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--- quantum-gate-single-qubit [August 26, 2026 at 16:40] – external edit 127.0.0.1
+++ quantum-gate-single-qubit [August 26, 2026 at 18:16] (current) – Ivan Janevski
@@ Line 3: / Line 3: @@
 **Single-qubit gates** are unitary operations that act on one qubit. They are represented by $2 \times 2$ unitary matrices and form the Lie group $\mathrm{SU}(2)$. Every quantum computation can be decomposed into single-qubit rotations and two-qubit entangling gates (e.g., CNOT), making single-qubit gates the fundamental building blocks of quantum circuits.
-A single-qubit gate is applied to a state $|\psi\rangle$ to produce a new state $U|\psi\rangle$. The gates compose: two gates in sequence form a new gate (matrix product). Some gates are self-inverse (like Pauli gates and Hadamard); others have finite order ($S^4 = I$, $T^8 = I$); some are parameterized (RX, RY, RZ with rotation angle $\theta$). Physically, all single-qubit gates correspond to rotations on the Bloch sphere; any unitary in $\mathrm{SU}(2)$ can be decomposed as a product of rotations around two perpendicular axes.
+A single-qubit gate is applied to a state $|\psi\rangle$ to produce a new state $U|\psi\rangle$. The gates compose: two gates in sequence form a new gate (matrix product). Some gates are self-inverse (like Pauli gates and Hadamard); others have finite order ($S^4 = I$, $T^8 = I$); some are parameterized (RX, RY, RZ with rotation angle $\theta$).
+All single-qubit gates correspond to rotations on the Bloch sphere; any unitary in $\mathrm{SU}(2)$ can be decomposed as a product of rotations around two perpendicular axes.
 ## 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
-Single-qubit gates act as rotations on the Bloch sphere. [[quantum-gate-pauli|Pauli gates]] (X, Y, Z) are 180° rotations around their respective axes. The [[quantum-gate-h|Hadamard]] swaps the x and z axes. [[quantum-gate-phase|Phase gates]] (S, S†, T, T†) are rotations around the z-axis. [[quantum-gate-rotation|Rotation gates]] ($R_X$, $R_Y$, $R_Z$) parameterize arbitrary angles around each axis. The [[quantum-gate-u|universal gate]] specifies an arbitrary rotation via three Euler angles: $U(\phi, \theta, \lambda)$.
+Single-qubit gates act as rotations on the Bloch sphere. [[quantum-gate-pauli|Pauli gates]] (X, Y, Z) are 180° rotations around their respective axes. The [[quantum-gate-h|Hadamard]] swaps the x and z axes. [[quantum-gate-phase|Phase gates]] (S, S†, T, T†) are rotations around the z-axis. [[quantum-gate-rotation|Rotation gates]] ($R_X$, $R_Y$, $R_Z$) parameterize arbitrary angles around each axis. [[quantum-gate-u|Universal gate]] specifies an arbitrary rotation via three Euler angles: $U(\phi, \theta, \lambda)$.
 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