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 d2. Zauner conjectured in 1999 that sets of d2 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 d2/4 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.