## Constructions of Complex Equiangular Lines

How many equiangular lines can be placed in *d*-dimensional complex space? This is a
highly challenging question, lying at the intersection of algebraic combinatorics and
quantum information theory.
A simple linear algebraic argument shows the answer to be at most
*d*^{2}. Zauner conjectured in 1999 that sets of
*d*^{2} equiangular lines indeed exist for every *d*,
and specified a potential construction method for such sets. His method
has been successfully applied for twenty dimensions *d*, but the
sets of lines it produces become enormously complicated as *d*
increases and the associated computations rapidly become infeasible.
It remains unclear whether Zauner's conjecture is true, and if so whether
his construction method can be successfully applied for infinitely
many values of *d*.

In 2014, Amy Wiebe and I proposed a radically different approach to the
construction of large sets of complex equiangular lines, involving the
modification of known combinatorial designs such as
complex Hadamard matrices
and
mutually unbiased bases derived from relative difference sets.
This new approach produces examples with transparent combinatorial structure,
including a simple set of *d*^{2}/4 equiangular lines for
infinitely many dimensions *d*. In 2015, we generalised the construction of
this set to give
large sets of complex and real equiangular lines.

The six diagonals of a regular icosahedron are equiangular in three-dimensional real space.