Quaternionic Perfect Sequences and Hadamard Matrices

TL;DR

This paper explores quaternionic perfect sequences and their connection to Hadamard matrices, providing a faster enumeration algorithm, new classifications, and insights into their properties relevant for quantum communication.

Contribution

It introduces a novel enumeration algorithm for quaternion-type Hadamard matrices, proves structural properties, and constructs new matrices with applications in quantum communication.

Findings

Enumeration of matrices up to order 21, surpassing previous limits

Blocks in quaternion-type Hadamard matrices are necessarily amicable

Constructed quaternionic Hadamard matrices are not equivalent to known types

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.