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 $R_z$:

enter image description here

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:

enter image description here

  • $\begingroup$ Thank you for your answer. $\endgroup$ – Ashy Dec 4 '19 at 8:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.