J. Comput. Appl. Math.

Density theorems with applications in quantum signal processing

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We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain [0,1] is physically desired, these polynomial families are defined by bound constraints not just in [0,1], but also with additional bound constraints outside [0,1]. One might wonder then if these additional constraints inhibit their approximation properties within [0,1]. The main result of this paper is that this is not the case — the additional constraints do not hinder the ability of these polynomial families to approximate arbitrarily well any continuous function f:[0,1]→[0,1] in the supremum norm, provided f also matches any polynomial in the family at 0 and 1. We additionally study the specific problem of approximating the step function on [0,1] (with the step from 0 to 1 occurring at [Formula presented]) using one of these families, and propose two subfamilies of monotone and non-monotone approximations. For the non-monotone case, under some additional assumptions, we provide an iterative heuristic algorithm that finds the optimal polynomial approximation.


01 Oct 2023


J. Comput. Appl. Math.