# How to prepare Unitary Coupled Cluster ansatz for VQE in a circuit?

I have read the following paper by Dumitrescu et al.

To make a Unitary Coupled Cluster (UCC) ansatz, one prepares with the following equation. $$| \psi_{\rm{UCC}} \rangle = U(\theta)|\mathrm{HF}\rangle$$ Here, $$U(\theta) = \exp[ T(\theta) - T^{\dagger}(\theta) ]$$ and $$|\mathrm{HF}\rangle$$ is the Hartree-Fock state.

Let me consider a simple system where $$T(\theta) = \theta a_0^{\dagger}a_1$$. Then one obtains \begin{align} U(\theta) &= \exp[ \theta(a_0^{\dagger}a_1 - a_1^{\dagger}a_0) ] \;. \\ &= \exp[ i(\theta/2) (X_0Y_1 - X_1Y_0) ] \end{align}

Here, one applies Jordan-Wigner transformation.

So far, I can understand but hereafter how can I make a circuit? Should I expand and calculate $$U(\theta)|\mathrm{HF}\rangle$$ with Baler-Campbell-Hausdorff formula or something?

You can create the unitary gate for operator $$U(\theta)=e^{-i\frac{\theta}{2}Z_{0}Z{1}}$$ using two $$CNOT$$ operations and single rotation gate $$R_z$$:
For operators which contain different tensor products of Pauli matrices beside the product of $$Z$$ you have to change basis using appropriate unitary transformation: $$R_y(-\frac{\pi}{2})$$ changes $$X$$ basis to $$Z$$ basis, and $$R_{x}(\frac{\pi}{2})$$ changes $$Y$$ basis to $$Z$$ basis. For example for operator $$U(\theta)=e^{-i\frac{\theta}{2}X_{0}X{1}}$$ you have the following cirquit: