  Quantum signal processing (QSP) is a framework which was proven to unify and
simplify a large number of known quantum algorithms, as well as discovering new
ones. QSP allows one to transform a signal embedded in a given unitary using
polynomials. Characterizing which polynomials can be achieved with QSP
protocols is an important part of the power of this technique, and while such a
characterization is well-understood in the case of univariate signals, it is
unclear which multivariate polynomials can be constructed when the signal is a
vector, rather than a scalar. This work uses a slightly different formalism
than what is found in the literature, and uses it to find simpler necessary
conditions for decomposability, as well as a sufficient condition - the first,
to the best of our knowledge, proven for a (generally inhomogeneous)
multivariate polynomial in the context of quantum signal processing.
