# Prepare superposition of quantum states weighted by fidelity with reference state

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.

• How are the input states given to you? Are they a list of quantum states in qubits, or are they classical descriptions of the quantum states, or just the descriptions of the unitaries? Aug 14 at 17:39

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.