  We re-analyse the Bu\v{z}ek-Hillery Universal Quantum Cloning machine
protocol and show that it allows better values for fidelity and Hilbert-Schmidt
norm than hitherto reported. This higher value for the fidelity is identical to
the maximum fidelity of phase covariant quantum cloning of Bru\ss
-Cinchetti-D'Ariano-Macchiavello. This value of fidelity has also been obtained
by Niu and Griffiths in their work without machine states. This is the maximum
possible fidelity obtainable in $1\rightarrow 2$ qubits cloning. We then
describe a different and new state dependent cloning protocol with four machine
states where all non-exact copies of input states are taken into account in the
output and we use the Hessian method of determining extrema of multivariate
functions. The fidelity for the best overall quantum cloning in this protocol
is $\bar{F}=0.847$ with an associated von-Neumann entropy of $\bar{S}=0.825$.
