Hamiltonian singular value transformation and inverse block encoding.

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.