Abstract 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. On the unit circle, this representation 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.
Operator spreading under stroboscopic time evolution due to a unitary is studied. An operator Krylov space is constructed and related to orthogonal polynomials on a unit circle (OPUC), as well as to the Krylov space of the edge operator of the Floquet transverse field Ising model with inhomogeneous couplings (ITFIM). The Verblunsky coefficients in the OPUC representation are related to the Krylov angles parameterizing the ITFIM. The relations between the OPUC and spectral functions are summarized and several applications are presented. These include derivation of analytic expressions for the OPUC for persistent
m-periodic dynamics, and the numerical construction of the OPUC for autocorrelations of the homogeneous Floquet-Ising model as well as the
Z3 clock model. The numerically obtained Krylov angles of the
Z3 clock model with long-lived period tripled autocorrelations show a spatial periodicity of six, and this observation is used to develop an analytically solvable model for the ITFIM that mimics this behavior.