Polynomial construction of complex Hadamard matrices with cyclic core

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c ≠ 1 to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order n and show that they all contain at least n entries. We conjecture that all trades in complex Hadamard matrices contain at least n entries.KeywordsHadamard matrixTradeRankMathematics Subject Classification (2010):05B2015B34

In this paper, a novel spectral technique based on the transform constructed from a Complex Hadamard matrix is introduced. The technique finds potential applications in various synthesis and optimization processes of logic circuits. Classification of switching function is shown here as one of the possible applications. The technique uses the Complex Hadamard matrix as the spectral transform, manipulates the complex spectral coefficients of the functions and classifies them into the NPN-equivalent and linearly separable functions. The method has proven to be a more efficient classification scheme than the existing classification method using the standard spectral technique based on the Walsh Transform. For efficient manipulation of switching functions, some new decision diagrams to represent Complex Hadamard matrices and their spectra of integervalued functions are proposed. With the intrinsic properties of the new transform in handling complex numbers, the presented decision diagrams lead to a more efficient and compact representation of both switching and complex functions.

Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2 ⊗ V and I 2 ⊗ W. We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.

A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on complex parameters. The method we use is via doubling the size of inverse complex conference matrices. When the free parameters take values on the unit circle the inverse orthogonal matrices transform into complex Hadamard matrices, and in this way we find new parametrizations of Hadamard matrices for dimensions n = 8, 10, and 12.

A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m > 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)

This paper discusses the factorization of a bent function type complex Hadamard matrix of order N=p/sup n/ with a prime p. The bent function type complex Hadamard matrix is the generalization of the bent function type binary Hadamard matrix. It is shown that the bent function type complex Hadamard matrix is closely related to the complex Hadamard matrix generalized the Sylvester type binary Hadamard matrix, and can be expressed by using 2n-2 matrices of order N with pN non-zero elements at most. The application of the bent function type complex Hadamard matrix to synchronous spread spectrum communication is also discussed.

Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we first determine the image of a certain rational map from the d-dimensional complex projective space to Cd(d+1)/2. Applying this result with d = 3, we give constructions of complex Hadamard matrices, and more generally, type-II matrices, in the Bose–Mesner algebra of a certain 3-class symmetric association scheme. In particular, we recover the complex Hadamard matrices of order 15 found by Ada Chan. We compute the Haagerup sets to show inequivalence of resulting type-II matrices, and determine the Nomura algebras to show that the resulting matrices are not decomposable into generalized tensor products.

In this paper we classify complex Hadamard matrices contained in the Bose-Mesner algebra of nonsymmetric 3-class association schemes. As a consequence of our classification, we have two infinite families and some small examples of complex Hadamard matrices contained in the Bose-Mesner algebra of a self-dual fission of a complete multipartite graph.

McNulty and Weigert (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908–25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any 6 × 6 complex Hadamard matrix whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of 2 × 2 complex Hadamard submatrices of any H 2-reducible matrix is not 10 , … , 16 , 18 . We also put forward some possible directions for further development.

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.

On the asymptotic existence of partial complex Hadamard matrices and related combinatorial objects

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for |q 0 |=...=|q N-1 |=1 the quantity Φ=∑ i+k=j+l q i q k q j q l satisfies Φ≥N 2 , with equality if and only if q=(q i ) is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of Φ, (2) the study of the critical points of Φ, and (3) the computation of the moments of Φ. We explore here these questions, with some results and conjectures.

In this letter, ternary semi-bent functions and a complex Hadamard matrix are used to construct spreading sequences for code-division multiple-access systems. An efficient assignment of these sequences to a lattice of regular hexagonal cells, which are mutually orthogonal within each cell and the adjacent cells, is achieved and most notably the number of sequences per cell is maximum possible. The large sequence reuse distances, being √27, and low cross-correlation between non-orthogonal sequences (in non-adjacent cells) make these sequences appealing for practical applications.

On the realizability of the critical points of a realizable list

A construction of shift sequence sets is proposed. Multiple distinct shift sequence sets are obtained by changing the parameters of the shift sequences. The shift sequences satisfy the conditions that P|L and P ≥ 2, where P is the length of the shift sequences, L is the length of the zero-correlation zone or low-correlation zone (ZCZ/LCZ). Then based on these shift sequence sets, many shift distinct ZCZ/LCZ sequence sets are constructed by using interleaving technique and complex Hadamard matrices. Furthermore, the new construction is optimal under the conditions proposed in this paper. Compared with previous constructions, the proposed construction extends the number of shift distinct ZCZ/LCZ sequence sets, so that more sequence sets are obtained for multi-cell quasi-synchronous code-division multiple access (QS-CDMA) systems.