A Solovay-Kitaev theorem for quantum signal processing

TL;DR

This paper establishes a Solovay-Kitaev type theorem for quantum signal processing, linking the density of circuit ansätze to the ability to efficiently approximate functions, thereby unifying aspects of QSP and gate approximation.

Contribution

It proves an SKT for QSP, providing a new framework that connects circuit density with function approximation, extending standard techniques and unifying quantum algorithms.

Findings

Proves an SKT for quantum signal processing.

Introduces 'lifted' variants of SKT proof techniques.

Establishes a formal link between QSP and gate approximation.

Abstract

Quantum signal processing (QSP) studies quantum circuits interleaving known unitaries (the phases) and unknown unitaries encoding a hidden scalar (the signal). For a wide class of functions one can quickly compute the phases applying a desired function to the signal; surprisingly, this ability can be shown to unify many quantum algorithms. A separate, basic subfield in quantum computing is gate approximation: among its results, the Solovay-Kitaev theorem (SKT) establishes an equivalence between the universality of a gate set and its ability to efficiently approximate other gates. In this work we prove an 'SKT for QSP,' showing that the density of parameterized circuit ans\"atze in classes of functions implies the existence of short circuits approximating desired functions. This is quite distinct from a pointwise application of the usual SKT, and yields a suite of independently interesting 'lifted' variants of standard SKT proof techniques. Our method furnishes alternative, flexible proofs for results in QSP, extends simply to ans\"atze for which standard QSP proof methods fail, and establishes a formal intersection between QSP and gate approximation.