An adversary bound for quantum signal processing

TL;DR

This paper introduces an adversary bound framework for quantum signal processing, providing a characterization of univariate protocols and extending the approach to multivariate cases, advancing quantum algorithm design.

Contribution

It recasts quantum signal processing as a state conversion problem and develops an adversary bound approach for multivariate extensions, offering a new analytical tool.

Findings

Adversary bound precisely characterizes univariate QSP protocols.

Feasible solutions to the adversary bound imply the existence of multivariate QSP protocols.

Protocol complexity can be reduced to a rank minimization problem.

Abstract

Quantum signal processing (QSP) and quantum singular value transformation (QSVT), have emerged as unifying frameworks in the context of quantum algorithm design. These techniques allow to carry out efficient polynomial transformations of matrices block-encoded in unitaries, involving a single ancilla qubit. Recent efforts try to extend QSP to the multivariate setting (M-QSP), where multiple matrices are transformed simultaneously. However, this generalization faces problems not encountered in the univariate counterpart: in particular, the class of polynomials achievable by M-QSP seems hard to characterize. In this work we borrow tools from query complexity, namely the state conversion problem and the adversary bound: we first recast QSP as a state conversion problem over the Hilbert space of square-integrable functions. We then show that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case. Motivated by this first result, we extend the formalism to several variables: the existence of a feasible solution to the adversary bound implies the existence of a M-QSP protocol, and the computation of a protocol of minimal space is reduced to a rank minimization problem involving the feasible solution space of the adversary bound.