Can we prepare a superposition of two many body state efficiently using quantum circuit?

Question

Let's say we have two quantum many body states $|\psi_1\rangle$ and $|\psi_2\rangle$(or equivalently, two quantum circuit $U_1$ and $U_2$ ), also an ancilla qubit $\alpha|0\rangle+\beta|1\rangle$. The goal is to prepare a state $\alpha|\psi_1\rangle+\beta|\psi_2\rangle$.

Can this be done efficiently using quantum circuits?

If $|\psi_1\rangle$ and $|\psi_2\rangle$ are logical state in quantum error correction. This is called encoding circuit. My question is, in general, could this be done?

Answer 1

Yes.

Given a circuit for a unitary $U$ on $\mathbb C^d$, we can always build a circuit for a controlled-$U$ gate acting on $\mathbb C^2\otimes C^d$, this is, a gate acting as \begin{align} |0\rangle|x\rangle&\mapsto |0\rangle |x\rangle\\ |1\rangle|x\rangle&\mapsto |1\rangle (U| x\rangle) \end{align} (i.e., $U$ acts on $\mathbb C^d$ if the control qubit is $|1\rangle$). Such a gate is obtained by replacing any elementary gate $G$ in the circuit for $U$ by a controlled-$G$ gate. (E.g., if the gate set is single-qubit rotations and CNOTs, replace them by controlled unitaries and Toffolis, for which efficient circuits are known.)

Now you can build your state by starting as follows:

1. Start with $(\alpha|0\rangle + \beta |1\rangle)|x\rangle$.
2. Apply $U_1$ to the second register.
3. Apply a controlled-$(U_2U_1^{-1})$ gate.