An adversary bound for quantum signal processing

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 $|0\rangle \mapsto (P, Q)$ state conversion in this space precisely identifies all and only the QSP protocols over $SU(2)$ in the univariate case. Motivated by this result, we extend the formalism to M-QSP: the computation of a M-QSP protocol of minimal space is thus reduced to a rank minimization problem involving the feasible solution space of the adversary bound.