Step Optimize1qGatesDecomposition in qiskit transpilation

Question

Is there a way to know what does it does qiskit in the step Optimize1qGatesDecomposition?

What I can see is that each time qiskit wants to represent a rotation in SU(2) like $U(\theta,\alpha,\gamma)$, in order to write it in terms of $R_z$ and $\sqrt{X}$ gates, it writes $$ R_z(\alpha)\sqrt{X} R(\theta+\pi)\sqrt{X} R_z(\gamma). $$

However, when one has two such matrix multiplications

$$ R_z(\alpha_1)\sqrt{X} R(\theta_1)\sqrt{X} R_z(\gamma_!) \times R_z(\alpha_2)\sqrt{X} R(\theta_2)\sqrt{X} R_z(\gamma_2) $$

the optimization step ends up with a matrix like

$$ R_z(\alpha_3)\sqrt{X} R(x)\sqrt{X} R_z(\gamma_4), $$

where the angle $x$ is not necessarily the sum of the previous $\theta_i$'s (sometimes is not even $\theta_1 + \theta_2 + \pi$).

Answer 1

The transpiler pass Optimize1qGatesDecomposition searches for sequences of single-qubit gates, then for each sequence:

• Gets the matrix representation of each gate and combines them into one matrix using matrix multiplication.
• Resynthesizes the result matrix using OneQubitEulerDecomposer into the proper basis.
• Replaces the original sequence with the resynthesized one.

The synthesize basis is selected based on the basis argument if exists. Otherwise, the target basis is used. In your case the basis ZSXX is used. So, the matrix

\begin{align}\begin{aligned}\newcommand{\th}{\frac{\theta}{2}}\\\begin{split} \begin{pmatrix} \cos\left(\th\right) & -e^{i\lambda}\sin\left(\th\right) \\ e^{i\phi}\sin\left(\th\right) & e^{i(\phi+\lambda)}\cos\left(\th\right) \end{pmatrix}\end{split}\end{aligned}\end{align} is synthesized into $$e^{i\gamma} R_Z(\phi+\pi).\sqrt{X}.R_Z(\theta+\pi).\sqrt{X}.R_Z(\lambda)$$

where $\gamma$ is a phase parameter.

---

Note: You can access the source code of Optimize1qGatesDecomposition from here.