Extended Hadamard Technique Overview

• Extended Hadamard Technique is a family of methods that generalize classical Hadamard objects by incorporating new structural criteria, recursions, and frameworks to preserve cancellation and orthogonality.
• It enhances practical applications such as single-pixel imaging by balancing low- and high-frequency information for sharper reconstructions using innovative sampling orders.
• It extends to diverse fields—including quantum information, combinatorics, coding theory, and asymptotic analysis—by providing formal constructions and optimizing algorithmic performance.

“Extended Hadamard Technique” is not a single universally fixed method; the expression is used for several technically distinct extensions of Hadamard-based ideas across imaging, quantum information, combinatorics, coding theory, asymptotic analysis, and fractional calculus. In each case, the extension starts from a standard Hadamard object—such as a Hadamard matrix, Hadamard unitary, Hadamard product, Hadamard code, or Hadamard expansion—and enlarges its scope by introducing a new structural criterion, a new recursion, a new geometric decomposition, or a new computational framework. A plausible unifying description is that these techniques preserve the cancellation, orthogonality, equal-amplitude, or correlation properties associated with Hadamard structures while adapting them to a broader class of problems (Sun et al., 2021).

1. Terminological scope and core mathematical motifs

Several mathematically different notions underlie the phrase. In linear algebra and combinatorics, a Hadamard matrix $H$ is a $\{\pm1\}$-matrix satisfying

$HH^T=nI_n,$

and extended techniques typically construct larger Hadamard matrices from smaller seeds, conference matrices, finite fields, generalized Boolean functions, or mutually unbiased bases (Djokovic, 2016). In matrix analysis for mixture models, the Hadamard Extension of an $n\times k$ matrix $Q$ is the $2^n\times k$ matrix whose rows are all Hadamard products of subsets of the rows of $Q$, with

$$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$

and full column rank of this extension is a necessary ingredient of identification algorithms for mixtures of product distributions on binary random variables (Gordon et al., 2021).

In quantum information, the extension is formulated through Hadamard unitaries with flat-modulus entries

$U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$

and a set of states is promoted to a “Hadamard set” when its Gram-Schmidt matrix can be diagonalized by such a unitary (Sun et al., 2021). In quantum algorithms, the Hadamard product of amplitude-encoded states is extended from the conventional postselection circuit to a Fourier-space method with localized truncation, shifting the relevant complexity parameter from the grid size $N$ to Fourier regularity (Huang et al., 2 Jun 2026). In asymptotic analysis, “extended Hadamard” designates a finite-segment steepest-descent expansion for the Airy function that replaces earlier non-systematic contour subdivisions by a small number of branch-point-controlled segments and introduces upper incomplete Gamma functions through tail integrals (Alvarez-Perez, 2019).

This diversity of usage shows that the term is best understood as a family resemblance rather than a single theorem. The common pattern is extension by structural preservation: orthogonality is preserved in matrix constructions, fixed marginals in quantum masking, rank information in Hadamard extensions, and convergence or accuracy in analytic and numerical variants.

2. Detail-enhanced sampling in Hadamard single-pixel imaging

In Hadamard single-pixel imaging (HSI), the extended technique appears as a detail-enhanced sampling strategy designed to improve low-sampling reconstructions by balancing low-frequency energy and high-frequency detail (Cai et al., 2022). Standard Hadamard ordering methods often prioritize coefficients according to pattern characteristics, whereas the reported difficulty is that the Hadamard spectrum energy is mostly concentrated in the upper left corner of the spectrum; sampling only this strongest low-frequency region reconstructs coarse structure but can also produce blurred edges, indistinguishable fine details, ringing artifacts, and loss of high-frequency structure (Cai et al., 2022).

The proposed strategy has two stages. First, a preliminary order (PO) is obtained from a Hadamard power spectrum built from 800 grayscale images from DIV2K after Hadamard transformation, normalization, and summation; descending sorting of the coefficients reveals the direction in the Hadamard spectrum where energy is arranged from large to small (Cai et al., 2022). Second, coefficients are sampled randomly along that order using the exponential probability function

$\{\pm1\}$0

where $\{\pm1\}$1, $\{\pm1\}$2 is the pattern size, and $\{\pm1\}$3 is the probability that point number $\{\pm1\}$4 is selected. The sampling ratio is

$\{\pm1\}$5

and, once the pattern size is fixed, $\{\pm1\}$6 varies only with the sampling ratio (Cai et al., 2022).

This PO+PF mechanism samples both low- and high-frequency parts proportionally, rather than only a compact low-frequency block. The paper then replaces the dataset-derived PO by a directly generated XY order. The XY order fixes the upper-left corner of the Hadamard spectrum as the coordinate origin, assigns each coefficient coordinates $\{\pm1\}$7, computes

$\{\pm1\}$8

and sorts basis patterns in ascending order of $\{\pm1\}$9 (Cai et al., 2022). Because this order is generated directly from spectral coordinates, without database scanning or feature-based evaluation, its reported generation time is 0.445 s for a $HH^T=nI_n,$0 Hadamard order, about three orders of magnitude shorter than CC (Cai et al., 2022).

Measurement uses the standard differential Hadamard procedure. If $HH^T=nI_n,$1 is a Hadamard pattern, then the binary DMD-compatible pair is

$HH^T=nI_n,$2

and the differential detector output is

$HH^T=nI_n,$3

For reconstruction, the HSI model is written in compressed-sensing form as $HH^T=nI_n,$4, with $HH^T=nI_n,$5 formed from selected Hadamard rows, and the reported reconstruction algorithm is TVAL3 (Cai et al., 2022).

Simulation used the USAF1951 resolution target and cameraman images, both resized to $HH^T=nI_n,$6 and corrupted with 1% Gaussian noise. Compared with Random, Natural, Walsh, RD, CC, and TV orders, the proposed methods preserved thinner fringes, sharper stripes, and clearer boundaries, while reducing ringing and jagged artifacts; at sampling ratios above 10%, they generally outperformed prior methods in PSNR and SSIM (Cai et al., 2022). Experimental validation used an LED parallel light source, a TI DLP7000 DMD, a CCD as a bucket detector, a $HH^T=nI_n,$7 pattern size, a DMD frame rate of 30 Hz, and a toy figurine with a USAF1951 target as background; the reported conclusion was that the method balances low-frequency and high-frequency information and maintains universality beyond the DIV2K dataset (Cai et al., 2022).

3. Algebraic and combinatorial extensions of Hadamard matrices and codes

A major strand of the extended Hadamard technique concerns constructive enlargement of Hadamard matrices. One such extension generalizes Scarpis’s 1898 theorem from primes to prime powers. If $HH^T=nI_n,$8 is a prime power and there exists a Hadamard matrix of order $HH^T=nI_n,$9, then there exists a Hadamard matrix of order $n\times k$0 (Djokovic, 2016). The construction normalizes the input matrix $n\times k$1, extracts its $n\times k$2 core $n\times k$3, and assembles a block matrix $n\times k$4 of order $n\times k$5 whose rows are arranged through a finite-field translation rule

$n\times k$6

using a bijection $n\times k$7. Orthogonality follows from the core identities

$n\times k$8

and from the uniqueness of solutions to affine equations over $n\times k$9 (Djokovic, 2016).

Another extension, framed as an extended Paley/Hadamard construction, uses a conference matrix

$Q$0

built from the extended quadratic character on $Q$1, together with repetition matrices $Q$2 formed from a known Hadamard matrix $Q$3 (Kumari et al., 2019). For $Q$4, the block matrix

$Q$5

satisfies $Q$6, producing Hadamard matrices of order $Q$7. For $Q$8, an analogous Sylvester-inflated construction yields order $Q$9 (Kumari et al., 2019). The paper also states a twin-prime extension producing order $2^n\times k$0 from a conference matrix of order $2^n\times k$1.

A different nonlinear extension starts from mutually unbiased bases. If $2^n\times k$2 are unitary $2^n\times k$3 matrices, then

$2^n\times k$4

is unitary, and when $2^n\times k$5 are chosen from a mutually unbiased family with suitable phasing, the construction yields new complex Hadamard matrices of size $2^n\times k$6 (Dita, 2010). In the $2^n\times k$7 case, the method produces $2^n\times k$8 families with five free phases, extending linear block-doubling by a nonlinear lower-right block.

Coding-theoretic extensions also adopt Hadamard structure. The $2^n\times k$9-linear family $Q$0 is defined from a generator matrix $Q$1 whose columns are all vectors of the form

$Q$2

and its Gray image $Q$3 is a binary $Q$4-code, i.e. a Hadamard code, with

$Q$5

The dual family $Q$6 yields extended perfect codes. The paper proves that if $Q$7, then there exist exactly $Q$8 pairwise nonequivalent $Q$9-linear Hadamard $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$0-codes and $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$1 pairwise nonequivalent $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$2-linear extended perfect $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$3-codes (0710.0199). The recurrent constructions

$$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$4

increment the parameters $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$5 and $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$6, respectively (0710.0199).

At the circuit level, recursive Hadamard-code encoders achieve exact lower bounds on XOR complexity. For the $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$7 Hadamard code, the paper proves a lower bound of $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$8 XORs and gives a matching recursive encoder; for systematic punctured Hadamard codes the exact bound is $$Q_S := r_{i_1}\odot \cdots \odot r_{i_\ell}, \qquad Q_\emptyset=1,$$9, and for non-systematic punctured Hadamard codes it is $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$0 (Li et al., 2018). By duality and the transposition principle, the corresponding Hamming and extended Hamming encoding bounds are $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$1 and $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$2, respectively (Li et al., 2018).

Protograph-based LDPC-Hadamard codes extend the classical LDPC-Hadamard framework by replacing SPC check nodes in a generalized protograph with Hadamard super-checks and then lifting the protograph (Zhang et al., 2020). If the row weight of a Hadamard check node is $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$3, the associated Hadamard order is

$U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$4

The modified PEXIT analysis supports degree-1 and punctured nodes, and the reported thresholds range from $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$5 to $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$6; at BER $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$7, the reported gaps to the ultimate Shannon limit range from $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$8 for rate $U_{jk}=\frac{1}{\sqrt n}e^{i\theta_{jk}},$9 to $N$0 for rate $N$1 (Zhang et al., 2020).

A recent matrix-sequence synthesis framework uses a $N$2 circulant Hadamard seed constructed from the generalized Boolean function

$N$3

and states that all 8 circulant Hadamard matrices of order 4 arise from parameter choices (Priyanshu et al., 14 Oct 2025). Recursive block doubling then produces Hadamard matrices of order $N$4, binary cross Z-complementary sets, binary Golay complementary sets, complete complementary codes, and binary optimal cross-Z complementary sequence sets (Priyanshu et al., 14 Oct 2025).

4. Quantum-information and quantum-computing extensions

In quantum information masking, the extension is a structural criterion for deterministic unitary masking. A unitary masks a set of states $N$5 if it maps them to bipartite states with identical local marginals,

$N$6

For a set of states with Gram-Schmidt matrix

$N$7

the paper defines a Hadamard set by the existence of a Hadamard unitary $N$8 such that

$N$9

Theorem 2 states that a Hadamard set $\{\pm1\}$00, with $\{\pm1\}$01, can be deterministically masked by a unitary operation (Sun et al., 2021). The masked states are constructed as

$\{\pm1\}$02

so the reduced density matrices

$\{\pm1\}$03

are independent of $\{\pm1\}$04 (Sun et al., 2021). Theorem 3 further states that a linear combination $\{\pm1\}$05 can be masked together with the set iff

$\{\pm1\}$06

In quantum compilation, Hadamard minimization is treated as an exact synthesis objective for Clifford+$\{\pm1\}$07 and Clifford+$\{\pm1\}$08 circuits (Vandaele et al., 2023). A Pauli-rotation sequence is encoded in a binary matrix

$\{\pm1\}$09

and the diagonalization problem H-Opt asks for a Clifford circuit with minimal Hadamard count. The central formula is

$\{\pm1\}$10

where $\{\pm1\}$11 is the commutativity matrix of the sequence (Vandaele et al., 2023). A more refined problem, Internal-H-Opt, minimizes Hadamards between the first and last $\{\pm1\}$12 gate, and the optimum is exactly

$\{\pm1\}$13

The paper motivates this by two resource effects: Hadamards can block $\{\pm1\}$14-count optimization, and each gadgetized Hadamard requires one ancilla qubit, one CZ gate, and one measurement (Vandaele et al., 2023).

A programming-language extension adds the Hadamard gate to the reversible classical language $\{\pm1\}$15. The resulting Hadamard-$\{\pm1\}$16 language has a sound and complete equational semantics and characterizes exact orthogonal matrices over

$\{\pm1\}$17

The construction proceeds via an auxiliary language Q-$\{\pm1\}$18 with both $\{\pm1\}$19 and $\{\pm1\}$20, then hides $\{\pm1\}$21 by simulating it as

$\{\pm1\}$22

A major technical result is a finite presentation and synthesis algorithm for the groups

$\{\pm1\}$23

using generators consisting of sign flips, swaps, and Hadamard-like two-level transformations (Fang et al., 7 Jun 2025).

In quantum algorithms for amplitude encoding, the Hadamard product state of two states

$\{\pm1\}$24

is conventionally prepared by a non-unitary postselection procedure whose success probability is

$\{\pm1\}$25

For amplitude-encoded samples of smooth functions on a uniform grid, the paper states this typically scales as $\{\pm1\}$26 (Huang et al., 2 Jun 2026). The extension moves to Fourier space, with coefficients

$\{\pm1\}$27

and retains only a localized window of size $\{\pm1\}$28 in one factor. The resulting success probability becomes approximately $\{\pm1\}$29, with

$\{\pm1\}$30

so the query complexity is $\{\pm1\}$31, independent of $\{\pm1\}$32; if either input has finitely many nonzero Fourier coefficients, the algorithm is exact with $\{\pm1\}$33 oracle queries and $\{\pm1\}$34 additional gates (Huang et al., 2 Jun 2026).

5. Analytic and geometric extensions: Hadamard expansions, Hadamard spaces, and Hadamard states

In asymptotic analysis, extended Hadamard expansions for the Airy function arise from a finite geometric splitting of the steepest-descent contour rather than the large number of non-systematic path subdivisions used in earlier Hadamard expansions (Alvarez-Perez, 2019). The phase is

$\{\pm1\}$35

and the steepest-descent parameterization

$\{\pm1\}$36

defines an inverse map $\{\pm1\}$37 on a three-sheeted Riemann surface with branch points at $\{\pm1\}$38, $\{\pm1\}$39, and $\{\pm1\}$40 (Alvarez-Perez, 2019). The contour is split into five segments: one central segment, two finite branch-point-centered segments, and two tails to infinity. Because two segments are semi-infinite, the expansion contains upper incomplete Gamma functions

$\{\pm1\}$41

whereas earlier Hadamard series involved only lower incomplete Gamma functions (Alvarez-Perez, 2019). The paper states that the resulting expansion is convergent for all values of the complex variable and gives a geometric interpretation of the Stokes phenomenon at $\{\pm1\}$42.

In metric geometry and nonlinear analysis, Hadamard spaces are complete CAT(0) spaces, and the Lie–Trotter–Kato formula has been extended to this setting (Bacak, 2013). For a proper convex lsc function $\{\pm1\}$43, the resolvent is

$\{\pm1\}$44

and the gradient flow semigroup is

$\{\pm1\}$45

If $\{\pm1\}$46, the product formula becomes

$\{\pm1\}$47

and, under a local compactness assumption on one domain,

$\{\pm1\}$48

The proof replaces ultrapower arguments by weak convergence in Hadamard spaces (Bacak, 2013).

In algebraic quantum field theory, Hadamard states are extended from normally hyperbolic operators to a broader class of decomposable Green-hyperbolic operators (Fewster, 16 Mar 2025). If the Pauli–Jordan propagator $\{\pm1\}$49 satisfies

$\{\pm1\}$50

the operator is called $\{\pm1\}$51-decomposable. A two-point function $\{\pm1\}$52 is then $\{\pm1\}$53-Hadamard if

$\{\pm1\}$54

This generalizes the standard null-cone microlocal spectrum condition and covers cases such as the Proca field, where the characteristic set is not the spacetime light cone (Fewster, 16 Mar 2025). The paper proves propagation under the equation of motion, stability under pullbacks and suitable pushforwards, preservation under tensor products and partial traces, and preservation under nonselective measurements. It also shows that for Proca states,

$\{\pm1\}$55

thereby identifying the generalized definition with earlier formulations (Fewster, 16 Mar 2025).

6. Convolution quadrature and statistical identifiability as extension frameworks

In fractional calculus, the convolution quadrature method developed for Riemann–Liouville operators is extended to Hadamard and Caputo–Hadamard fractional calculus by the logarithmic change of variables

$\{\pm1\}$56

which transforms a uniform mesh in $\{\pm1\}$57 into the exponential mesh

$\{\pm1\}$58

in physical time (Yin et al., 2023). The Hadamard fractional integral is

$\{\pm1\}$59

and the Caputo–Hadamard derivative is

$\{\pm1\}$60

with $\{\pm1\}$61 (Yin et al., 2023). On the transformed mesh, BDF-$\{\pm1\}$62 convolution quadrature with $\{\pm1\}$63 is imported through

$\{\pm1\}$64

yielding the discrete Hadamard operator

$\{\pm1\}$65

Because singular source terms destroy the nominal high-order accuracy, the method adds starting-step corrections through coefficients $\{\pm1\}$66 and $\{\pm1\}$67; the paper states that the corrected scheme recovers the designed $\{\pm1\}$68-th order temporal accuracy, whereas the uncorrected scheme is only first-order accurate in the singular-source setting (Yin et al., 2023).

In mixture identification, the Hadamard Extension is not a numerical or geometric approximation but a rank-completion device. For an $\{\pm1\}$69 matrix $\{\pm1\}$70 with rows $\{\pm1\}$71, the extension $\{\pm1\}$72 collects all subsetwise coordinate products

$\{\pm1\}$73

and full column rank of $\{\pm1\}$74 is necessary for injectivity of the source-identification map $\{\pm1\}$75 in mixture-of-products models (Gordon et al., 2021). The paper proves a “Knockdown theorem”: if $\{\pm1\}$76 has full column rank, then there exists a subset $\{\pm1\}$77 of at most $\{\pm1\}$78 rows such that $\{\pm1\}$79 also has full column rank. It also introduces the NAE condition, a combinatorial sufficient condition ensuring $\{\pm1\}$80 (Gordon et al., 2021). This use of “Hadamard extension” differs from matrix-construction uses but retains the same multiplicative coordinatewise algebra.

7. Common structure, contrasts, and recurring misconceptions

Across these domains, the extended Hadamard technique is characterized less by a single formal definition than by recurring operational themes. One theme is controlled enlargement of a Hadamard object: Scarpis–Paley–circulant constructions enlarge matrix order; MUB-based doubling enlarges dimension; $\{\pm1\}$81-linear recursions enlarge code length; and protograph lifting enlarges local Hadamard constraints into long low-rate codes (Djokovic, 2016). A second theme is structural diagonalization or flattening: Hadamard unitaries diagonalize Gram matrices in masking, Hadamard gates diagonalize Pauli products in Clifford synthesis, and Fourier localization converts a Hadamard-product preparation problem into a truncated convolution structure (Sun et al., 2021). A third theme is singularity or correlation control: finite contour segmentation controls Stokes transitions, exponential-mesh CQ controls logarithmic fractional singularities, and decomposable propagators control generalized Hadamard wavefront sets (Alvarez-Perez, 2019).

A common misconception is to treat the term as if it always referred to Hadamard matrices. The provided literature does not support that restriction. Some uses are matrix-theoretic, but others concern Hadamard products, Hadamard gates, Hadamard spaces, Hadamard states, or Hadamard sampling (Cai et al., 2022). Another misconception is that these extensions are merely heuristic embellishments. In the cited works, the extensions are formalized by explicit criteria or constructions: $\{\pm1\}$82 in generalized Hadamard state theory, $\{\pm1\}$83 in optimal Hadamard-count synthesis, $\{\pm1\}$84 in detail-enhanced HSI sampling, and $\{\pm1\}$85 in Fourier-regularity-driven quantum Hadamard products (Fewster, 16 Mar 2025).

A plausible implication is that the phrase persists because Hadamard structures occupy an intermediate level of abstraction: they are concrete enough to yield implementable constructions, yet general enough to transfer between orthogonality, correlation cancellation, equal-amplitude interference, and microlocal or variational decomposition. The literature surveyed here supports that interpretation, while also making clear that “Extended Hadamard Technique” should be read contextually rather than as a single standardized method name.