Inner product as unitary operation

Question

Inner products of two states $\psi$ and $\phi$ are usually performed at the end of a quantum algorithm where we measure the final state, e.g. using the swap test. However, this operation is not unitary. We cannot "undo" the inner product to recover the $\psi$ and $\phi$ states -- for the swap test, we must prepare $\psi$ and $\phi$ multiple times independently to get a good probability estimate and hence the value of inner product.

Has anyone developed a "unitary" way to perform the inner product? If so, we can use it as an intermediate step in some quantum algorithm, as it would be a unitary and reversible operation.

Answer 1

No, this violates the Holevo-Helstrom theorem.

Suppose you had a black box that computes the inner product between two states but leaves the states alone. You could create an algorithm using that black box to implement a state discrimination algorithm that always produces the correct state, even if the two states are not orthogonal.

Note that even in the swap test, you can't compute the inner product directly, instead you need multiple copies and perform the swap test multiple times and the average will be close to the inner product.