# Rotation of multi-body interaction in quantum circuit

In quantum circuit, how do you implement the rotation of multi-body interaction, such as $$e^{-i\theta\sigma_z^1\sigma_z^2\sigma_z^3}$$? I already know the case of less than two-body interaction, but I cannot find any textbook about more than three-body interaction.

• You may refer to answer given here – Omkar Apr 9 at 4:30
• @Omkar Thank you very much – sotowa Apr 13 at 3:36

## 1 Answer

Here's an image from a previous answer of mine: If you replace the $$\sigma_1\otimes\sigma_2\otimes\ldots\otimes\sigma_n$$ with the tensor product of operators that you want (a single tensor product; a sum of terms needs some extra techniques based on, at its most simplistic, a Trotter expansion), and set the phase of the phase gate, $$t$$ equal to $$-2\theta$$, this will do the job up to a global phase.

Basically, what the circuit does is it entangles the register with the ancilla such that the ancilla is in the state 0/1 or the other register is in a +1 or -1 eigenstate of the operator respectively. That means you can use the ancilla to decide if you need to acquire a phase, before undoing the entanglement.