quantum gate directory

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The magic gate is a fixed two-qubit unitary that performs a basis change to the magic basis (a Bell basis with specific phases).

$$ M = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & i & 0 & 0 \\ 0 & 0 & i & 1 \\ 0 & 0 & i & -1 \\ 1 & -i & 0 & 0 \end{bmatrix} $$

Properties

• Columns are (phase-adjusted) Bell states, so it maps the computational basis to the magic basis: $M|00\rangle = \tfrac{|00\rangle + |11\rangle}{\sqrt{2}}$, $M|01\rangle = \tfrac{i(|00\rangle - |11\rangle)}{\sqrt{2}}$, $M|10\rangle = \tfrac{i(|01\rangle + |10\rangle)}{\sqrt{2}}$, $M|11\rangle = \tfrac{|01\rangle - |10\rangle}{\sqrt{2}}$.
• If $U \in \mathsf{SO}(4)$, then $M U M^\dagger = A \otimes B$ for some $A,B \in \mathsf{SU}(2)$ (the $\mathsf{SO}(4) \simeq (\mathsf{SU}(2)\times\mathsf{SU}(2))/\mathbb{Z}_2$ correspondence).