  We study mechanisms that allow one to synchronize the quantum phase of two
qubits relative to a fixed basis. Starting from one qubit in a fixed reference
state and the other in an unknown state, we find that contrary to the
impossibility of perfect quantum cloning, the quantum-phase can be synchronized
perfectly through a joined unitary operation. When both qubits are initially in
a pure unknown state, perfect quantum-phase synchronization through unitary
operations becomes impossible. In this situation we determine the maximum
average quantum-phase synchronization fidelity, the distribution of relative
phases and fidelities, and identify optimal quantum circuits that achieve this
maximum fidelity. A subset of these optimal quantum circuits enable perfect
quantum-phase synchronization for a class of unknown initial states restricted
to the equatorial plane of the Bloch sphere.
