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--- quantum-gate-pauli [August 26, 2026 at 15:47] – Ivan Janevski
+++ quantum-gate-pauli [August 26, 2026 at 15:58] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
 # Pauli gates
-**Pauli gates** (I, X, Y, Z) form the fundamental set of single-qubit gates that generate the Pauli group. They are Clifford gates and commute/anticommute in well-defined ways, making them essential for quantum error correction and circuit decomposition.
+**Pauli gates** (I, X, Y, Z) form the fundamental set of single-qubit gates that generate the Pauli group. They are [[quantum-gate-clifford|Clifford gates]] with eigenvalues ±1 and well-defined commutation relations, making them essential for quantum error correction, measurement, and circuit decomposition. Any single-qubit unitary can be expressed as a linear combination of Paulis.
-## The four pauli gates
+## The four Pauli gates
-- **[[quantum-gate-i|Identity (I)]]**: no-op; eigenvalues ±1
+- **[[quantum-gate-i|Identity (I)]]**: no-op; leaves all states unchanged
-- **[[quantum-gate-x|Pauli X (NOT)]]**: bit flip around x-axis
+- **[[quantum-gate-x|Pauli X (NOT)]]**: bit flip; swaps $|0\rangle \leftrightarrow |1\rangle$
-- **[[quantum-gate-y|Pauli Y]]**: rotation around y-axis; composition of X and Z
+- **[[quantum-gate-y|Pauli Y]]**: combined bit and phase flip; $Y = iXZ$
-- **[[quantum-gate-z|Pauli Z]]**: phase flip; $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
+- **[[quantum-gate-z|Pauli Z]]**: phase flip; applies $-1$ to $|1\rangle$ state
-## Algebraic properties
+## Matrix representations
-- **Involution**: All Paulis are self-inverse ($P^2 = I$ for all P)
+$$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \quad X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
-- **Anticommutation**: $\{P_i, P_j\} = 0$ for $i \neq j$ (any two distinct Paulis anticommute)
-- **Eigenvalues**: All Paulis have eigenvalues ±1
+X and Z have eigenvalues $\pm 1$ with eigenvectors in the $x$ and $z$ bases; Y has eigenvalues $\pm 1$ with eigenvectors in the $x$-$z$ diagonal basis.
-- **Completeness**: Any single-qubit unitary can be decomposed as a linear combination of Paulis
+## Commutation relations
+The Pauli matrices satisfy:
+- **Same axis**: $[P_i, P_i] = 0$ (commute with themselves)
+- **Different axes**: $\{P_i, P_j\} = 2\delta_{ij}$ for $i \neq j$ (anticommute): $XY = iZ$, $YZ = iX$, $ZX = iY$
+- **Reverse**: $P_i P_j = -P_j P_i$ for $i \neq j$
+## Key properties
+- **Involution**: All Paulis are self-inverse: $P^2 = I$ for all $P \in \{I, X, Y, Z\}$
+- **Unitary**: All Paulis are unitary; $P^\dagger = P$
+- **Trace**: $\mathrm{Tr}(P) = 0$ for $X, Y, Z$; $\mathrm{Tr}(I) = 2$
+- **Determinant**: $\det(X) = \det(Y) = \det(Z) = -1$; $\det(I) = 1$
+- **Completeness**: Any $2 \times 2$ unitary can be written as $U = a_0 I + a_1 X + a_2 Y + a_3 Z$ for complex $a_i$
+## Special cases and relations to rotations
+Pauli gates are special cases of [[quantum-gate-rotation|rotation gates]] at $\theta = \pi$ (up to global phase):
+- $X = R_X(\pi) \cdot e^{i\pi/2}$ (equivalently, $X = -i R_X(\pi)$)
+- $Y = R_Y(\pi) \cdot e^{i\pi/2}$ (equivalently, $Y = -i R_Y(\pi)$)
+- $Z = R_Z(\pi) \cdot e^{i\pi/2}$ (equivalently, $Z = -i R_Z(\pi)$)
+More generally, Pauli matrices appear as generators in rotation exponentials: $R_i(\theta) = e^{-i\theta \sigma_i/2}$ where $\sigma_i$ is the corresponding Pauli.
 ## Uses
-- **Error correction**: Syndrome measurements detect which Pauli error occurred
+- **Error correction**: Pauli syndrome measurements in quantum error-correcting codes detect which error type occurred
-- **Tomography**: Measuring expectations of all three X, Y, Z gives complete state info
+- **State tomography**: Measuring expectations $\langle X \rangle$, $\langle Y \rangle$, $\langle Z \rangle$ completely characterizes a qubit state
-- **Circuit synthesis**: Arbitrary unitaries decompose into Pauli rotations (RX, RY, RZ)
+- **Measurement basis**: Rotate before measurement to measure in different bases (e.g., X-basis requires $H$ rotation before Z measurement)
+- **Circuit decomposition**: Arbitrary single-qubit gates decompose into Pauli products and rotations
+- **Stabilizer formalism**: Stabilizer codes use Pauli group measurements for error detection without measuring the state directly
+## Decomposition and completeness
+Any $2 \times 2$ Hermitian matrix can be decomposed as $H = a_0 I + a_1 X + a_2 Y + a_3 Z$. For Hermitian observables, this means any single-qubit measurement decomposes into Pauli measurements:
+$$\langle \psi | H | \psi \rangle = a_0 + a_1 \langle X \rangle + a_2 \langle Y \rangle + a_3 \langle Z \rangle$$
+This is the basis for Pauli measurement grouping in quantum algorithms.
+## Implementation
+- **Measurement**: X and Y basis measurements require basis rotation before measuring in Z (computational) basis
+- **Superconducting qubits**: Paulis are realized as single microwave pulses or virtual operations (phase shifts)
+- **Trapped ions**: State-selective measurements (e.g., fluorescence) give Z measurements; other bases via basis rotation
+- **Photonic**: Measurement via beam splitters and detectors; basis rotation via wave plates
+- **Fidelity**: Pauli measurements ~99%+ for state-of-the-art platforms
+## Relations
+- [[quantum-gate-rotation|Rotation gates]]: Paulis are special cases of $R_i(\pi)$ up to global phase
+- [[quantum-gate-clifford|Clifford gates]]: All Pauli gates are Clifford
+- [[quantum-gate-single-qubit|Single-qubit gates]]: Paulis generate $\mathbb{C}^{2 \times 2}$ via linear combinations
+- [[quantum-gate-i|Identity (I)]], [[quantum-gate-x|X gate]], [[quantum-gate-y|Y gate]], [[quantum-gate-z|Z gate]]: Individual gate articles
+- [[quantum-gate-stabilizer-formalism|Stabilizer formalism]]: Uses Pauli group for error detection
+- Error correction: Pauli syndromes in quantum codes like surface codes and toric codes