Measure decomposer

The measure decomposer (MeasureDecomposer) decomposes measurements along an arbitrary axis into at most 2 single-qubit gates followed by a measurement along the Z-axis.

Theory

For a measurement axis defined by the unit vector $$\hat{n} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$$

where $\theta = \arccos(n_z)$ and $\phi = \arctan2(n_y, n_x)$, we construct the unitary transformation: $$U = R_z(\phi) R_y(\theta)$$

This unitary maps Z-axis eigenstates to eigenstates along the $\hat{n}$ direction. Measuring in the $\hat{n}$ basis is equivalent to applying the inverse transformation $U^\dagger = R_y(-\theta) R_z(-\phi)$ and then measuring along Z.

Decomposition steps

Apply $R_z(-\phi)$
Apply $R_y(-\theta)$
Measure in the Z basis

Common examples

Measurement axis	Decomposition
Z	$I$
X	$R_y(-\pi/2)$
Y	$R_y(-\pi/2) \cdot R_z(-\pi/2)$
H	$R_y(-\pi/4)$

Measurement axis	Decomposition
Z	\(I\)
X	\(R_y(-\pi/2)\)
Y	\(R_y(-\pi/2) \cdot R_z(-\pi/2)\)
H	\(R_y(-\pi/4)\)