The question boils down to whether it is possible to compare a given unknown state with a known state, without disturbing it. The answer is clearly a no. Because, doing so is equivalent to determining the complete state.
The measure used to say how close is a state to a given state is fidelity. If $\psi$ is the unknown arbitrary state, and $\phi$ is a known state, then the fidelity of $\psi$ with respect to $\phi$ is $|\langle\psi|\phi\rangle|^2$, which is $Tr[|\psi\rangle\langle\psi|\ |\phi\rangle\langle\phi|]$(that is the expectation value of $|\phi\rangle\langle\phi|$, if you like). If it were possible to measure this quantity using a single copy of $\psi$, it means that it is possible to measure the expectation value of any observable; since every hermitian operator is a sum of such projections, weighted with the eigen values.
And since measurement is just a comparison, finding the closest state to a given state equivalent to measuring it's fidelity with every state.
So, just like cloning, comparison is also impossible.