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arXiv:2609.16813

TSS Graphs for Hadamard Matrices: Real vs Complex

Wesley Lewis, Darsh Pareek, Ravi Janjam

quant-phmath-phSubmitted Sep 15, 2026• Announced Sep 16, 2026• 0 citations
v1 · Sep 15, 2026v2 · Sep 16, 2026
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Abstract

We have observed that Hadamard matrices whether real or complex lead to a dense output of identical probabilities for any given single input state. Complex Hadamard matrices have unusual orders and introduce phase which prove useful when controlled phase transformation gates are used. By taking a superposed combinations of input states, we discovered that the Hadamards generate non-uniform probabilities which is practically significant towards amplitude amplification without the need for manual parameterization like Grover's operator~\cite{grover1996}. A variety of graph theoretic properties are applied to TSS graphs and their trends were explored. Additionally, graphs turn out to be nearly isomorphic for the same set of input states which has potential applications for developing Quantum Algorithms. Keywords: Quantum Algorithms, Hadamard Matrices, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory

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We have observed that Hadamard matrices whether real or complex lead to a dense output of identical probabilities for any given single input state. Complex Hadamard matrices have unusual orders and introduce phase which prove useful when controlled phase transformation gates are used. By taking a superposed combinations of input states, we discovered that the Hadamards generate non-uniform probabilities which is practically significant towards amplitude amplification without the need for manual parameterization like Grover's operator~\cite{grover1996}. A variety of graph theoretic properties are applied to TSS graphs and their trends were explored. Additionally, graphs turn out to be nearly isomorphic for the same set of input states which has potential applications for developing Quantum Algorithms.
Keywords: Quantum Algorithms, Hadamard Matrices, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory
Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic framework for analyzing quantum operators by mapping unitary matrices to directed graphs; we term these structures \emph{Imperfect Graphs} or more formally as \emph{Topological Structure of Superpositions}(TSS) as a tool to devise better Quantum Algorithms. In this framework, we represent computational basis states as vertices. A directed edge exists between two vertices if and only if there is a non-zero amplitude transition between them, effectively mapping the support of the unitary operator. In this paper we deliberately discard probability amplitudes and phase information to isolate the connectivity and reachability properties of the operator. We demonstrate how TSS intuitively helps describe gates such as the Hadamard, Pauli-(X,Y,Z) gates, etc \cite{nielsen2000}. This framework provides a novel perspective for viewing quantum circuits as discrete dynamical systems \cite{childs2009,aharonov2001} Keywords: Quantum Algorithms, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory
We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of [Formula: see text] subsystems with [Formula: see text] levels each to the set of complex Hadamard matrices of order [Formula: see text]. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ([Formula: see text] or [Formula: see text]), two-unitary ([Formula: see text] and [Formula: see text] are unitary), or [Formula: see text]-unitary. Here [Formula: see text] denotes reshuffling of a matrix [Formula: see text] describing a bipartite system, and [Formula: see text] its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in [Formula: see text] dimensions.
Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţă's matrix D6. This answers a recent question of Bengtsson et al. [3].
We study the continuous-time quantum walks on graphs in the adjacency algebra of the n -cube and its related distance regular graphs. For k ≥ 2 , we find graphs in the adjacency algebra of ( 2 k + 2 - 8 ) -cube that admit instantaneous uniform mixing at time π / 2 k and graphs that have perfect state transfer at time π / 2 k . We characterize the folded n -cubes, the halved n -cubes and the folded halved n -cubes whose adjacency algebra contains a complex Hadamard matrix. We obtain the same conditions for the characterization of these graphs admitting instantaneous uniform mixing.
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.
Towards a classification of 6×66\times 66×6 complex Hadamard matrices
Complex Hadamard matrices have received considerable attention in the past few years due to their appearance in quantum information theory. While a complete characterization is currently available only up to order 5 (in \cite{haagerup}), several new constructions of higher order matrices have appeared recently \cite{dita, karol, BN, MM, sz}. In particular, the classification of {\it self-adjoint} complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in \cite{BN}, providing a previously unknown non-affine one-parameter orbit. In this paper we classify all \emph{dephased, symmetric} complex Hadamard matrices \emph{with real diagonal} of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6F_6F6​ and Diţă's matrix D6D_6D6​. This answers a recent question of Bengtsson & al. in \cite{BB}.