  In this paper an alternative version of the quantum phase estimation is
proposed, in which the Hadamard gates at the beginning are substituted by a
quantum Fourier transform. This new circuit coincides with the original one,
when the ancilla is initialized with $|0\rangle$. With the help of a
projection-based tensor decomposition and closed-form expressions of its
exponential, this new method can be interpreted as a multiplier coupled to the
Hamiltonian of the corresponding target unitary operator. Based on this
observation a recursive decomposition is derived.
