Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2204.11727 · Journal of Mathematical Physics

Block-circulant complex Hadamard matrices

Wojciech Bruzda

Published May 1, 2023• 3 citations
arXivHTMLPDF

Abstract

A new method of obtaining a sequence of isolated complex Hadamard matrices (CHM) for dimensions N ⩾ 7, based on block-circulant structures, is presented. We discuss several analytic examples resulting from a modification of the Sinkhorn algorithm. In particular, we present new isolated matrices of orders 9, 10, and 11, which elements are not roots of unity, and also several new multiparametric families of order 10. We note novel connections between certain eight-dimensional matrices and provide new insights toward the classification of CHM for N ⩾ 7. These contributions can find real applications in quantum information theory and constructions of new families of mutually unbiased bases or unitary error bases.

DOI
10.1063/5.0144667

Related papers

Isolated Hadamard Matrices from Mutually Unbiased Product Bases

Daniel McNulty, Stefan Weigert

A new construction of complex Hadamard matrices of composite order d=pq, with primes p,q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the method by deriving many previously unknown complex Hadamard matrices. We obtain at least 12 new isolated matrices of Butson type, with orders ranging from 9 to 91.

•Jan 1, 2012•12 citations

Isolated Hadamard matrices from mutually unbiased product bases

Daniel McNulty, Stefan Weigert

A new construction of complex Hadamard matrices of composite order d = pq, with primes p, q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the method by deriving many previously unknown complex Hadamard matrices. We obtain at least 12 new isolated matrices of Butson type, with orders ranging from 9 to 91.

arXiv:1208.1057•Journal of Mathematical Physics•Nov 28, 2012•13 citations

Hadamard matrices from mutually unbiased bases

P. Diţă

An analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the n=4 case that leads to a rich family of eight-dimensional Hadamard matrices that depend on five arbitrary parameters whose modulus is equal to unity.

arXiv:1002.4933•Journal of Mathematical Physics•Jul 1, 2010•15 citations

Two-unitary complex Hadamard matrices of order 36

Wojciech Bruzda, Karol Życzkowski

Abstract A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

arXiv:2401.01671•Special Matrices•Jan 1, 2024•6 citations

A Complete Classification of Complex Hadamard Matrices of Order Six

Mateo Cárdenes Wuttig, Joseph Tindall

Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased $3 \times 3$ corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.

quant-pharXiv:2608.18053•Aug 19, 2026

Unbiased Hadamard matrices and bases

Hadi Kharaghani

Mutually unbiased unit Hadamard (MUUH) matrices have been studied for almost 40 years. Recent interest in such matrices has been motivated by their applications to quantum information theory. In this paper, we introduce the class of mutually unbiased complex Hadamard (MUCH) matrices having as sntries fourth roots of unity. The number of MUCH matrices of order 2n, n odd, is at most 2, and the bound is attained for n = 1,5,9. Certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. We then turn our attention to the class of mutually unbiased real Hadamard matrices of order 16n2which seem to be the most interesting ones. Mutually unbiased weighing (MUW) matrices are introduced for the first time. Further study of these objects look quite promising.

NATO science for peace and security series. D, Information and communication security•Jan 1, 2011•1 citation
A new method of obtaining a sequence of isolated complex Hadamard matrices (CHM) for dimensions N ⩾ 7, based on block-circulant structures, is presented. We discuss several analytic examples resulting from a modification of the Sinkhorn algorithm. In particular, we present new isolated matrices of orders 9, 10, and 11, which elements are not roots of unity, and also several new multiparametric families of order 10. We note novel connections between certain eight-dimensional matrices and provide new insights toward the classification of CHM for N ⩾ 7. These contributions can find real applications in quantum information theory and constructions of new families of mutually unbiased bases or unitary error bases.
Abstract A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.
Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased 3×33 \times 33×3 corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.
Mutually unbiased unit Hadamard (MUUH) matrices have been studied for almost 40 years. Recent interest in such matrices has been motivated by their applications to quantum information theory. In this paper, we introduce the class of mutually unbiased complex Hadamard (MUCH) matrices having as sntries fourth roots of unity. The number of MUCH matrices of order 2n, n odd, is at most 2, and the bound is attained for n = 1,5,9. Certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. We then turn our attention to the class of mutually unbiased real Hadamard matrices of order 16n2which seem to be the most interesting ones. Mutually unbiased weighing (MUW) matrices are introduced for the first time. Further study of these objects look quite promising.