  Quantum signal processing (QSP) and the quantum singular value transformation
(QSVT) are pivotal tools for simplifying the development of quantum algorithms.
These techniques leverage polynomial transformations on the eigenvalues or
singular values of block-encoded matrices, achieved with the use of just one
control qubit. In contexts where the control qubit is used to extract
information about the eigenvalues or singular values, the amount of extractable
information is limited to one bit per protocol. In this work, we extend the
original QSP ansatz by introducing multiple control qubits. We show that, much
like in the single-qubit case, nearly any vector of polynomials can be
implemented with a multi-qubit QSP ansatz, and the gate complexity scales
polynomially with the dimension of such states. Moreover, assuming that powers
of the matrix to transform are easily implementable - as in Shor's algorithm -
we can achieve polynomial transformations with degrees that scale exponentially
with the number of control qubits. This work aims to provide a partial
characterization of the polynomials that can be implemented using this
approach, with phase estimation schemes and discrete logarithm serving as
illustrative examples.
