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 \(\phi = \text{arctan2}(n_y, n_x)\), 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

1. Apply \(R_z(-\phi)\)
2. Apply \(R_y(-\theta)\)
3. Measure in the Z basis

Common examples

Measurement axis	Axis	Decomposition
Z	\((0, 0, 1)\)	\(I\)
X	\((1, 0, 0)\)	\(R_y(-\pi/2)\)
Y	\((0, 1, 0)\)	\(R_y(-\pi/2) \cdot R_z(-\pi/2)\)
H	\(\tfrac{1}{\sqrt{2}}(1, 0, 1)\)	\(R_y(-\pi/4)\)