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:2412.01334 · Quantum Information and Computation

Order-Six CHMs Containing Exactly Three Distinct Elements

Yanzu Huang, Mengfan Liang, Lin Chen

Published Mar 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract Complex Hadamard matrices (CHMs) are intimately related to the number of distinct matrix elements. We investigate CHMs containing exactly three distinct elements, which is also the least number of distinct elements. In this paper, we show that such CHMs can only be complex equivalent to two kinds of matrices, one is H 2 -reducible and the other is the Tao matrix. Using our result one can further narrow the range of MUB trio (a set of four MUBs in ℂ 6 consists of an MUB trio and the identity) since we find that neither of the two CHMs belong to any MUB trio. Our results may lead to the more complete classification of 6 × 6 CHMs whose elements in the first row are all 1.

DOI
10.2478/qic-2026-0001

Related papers

Eigenvalues of Two-Parameter Complex Hadamard Matrices of Order Six

Yanzu Huang, Lin Chen

Abstract Characterizing complex Hadamard matrices (CHMs) of order six is an open problem in both algebra and quantum information theory. Analyzing the eigenvalues of CHMs provides a novel approach to this problem. We construct a class of two-parameter CHMs of order six with two sorts of eigenvalues. All matrices in this class are H 2-reducible, and the Diţă matrix, Haagerup matrices and Hermitian matrices are both included. This class turns out to be complex equivalent to the class of all CHMs in the dephased form whose eigenvalues are of the form 6 \sqrt 6 , – 6 \sqrt 6 , λ 1 , λ 1 , λ 2 , λ 2 with λ j of modulus 6 \sqrt 6 excluding Tao matrix S 6 ( 0 ) S_6^{(0)} . We further show that some CHMs including all Hermitian CHMs in the former class have Schmidt rank at most three up to complex equivalence, and they are actually controlled unitary gates implementable in experiments. Our result demonstrates the feasibility of using eigenvalues for characterizing order-six CHMs and their application in quantum circuits.

Quantum Information and Computation•Mar 1, 2026•0 citations

Eigenvalues and eigenvectors of complex Hadamard matrices

Mengfan Liang, Lin Chen

Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs.

arXiv:2408.10471•Aug 20, 2024•0 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

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

Block-circulant complex Hadamard matrices

Wojciech Bruzda

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.

arXiv:2204.11727•Journal of Mathematical Physics•May 1, 2023•3 citations

On 6× 6 complex Hadamard matrices containing two nonintersecting identical 3× 3 submatrices

Lin Chen

It is conjectured that four mutually unbiased bases in dimension 6 do not exist in quantum information. The conjecture is equivalent to the nonexistence of some three 6×6 complex Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate the 6×6 CHM U of Schmidt rank 3 containing two nonintersecting identical 3×3 submatrices V, i.e. U=12[WVVX]. We show that such U exists, V, W, X have rank 2 or 3, and they have rank 2 at the same time. We construct the analytical expressions of U when V is, respectively, of rank 2, unitary and normal. We apply our results to the conjecture by showing that U with some normal V is not one of the three 6×6 CHMs.

Linear and Multilinear Algebra•Dec 16, 2021•0 citations
Abstract Complex Hadamard matrices (CHMs) are intimately related to the number of distinct matrix elements. We investigate CHMs containing exactly three distinct elements, which is also the least number of distinct elements. In this paper, we show that such CHMs can only be complex equivalent to two kinds of matrices, one is H 2 -reducible and the other is the Tao matrix. Using our result one can further narrow the range of MUB trio (a set of four MUBs in ℂ 6 consists of an MUB trio and the identity) since we find that neither of the two CHMs belong to any MUB trio. Our results may lead to the more complete classification of 6 × 6 CHMs whose elements in the first row are all 1.
Abstract Characterizing complex Hadamard matrices (CHMs) of order six is an open problem in both algebra and quantum information theory. Analyzing the eigenvalues of CHMs provides a novel approach to this problem. We construct a class of two-parameter CHMs of order six with two sorts of eigenvalues. All matrices in this class are H 2-reducible, and the Diţă matrix, Haagerup matrices and Hermitian matrices are both included. This class turns out to be complex equivalent to the class of all CHMs in the dephased form whose eigenvalues are of the form 6 \sqrt 6 , – 6 \sqrt 6 , λ 1 , λ 1 , λ 2 , λ 2 with λ j of modulus 6 \sqrt 6 excluding Tao matrix S 6 ( 0 ) S_6^{(0)} . We further show that some CHMs including all Hermitian CHMs in the former class have Schmidt rank at most three up to complex equivalence, and they are actually controlled unitary gates implementable in experiments. Our result demonstrates the feasibility of using eigenvalues for characterizing order-six CHMs and their application in quantum circuits.
Characterizing the 6×66\times 66×6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any n×nn\times nn×n CHM with dephased form has two constant eigenvalues ±n\pm\sqrt{n}±n​ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of 6×66\times 66×6 CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no 6×66\times 66×6 CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of 6×66\times 66×6 CHMs will lead to the complete classification of 6×66\times 66×6 CHMs.
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.
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.
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.
It is conjectured that four mutually unbiased bases in dimension 6 do not exist in quantum information. The conjecture is equivalent to the nonexistence of some three 6×6 complex Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate the 6×6 CHM U of Schmidt rank 3 containing two nonintersecting identical 3×3 submatrices V, i.e. U=12[WVVX]. We show that such U exists, V, W, X have rank 2 or 3, and they have rank 2 at the same time. We construct the analytical expressions of U when V is, respectively, of rank 2, unitary and normal. We apply our results to the conjecture by showing that U with some normal V is not one of the three 6×6 CHMs.