  The quantum singular value transformation is a powerful quantum algorithm
that allows one to apply a polynomial transformation to the singular values of
a matrix that is embedded as a block of a unitary transformation. This paper
shows how to perform the quantum singular value transformation for a matrix
that can be embedded as a block of a Hamiltonian. The transformation can be
implemented in a purely Hamiltonian context by the alternating application of
Hamiltonians for chosen intervals: it is an example of the Quantum Alternating
Operator Ansatz (generalized QAOA). We also show how to use the Hamiltonian
quantum singular value transformation to perform inverse block encoding to
implement a unitary of which a given Hamiltonian is a block. Inverse block
encoding leads to novel procedures for matrix multiplication and for solving
differential equations on quantum information processors in a purely
Hamiltonian fashion.
