  Quantum computing can provide speedups in solving many problems as the
evolution of a quantum system is described by a unitary operator in an
exponentially large Hilbert space. Such unitary operators change the phase of
their eigenstates and make quantum algorithms fundamentally different from
their classical counterparts. Based on this unique principle of quantum
computing, we develop a new algorithmic toolbox "quantum phase processing" that
can directly apply arbitrary trigonometric transformations to eigenphases of a
unitary operator. The quantum phase processing circuit is constructed simply,
consisting of single-qubit rotations and controlled-unitaries, typically using
only one ancilla qubit. Besides the capability of phase transformation, quantum
phase processing in particular can extract the eigen-information of quantum
systems by simply measuring the ancilla qubit, making it naturally compatible
with indirect measurement. Quantum phase processing complements another
powerful framework known as quantum singular value transformation and leads to
more intuitive and efficient quantum algorithms for solving problems that are
particularly phase-related. As a notable application, we propose a new quantum
phase estimation algorithm without quantum Fourier transform, which requires
the fewest ancilla qubits and matches the best performance so far. We further
exploit the power of our method by investigating a plethora of applications in
Hamiltonian simulation, entanglement spectroscopy and quantum entropies
estimation, demonstrating improvements or optimality for almost all cases.
