Polylogarithmic quantum implementation of general orthogonal polynomial transforms

Establish whether the remaining steps in the DriscollHealyRockmore construction—converting between the Chebyshev basis and a general orthogonal-polynomial basis through its three-term recurrence—can be implemented with polylogarithmic cost.

Background

The paper constructs a quantum non-uniform Chebyshev transform by reducing it to two non-uniform quantum Fourier transforms. This is presented as the first step toward quantum versions of the broader discrete polynomial-transform pipeline of Driscoll, Healy, and Rockmore.

That broader pipeline applies to families of orthogonal polynomials satisfying a three-term recurrence and uses Chebyshev transforms as basic operations. The unresolved issue is whether the additional basis-conversion operations required for a general orthogonal family can retain polylogarithmic quantum complexity.