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--- quantum-gate-single-qubit [August 26, 2026 at 16:45] – Ivan Janevski
+++ quantum-gate-single-qubit [August 26, 2026 at 18:16] (current) – Ivan Janevski
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 Composing rotations around different axes yields a rotation around a third axis. The $2 \times 2$ unitary matrices $\mathrm{SU}(2)$ are isomorphic to the unit quaternions $\mathrm{Sp}(1)$, which naturally represent 3D rotations via quaternion multiplication. This explains the double-cover relationship between $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$: two distinct quantum gates (differing by a global phase factor of $-1$) represent the same Bloch sphere rotation.
+## Dagger
+The dagger operator (†) computes the conjugate transpose of a gate's matrix. For unitary gates, $U^\dagger = U^{-1}$, meaning the dagger is the inverse: applying $U^\dagger$ undoes $U$ ($U^\dagger U = I$). On the Bloch sphere, dagger reverses the direction of rotation: a rotation by angle $\theta$ becomes a rotation by angle $-\theta$. This explains why $U^\dagger U = I$—two opposite rotations cancel.
+## Square root
+Taking the square root of a gate produces a gate that, applied twice, yields the original. Examples: $T^2 = S$ (T is the square root of S), $S^2 = Z$ (S is the square root of Z). On the Bloch sphere, square root halves the rotation angle: a gate rotating by $\theta$ yields a square root rotating by $\theta/2$. This explains why square roots compose: $\sqrt{U}$ applied twice returns to $U$.
 ## Matrix representations