  Quantum signal processing is a framework for implementing polynomial
functions on quantum computers. To implement a given polynomial $P$, one must
first construct a corresponding complementary polynomial $Q$. Existing
approaches to this problem employ numerical methods that are not amenable to
explicit error analysis. We present a new approach to complementary polynomials
using complex analysis. Our main mathematical result is a contour integral
representation for a canonical complementary polynomial. The integral
representation on the unit circle has a particularly simple and efficacious
Fourier analytic interpretation, which we use to develop a Fast Fourier
Transform-based algorithm for the efficient calculation of $Q$ in the monomial
basis with explicit error guarantees. Numerical evidence that our algorithm
outperforms the state-of-the-art optimization-based method for computing
complementary polynomials is provided.
