# Decomposition of 2-qubit Hamiltonian into standard gate set for QAOA

I try to decompose ansatz into gate set in order to create a circuit in qiskit for QAOA algorithm. I don't understand how represent parametrized 2 qubit ansatz as circuit.

$$H{_B} = \sum_{j=1}^{n} {\sigma_j^x}$$

$$H_{A} = \frac{1}{2}\sigma_z^1 + \frac{1}{2}\sigma_z^1\otimes\sigma_z^2$$

Ansatz for p=1

$$\left| \gamma_1,\beta_1 \right\rangle = e^{-i\beta_1H_B} e^{-i\gamma_1H_A} \left| ++ \right\rangle$$

It is clear how a circuit for $$e^{-i\beta_1H_B}$$ looks like, but I stuck in decomposing $$e^{-i\gamma_1H_A}$$(more precisely it second term) into parametrised circuit acting on both qubits and depends on $$\gamma_1$$

Any help would be appreciated as well as any insight on multiple qubit decomposition.

Since $$\sigma_z^1 = I \otimes Z$$ and $$\sigma_z^1 \otimes \sigma_z^2 = Z \otimes Z$$ are commute with one another, that is $$[\sigma_z^1 , \sigma_z^1 \otimes \sigma_z^2 ] = \sigma_z^1 \cdot \sigma_z^1 \otimes \sigma_z^2 - \sigma_z^1 \otimes \sigma_z^2 \cdot \sigma_z^1 = \boldsymbol{0}$$ we have that $$e^{i\gamma_1 H_a} = e^{i \gamma \frac{1}{2}(\sigma_z^1 + \sigma_z^1 \otimes \sigma_z^2 ) } = e^{i \gamma \frac{1}{2}\sigma_z^1 } e^{i \gamma \frac{1}{2}\sigma_z^1 \otimes \sigma_z^2 }$$ and now note that $$e^{i \gamma \frac{1}{2}\sigma_z^1 }$$ has circuit construction as: (look here and here page 7 and 8)
and similarly, $$e^{i \gamma \frac{1}{2}\sigma_z^1 \otimes \sigma_z^2 }$$ have the circuit construction as:
and put them together, we have the circuit construction for $$e^{i \gamma \frac{1}{2}(\sigma_z^1 + \sigma_z^1 \otimes \sigma_z^2 ) }$$ as:
• @Masamune sorry, I got it drawn backward... I drew it as $IZ$ instead of $ZI$... I wrote $\sigma_z^1 = IZ$ instead of $\sigma_z^1 = ZI$... Commented Dec 14, 2020 at 0:45