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--- quantum-gate-single-qubit [August 26, 2026 at 14:59] – Ivan Janevski
+++ 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 $\theta$).
-## Universality
+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.
-Any single-qubit unitary matrix belongs to $\mathrm{SU}(2)$ and can be decomposed into rotations around chosen axes. The universal gate $U(\theta, \phi, \lambda)$ parameterizes all single-qubit unitaries; any gate in the list below can be recovered by setting its three angles appropriately.
+## List of gates
-- [[quantum-gate-u|Universal single-qubit (U)]]
+  - [[quantum-gate-pauli]]
+    - [[quantum-gate-i|Identity (I)]]
+    - [[quantum-gate-x|Pauli X (NOT)]]
+    - [[quantum-gate-y|Pauli Y]]
+    - [[quantum-gate-z|Pauli Z]]
+  - [[quantum-gate-h|Hadamard (H)]]
+  - [[quantum-gate-phase]]
+    - [[quantum-gate-s|S (phase) gate]]
+    - [[quantum-gate-s-dagger|S† (inverse phase gate)]]
+    - [[quantum-gate-t|T gate]]
+    - [[quantum-gate-t-dagger|T† gate (inverse T gate)]]
+  - [[quantum-gate-rotation]]
+    - [[quantum-gate-rx|Rx (rotation around X)]]
+    - [[quantum-gate-ry|Ry (rotation around Y)]]
+    - [[quantum-gate-rz|Rz (rotation around Z)]]
+    - [[quantum-gate-u|U (universal single-qubit)]]
-## Decomposition
+## Bloch sphere action
-A common decomposition is the **ZYZ decomposition**: any single-qubit unitary can be written as $U = R_Z(\phi) R_Y(\theta) R_Z(\lambda)$ for angles $\phi, \theta, \lambda$. Other decompositions exist (e.g., XYX); the choice depends on available native gates in hardware.
+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: $U(\phi, \theta, \lambda)$.
-- [[quantum-gate-rx|RX (rotation around X)]]
+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.
-- [[quantum-gate-ry|RY (rotation around Y)]]
-- [[quantum-gate-rz|RZ (rotation around Z)]]
-- [[quantum-gate-u|Universal single-qubit (U)]]
-## Clifford
+## Dagger
-**Clifford gates** (X, Y, Z, H, S) map Pauli strings to Pauli strings and can be simulated classically in polynomial time. They form the stabilizer code formalism and are cheap to implement on most quantum hardware.
+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.
-- [[quantum-gate-x|Pauli X (NOT)]]
+## Square root
-- [[quantum-gate-y|Pauli Y]]
-- [[quantum-gate-z|Pauli Z]]
-- [[quantum-gate-h|Hadamard (H)]]
-- [[quantum-gate-s|S (Phase) gate]]
-**Non-Clifford gates** (primarily the T gate) are required for universal quantum computation. T gates are expensive in fault-tolerant quantum error correction, making their count a key resource metric. The magic state distillation protocol converts many noisy non-Clifford states into fewer high-fidelity ones.
+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 $U$.
-- [[quantum-gate-t|T gate]]
+## Matrix representations
-## Parametrized rotation (RX, RY, RZ)
+**Pauli gates:**
-The rotation gates $R_X(\theta)$, $R_Y(\theta)$, $R_Z(\theta)$ are parameterized rotations around the x, y, z axes by angle $\theta$. The Lie group $\mathrm{SU}(2)$ is isomorphic to the rotation group $\mathrm{SO}(3)$, so these three families generate all single-qubit rotations; any two rotations around different axes compose to form a rotation around a third axis.
+$$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \quad X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
-- [[quantum-gate-rx|RX (rotation around X)]]
+**Hadamard:**
-- [[quantum-gate-ry|RY (rotation around Y)]]
-- [[quantum-gate-rz|RZ (rotation around Z)]]
-## Quaternion rotation (U gate)
+$$H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$
-The universal gate $U(\theta, \phi, \lambda)$ represents a quaternion rotation on the Bloch sphere. The Lie group $\mathrm{SU}(2)$ is isomorphic to the group of unit quaternions $\mathrm{Sp}(1)$, which parameterize all 3D rotations; the three parameters of $U$ encode the axis and angle of rotation. Pauli and Hadamard gates are special cases of $U$ with specific angle choices.
+**Clifford phase gates:**
-- [[quantum-gate-u|Universal single-qubit (U)]]
+$$S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \quad S^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}$$
-## Gates
+**Non-Clifford phase gates:**
+$$T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix} \quad T^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & e^{-i\pi/4} \end{pmatrix}$$
+**Parametric rotation gates (angle $\theta$):**
+$$R_X(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix} \quad R_Y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix} \quad R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$
+Note: T and T† are special cases ($T = R_Z(\pi/4)$, $T^\dagger = R_Z(-\pi/4)$).
+## Clifford vs non-Clifford
+Single-qubit gates partition into two classes:
+**Clifford gates** (24 total) conjugate [[quantum-gate-pauli|Pauli operators]] to Paulis: if $C$ is Clifford and $P$ is Pauli, then $C P C^\dagger$ is also Pauli (up to global phase). Single-qubit Cliffords include: Pauli gates (I, X, Y, Z), Hadamard, S, and S†. Examples:
+- $H X H = Z$, $H Z H = X$
+- $S X S^\dagger = Y$, $S Z S^\dagger = Z$
+Clifford circuits are efficiently simulatable classically using the stabilizer tableau method in $O(n^3)$ time; they cannot provide quantum advantage alone.
+**Non-Clifford gates** (T, T†) break the Pauli closure property:
+$$T X T^\dagger = \frac{1}{\sqrt{2}}(X + Y), \quad T Z T^\dagger = Z$$
+The result is a superposition of Paulis, not a Pauli itself. This breakdown is precisely what enables universal quantum computation and exponential speedup. However, non-Clifford gates require [[quantum-gate-stabilizer-formalism|magic state distillation]] in fault-tolerant systems, making T-count the dominant cost metric.
+## Group structure
+Single-qubit gates form the Lie group $\mathrm{SU}(2)$, the group of $2 \times 2$ unitary matrices with determinant 1. This group is isomorphic to $\mathrm{Sp}(1)$ (compact symplectic group) and to $\mathrm{SO}(3)$ (3D rotations), explaining why single-qubit gates correspond to Bloch sphere rotations.
+The Clifford subgroup $\mathcal{C}_1$ has exactly 24 elements. These are generated by Hadamard and S: $(HS)^3 = I$. Any Clifford can be decomposed into H, S, and CNOT gates (when acting on multiple qubits).
+The group $\mathrm{SU}(2)$ is continuous and infinite-dimensional. Single-qubit rotations at arbitrary angles form a dense subset; any unitary can be approximated to desired accuracy using the Solovay-Kitaev theorem with $O(\log^c(1/\epsilon))$ gates.
+## Euler angle decomposition
+Any single-qubit unitary can be expressed as a product of rotations around two perpendicular axes. The standard parametrization is:
+$$U(\phi, \theta, \lambda) = R_Z(\phi) R_Y(\theta) R_Z(\lambda)$$
+where $\phi, \theta, \lambda \in [0, 2\pi)$. Alternative decompositions exist (e.g., $R_X R_Y R_Z$) depending on axis ordering. The [[quantum-gate-u|U gate]] implements this parametrization directly.
+This decomposition proves that any single-qubit unitary requires at most three rotations. Combined with two-qubit gates (like CNOT), the Euler angles form a universal gate set.
+## Uses
+- **State preparation**: Rotate computational basis states to arbitrary points on the Bloch sphere
+- **Circuit synthesis**: Decompose arbitrary single-qubit unitaries into rotation sequences
+- **Variational algorithms**: [[cuda-q-vqe|VQE]] and [[cuda-q-qaoa|QAOA]] use parameterized rotation layers as ansatze
+- **Measurement basis rotation**: Apply single-qubit gates before measurement to extract different observables (e.g., H before Z-basis measurement gives X-basis measurement)
+- **Quantum error correction**: Pauli and Hadamard gates measure stabilizers in different bases without collapsing encoded information
+- **Basis switching**: Prepare for operations in rotated bases (Hadamard for X basis, S for Y basis)
+## Implementation
+Gate performance varies by platform:
+**Superconducting qubits**:
+- Pauli rotations (RX, RY): ~20–40 ns; achieved via microwave pulses
+- RZ gates: typically virtual (reference frame adjustment), free cost
+- Hadamard: ~20–50 ns; often decomposed as RZ + RX + RZ
+- S, S† gates: virtual on most systems; no pulse needed
+- T, T† gates: ~20–30 ns when implemented as RZ rotations; bottleneck is magic state distillation for fault tolerance, not gate time
+- Fidelity: typically 99–99.9%
+**Trapped ions**:
+- Single-qubit gates: ~1–5 μs via laser pulses at multiple frequencies
+- RZ gates: detuned pulse (leaves Rabi frequency small)
+- Fidelity: 99.9%+ achievable
+**Photonic**:
+- Rotations via beam splitters and optical wave plates
+- RZ via optical phase shifters
+- Fidelity: ~99% (limited by optical component precision)
+## Relations
+- [[quantum-gate-pauli|Pauli gates]]: X, Y, Z are 180° Bloch rotations and Clifford gates
+- [[quantum-gate-h|Hadamard (H)]]: basis rotation; single most important Clifford gate
+- [[quantum-gate-phase|Phase gates]]: S, S†, T, T† are rotations around z-axis
+- [[quantum-gate-rotation|Rotation gates]]: $R_X$, $R_Y$, $R_Z$ parametrize Bloch rotations
+- [[quantum-gate-u|U gate]]: universal single-qubit; implements Euler angle decomposition
+- [[quantum-gate-clifford|Clifford gates]]: single-qubit Cliffords form 24-element subgroup of $\mathrm{SU}(2)$
+- [[quantum-gate-non-clifford|Non-Clifford gates]]: T and T† enable universality but require magic state distillation
+- [[quantum-gate-two-qubit|Two-qubit gates]]: single-qubit gates plus CNOT form universal gate set
+- $\mathrm{SU}(2)$: group structure of single-qubit unitaries
+- Bloch sphere: geometric interpretation of single-qubit operations
-- [[quantum-gate-i|Identity (I)]]
-- [[quantum-gate-x|Pauli X (NOT)]]
-- [[quantum-gate-y|Pauli Y]]
-- [[quantum-gate-z|Pauli Z]]
-- [[quantum-gate-h|Hadamard (H)]]
-- [[quantum-gate-s|S (Phase) gate]]
-- [[quantum-gate-t|T gate]]
-- [[quantum-gate-rx|RX (rotation around X)]]
-- [[quantum-gate-ry|RY (rotation around Y)]]
-- [[quantum-gate-rz|RZ (rotation around Z)]]
-- [[quantum-gate-u|Universal single-qubit (U)]]