  We find an optimal quantum cloning machine, which clones qubits of arbitrary
symmetrical distribution around the Bloch vector with the highest fidelity. The
process is referred to as phase-independent cloning in contrast to the standard
phase-covariant cloning for which an input qubit state is a priori better
known. We assume that the information about the input state is encoded in an
arbitrary axisymmetric distribution (phase function) on the Bloch sphere of the
cloned qubits. We find analytical expressions describing the optimal cloning
transformation and fidelity of the clones. As an illustration, we analyze
cloning of qubit state described by the von Mises-Fisher and Brosseau
distributions. Moreover, we show that the optimal phase-independent cloning
machine can be implemented by modifying the mirror phase-covariant cloning
machine for which quantum circuits are known.
