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--- quantum-gate-single-qubit [August 26, 2026 at 16:41] – Ivan Janevski
+++ quantum-gate-single-qubit [August 26, 2026 at 18:16] (current) – Ivan Janevski
@@ Line 9: / Line 9: @@
 ## List of gates
-- **Pauli gates**
+  - [[quantum-gate-pauli]]
-  - [[quantum-gate-i|Identity (I)]]
+    - [[quantum-gate-i|Identity (I)]]
-  - [[quantum-gate-x|Pauli X (NOT)]]
+    - [[quantum-gate-x|Pauli X (NOT)]]
-  - [[quantum-gate-y|Pauli Y]]
+    - [[quantum-gate-y|Pauli Y]]
-  - [[quantum-gate-z|Pauli Z]]
+    - [[quantum-gate-z|Pauli Z]]
-- **Hadamard**
   - [[quantum-gate-h|Hadamard (H)]]
-- **Phase gates**
+  - [[quantum-gate-phase]]
-  - [[quantum-gate-s|S gate (phase)]]
+    - [[quantum-gate-s|S (phase) gate]]
-  - [[quantum-gate-s-dagger|S† (inverse phase)]]
+    - [[quantum-gate-s-dagger|S† (inverse phase gate)]]
-  - [[quantum-gate-t|T gate]]
+    - [[quantum-gate-t|T gate]]
-  - [[quantum-gate-t-dagger|T† (inverse T)]]
+    - [[quantum-gate-t-dagger|T† gate (inverse T gate)]]
-- **Parametric rotations**
+  - [[quantum-gate-rotation]]
-  - [[quantum-gate-rx|RX (rotation around X)]]
+    - [[quantum-gate-rx|Rx (rotation around X)]]
-  - [[quantum-gate-ry|RY (rotation around Y)]]
+    - [[quantum-gate-ry|Ry (rotation around Y)]]
-  - [[quantum-gate-rz|RZ (rotation around Z)]]
+    - [[quantum-gate-rz|Rz (rotation around Z)]]
-- **Universal**
+    - [[quantum-gate-u|U (universal single-qubit)]]
-  - [[quantum-gate-u|U (universal single-qubit)]]
 ## Bloch sphere action
@@ Line 33: / Line 31: @@
 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