Prepare superposition of quantum states weighted by fidelity with reference state

Question

Given a list of $m$ quantum states $$|\phi_0\rangle, |\phi_1\rangle, ... |\phi_{m-1}\rangle$$

each on $n$ qubits, with unitaries to prepare these ($U_0, U_1, ...$), I'd like to prepare a superposition of these states weighted by their overlap with another quantum state ${|\psi\rangle}$, such that the resultant state is:

$$ \frac{1}{\sum_i |\langle \psi | \phi_i \rangle|^2} \sum_i |\langle \psi | \phi_i \rangle| | \phi_i \rangle $$

Is there a simple method to prepare such a state where the postselection cost is not exponential in $n$ or $m$?

I've looked at arXiv:1401.2142, but this requires amplitude estimation which I'd like to avoid.

Answer 1

Hm. If you removed the absolute value from the coefficients, you'd have $\sum_i \langle \phi_i|\psi\rangle|\phi_i\rangle$, which equals $|\psi\rangle$ if it is in the span of the $|\phi_i\rangle$ or else the projection of $|\psi\rangle$ onto that subspace. So maybe you want an algorithm that changes the phases of the coefficients. Now that is easy to do, if the states $|\phi_i\rangle$ are orthonormal: apply $$\exp(-i\arg \langle \phi_0|\psi\rangle)\exp(-i\arg \langle \phi_1|\psi\rangle)\cdots \exp(-i\arg \langle \phi_{m-1}|\psi\rangle)$$ to your state and you're done.

If all of these conditions are met (i.e. the states are orthonormal and $|\psi\rangle$ lies within their span) then all you need is to know/measure/learn the phase arguments of the overlaps $ \langle \phi_0|\psi\rangle$ and then they can all be fixed with unitaries.