Suppose I have two quantum states $|\psi_1\rangle,|\psi_2\rangle$ that I know how to prepare from some fixed initial state, say $|0\rangle^{\otimes N}$ on some $N$ qubits with known quantum circuits $U_1, U_2$.

My question is, can I leverage this information to create a third quantum circuit that performs arbitrary rotations in the two dimensional plane spanned by $|\psi_1\rangle,|\psi_2\rangle$? Such a transformation is perfectly well defined as my two states are simply two vectors in a complex vector space with the same norm and linked by a family of unitary transformations, with the geodesic path being the desired one that rotates $|\psi_1\rangle$ through $|\psi_2\rangle$, but creating a quantum circuit that performs this specific transformation seems rather tricky. I could do something with a Grover-like search but this is exponentially expensive in the worst case (which is the one I actually expect - in general my two states will be nearly orthogonal) and doesn't take advantage of the fact that I know my 'solution' state $|\psi_2\rangle$ nor will it give me the choice of rotation angle, only a number of steps I should apply my Grover iterate to approximate the desired angle.

This answer for creating arbitrary superpositions using 'coherent addition' seems applicable as does the linked paper but there one has two unknown quantum states which incurs a sampling cost. It is not clear to me if one can use the known preparation circuits $U_1, U_2$ in that protocol to gain, say, a deterministic quantum circuit that achieves the same thing.



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