 Solution

Construct an economical two-state quantum cloner by moving both actual outputs slightly toward one another relative to the two desired copies. Set $|a⟩=|00⟩$ and $|b⟩=|++⟩$. Their overlap is $1/2$, so their sum and difference yield the orthonormal vectorsThe desired outputs have the symmetric expressionsChoose the actual outputs to beThey are normalized and satisfyConsequently the prescription $|00⟩\mapsto |c_0⟩$, $|+0⟩\mapsto |c_{+}⟩$ preserves the Gram matrix of the two inputs and defines a linear isometry on their span. The unitary extension of a finite-dimensional isometry gives a unitary operator on the same four-dimensional, two-qubit Hilbert space; no extra workspace is required.

For an explicit full-space construction, define $c=\cos (\pi /8)$, $s=\sin (\pi /8)$ and the orthonormal basisChoose two further orthonormal vectors perpendicular to $u,v$,Thenmaps one orthonormal basis to another and is therefore unitary. Since $|00⟩=c|p⟩+s|m⟩$ and $|+0⟩=c|p⟩-s|m⟩$, it has exactly the required two outputs.

Finally,Both inputs therefore have the same global cloning fidelity, and so does their average:This proves the requested existence with a two-qubit unitary gate.