Quantum algorithm to add zeroes between two halves of a vector

Question

Is there a quantum gate or circuit that is capable of inputting any quantum state with $N$ qubits called $\left | \psi \right >$ and a state in $\left | 0 \right >$, and outputting a value that, in English, is “inserting $2^N$ zeroes in between the halves of $\psi$?”

More formally, I’ll state it like this.

Given a state $\psi = \left | 0, \psi_1 \right > + \left | 1, \psi_2 \right >$, is there a circuit, $C$, such that $C \left | \psi, 0 \right > = \left | 00, \psi_1 \right > + \left | 11, \psi_2 \right >$ ?

Answer 1

Yes, should be possible.

Using your notation, the state your starting with (on n+1 qubits) is: $$|0,\psi_1,0 \rangle + |1, \psi_2, 0 \rangle $$

You can apply n-1 swap gates in the following order:

• between qubit n+1 and n
• between qubit n and n-1
• between qubit n-1 and n-2 and so on, till
• between qubit 3 and 2

This leaves you in: $$|0,0,\psi_1\rangle + |1,0, \psi_2 \rangle $$

Then you can apply CNOT on qubit 1 and 2, which gives you

$$|0,0,\psi_1\rangle + |1,1, \psi_2 \rangle $$

as desired