  Quantum signal processing and quantum singular value transformation are
powerful tools to implement polynomial transformations of block-encoded
matrices on quantum computers, and has achieved asymptotically optimal
complexity in many prominent quantum algorithms. We propose a framework of
quantum signal processing and quantum singular value transformation on $U(N)$,
which realizes multiple polynomials simultaneously from a block-encoded input,
as a generalization of those on $U(2)$ in the original frameworks. We also
perform a comprehensive analysis on achievable polynomials and give a recursive
algorithm to construct the quantum circuit that gives the desired polynomial
transformation. As an example application, the new framework is used to
construct the quantum amplitude estimation algorithm with asymptotically
optimal query complexity. Based on this framework, we also propose a framework
to realize multi-variate polynomial functions for commutative variables.
