Decompose a general two-qubit gate into general controlled-qubit gates
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We often seek to decompose multi-qubit unitaries into single-qubit rotations and controlled-rotations, minimising the latter or restricting to gates like CNOTs. I'm interested in expressing a general 2-qubit unitary in the minimum total number of gates, which can include controlled general unitaries. That is, express $U_{4}$ with as few as possible gates in ${U_2,; |0⟩⟨0|mathbb{1} + |1⟩⟨1|U_2}$ . While I could simply take the shortest decomposition to CNOTs and rotations (Vatan et al) and bring some rotations into the CNOTs, I suspect another formulation could add more control-unitaries to achieve fewer total gates. How can I go about performing this decomposition algorithmically for any 2-qubit unitary? This decomposition could be useful for easily extending distributed quantum simulator...
