Applications for Fall 2025 are closed. Selected students will be notified in about 10 days following the application deadline.
WEAVING LOW-EXCESS CLASSES OF COMPLEX HADAMARD MATRICES
Supervisor: Robert Craigen
Level: 3000
Prerequisites: MATH 2090 or equivalent, MATH 1240, at least one CS course at the 2000+ year level focussing on advanced programming practice such as CS 2080 or CS 2160
Programming Skills: LaTeX familiarity desired but not required, C, Python or other similar modern general programming language, familiarity with at least one mathematical programming suite with Linear Algebraic tools such as Mathematica, Maple, Matlab, Magma or Gap
Project Description: A complex, or unit, Hadamard matrix H = UH(n) is an n x n matrix of complex numbers of unit modulus whose rows are orthogonal under the Euclidean inner product. The excess of a matrix is the sum of its entries.
Two matrices are said to be equivalent if one can be obtained from the other by permuting rows, permuting columns and multiplying rows and/or columns by units. The equivalence class of a matrix is its orbit under such operations. The maximum excess of an equivalence class is the largest absolute value of the excess of members of that class.
The excess of a class of UH(n)s cannot exceed n^(3/2), a value that can be attained in the real case in infinitely many, but not all, orders. For 50 years mathematicians have studied the largest values attainable in other (real) orders; relatively little work has been done in the complex case.Â
But in recent years work in Quantum Information Theory has spawned interest in the opposite question: How low can you go? Classes having low maximum excess are related to theoretical constructs permitting improved use of quantum information processing making better use of the so-called quantum advantage.
Weaving is a recursive system of matrix construction (developed in 1991 by the supervisor of this project) that is well-suited to constructing relatively large UH(n)s having unusually low maximum excess, potentially improving on all prior approaches.
We plan to implement this system in a computational environment to produce a small library of useful examples of such matrices for use in current work on quantum computer design. This will require maintaining large collections of data and mining these algorithmically for particularly useful components to be used in weaving.
Expected Outcome: We seek to fill out a library of examples, as complete as possible in low orders, of complex (unit) Hadamard matrices belonging to classes having maximum excess significantly lower than the theoretical maximum.
Additional Information: The first month of this project will be devoted to students mastering the mathematical components and literature on the subject, and doing initial explorations; meetings will be more frequent. Subsequently meetings will be less frequent and students will work more autonomously, with much direction via email and a shared Overleaf project to which all will contribute. Meetings will be a mixture of online and in-person.