Quaternionic Perfect Sequences and Hadamard Matrices
Aidan Bennett, Curtis Bright, Paul Colinot, Ashwin Nayak
TL;DR
The paper studies quaternionic perfect sequences and their one-to-one correspondence with quadruples of QT sequences, enabling a fast, symmetry-free enumeration of quaternionic Hadamard matrices with circulant blocks. By proving an equivalence between QT sequences and Williamson-type sequences in the circulant-block setting, the authors implement a highly efficient algorithm and exhaustively enumerate QT sequences up to length $n\le 21$, yielding new nonequivalent QHMs and confirming the abundance of such matrices for larger orders. They also develop analytical results on circulant-block Williamson-type matrices, construct infinite families of QHMs via quaternion automorphisms, and analyze small-order cases (notably order $5$ and $7$) to show nonequivalences with prior classes. The findings have implications for quantum information applications (e.g., MUMs) and suggest a rich landscape of quaternionic Hadamard matrices beyond previously known families, including fixed-entry-pattern characterizations.
Abstract
A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in Williamson's construction of quaternion-type Hadamard matrices. Using this correspondence, we devise an enumeration algorithm that is significantly faster than previously used algorithms and does not require the sequences to be symmetric. We implement our algorithm and use it to enumerate all circulant and possibly non-symmetric Williamson-type matrices of orders up to 21; previously, the largest order exhaustively enumerated was 13. We prove that when the blocks of a quaternion-type Hadamard matrix are circulant, the blocks are necessarily pairwise amicable. This dramatically improves the filtering power of our algorithm: in order 20, the number of block pairs needing consideration is reduced by a factor of over 25,000. We use our results to construct quaternionic Hadamard matrices of interest in quantum communication and prove they are not equivalent to those constructed by other means. We also study the properties of quaternionic Hadamard matrices analytically, and demonstrate the feasibility of characterizing quaternionic Hadamard matrices with a fixed pattern of entries. These results indicate a richer set of properties and suggest an abundance of quaternionic Hadamard matrices for sufficiently large orders.