# Arbitrary $2\times 2$ unitary matrix using waveplates

I'm studying about the quantum computation using quantum optics and I wonder about how to implement arbitrary $$2\times 2$$ unitary operation using waveplates such as half wave plates and quarter wave plates.

Is it possible to implement arbitrary $$2\times 2$$ single qubit unitary operation using waveplates?

Short-answer: yes, it is! The whole business about using light's degrees of freedom for encoding information on qubits relies on using stuff like half and quarter-wave plates for changing polarization, or mode converters for working with transverse modes.

A bit longer-answer: A general SU(2) operation can be represented by three independent real parameters on the hypersphere $$S^3$$, and there are different types of parametrization. For example, it can be characterized as a rotation by an angle $$\psi \in [0, \pi]$$ around a unit vector $$\mathbf{\hat{n}}$$ oriented along the direction given by the sagittal $$(\theta \in [0, \pi])$$ and azimuthal $$(\phi \in [0, 2\pi])$$ angles, which gives $$\begin{eqnarray} U &=& R_{\mathbf{\hat{n}}}(\psi) = e^{-i\,\psi\,\mathbf{\hat{n}}\,\cdot\,\pmb{\sigma}} =\cos\psi\,I - i\sin\psi\, \mathbf{\hat{n}}\cdot\pmb{\sigma}\;, \nonumber\\ u &=& \cos\psi - i\cos\theta\sin\psi\;, \nonumber\\ w &=& -i\sin\theta\sin\psi \,e^{i\phi}\;. \label{eq:UR} \end{eqnarray}$$

Another possible parametrization uses the Euler angles, which are suitable for optical implementations with polarization components. In this parametrization, a general SU(2) matrix can be written as a sequence of three rotations characterized by the Euler angles $$(\varphi, \xi, \zeta)$$ % $$\begin{eqnarray} && U(\varphi,\xi,\zeta) = R_y(\varphi)\, R_z(-\xi)\, R_y(\zeta)\;, \nonumber\\ && u = \cos\xi\cos(\varphi+\zeta) + i \sin\xi\cos(\varphi-\zeta)\;, \nonumber\\ && w = \cos\xi\sin(\varphi+\zeta) + i \sin\xi\sin(\varphi-\zeta)\;. \label{SU2Euler} \end{eqnarray}$$

It provides a simple relationship between the group parameters and the orientations of retardation devices, such as half- and quarter-wave plates used for polarization transformations, or mode converters used for transverse mode operations.

According to the Jones matrix representation, the most general SU(2) operator can be decomposed as a product of waveplate operations in the following way:

$$\begin{eqnarray}\label{eq:QWP+HWP+QWP-Matrix} U &=& \text{QWP}(\eta_1) \, \text{HWP}(\tau)\, \text{QWP}(\eta_2)\;, \end{eqnarray}$$

where

$$\begin{eqnarray}\label{eq:QWP+HWP+QWP-Matrix2} \text{QWP}(\eta) &=& R_y(\eta)\,Q_0\,R_y(-\eta)\;, \nonumber\\ \text{HWP}(\tau) &=& R_y(\tau)\,H_0\,R_y(-\tau)\;, \nonumber\\ Q_0 &=&\left( \begin{array}{cc} 1 & \,\,\,0\\ 0 & \,\,\,i\\ \end{array} \right)\;, \nonumber \\ H_0 &=& \left( \begin{array}{cc} 1 & \,\,\,0\\ 0 & -1\\ \end{array} \right)\;. \end{eqnarray}$$

For a complete description and example take a look at this paper: Spin-orbit implementation of the Solovay-Kitaev decomposition of single-qubit channels