19(10):2401110, 2025

    2025

  33. [41]

    Quantum state transfer via noisy photonic and phononic waveguides

    Beno ˆıt Vermersch, P-O Guimond, Hannes Pichler, and Peter Zoller. Quantum state transfer via noisy photonic and phononic waveguides. Physical Review Letters, 118(13):133601, 2017

    2017

  34. [42]

    Effect of data encoding on the expressive power of variational quantum-machine-learning models

    Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. Effect of data encoding on the expressive power of variational quantum-machine-learning models. Physical Review A, 103(3):032430, 2021

    2021

  35. [43]

    Exponential data encoding for quantum super- vised learning

    Seongwook Shin, Yong-Siah Teo, and Hyunseok Jeong. Exponential data encoding for quantum super- vised learning. Physical Review A, 107(1):012422, 2023. 42

    2023

  36. [44]

    Efficient quantum amplitude encoding of polynomial functions

    Javier Gonzalez-Conde, Thomas W Watts, Pablo Rodriguez-Grasa, and Mikel Sanz. Efficient quantum amplitude encoding of polynomial functions. Quantum, 8:1297, 2024

    2024

  37. [45]

    Digital signal processing: principles, algorithms, and applications, 4/E

    John G Proakis. Digital signal processing: principles, algorithms, and applications, 4/E. Pearson Education India, 2007

    2007

  38. [46]

    Multi-controlled quantum gates in linear nearest neighbor

    Ben Zindorf and Sougato Bose. Multi-controlled quantum gates in linear nearest neighbor. arXiv preprint arXiv:2506.00695, 2025. 43

    2025 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.