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--- pauli-gate [May 11, 2026 at 15:18] – created yanevskiv
+++ pauli-gate [May 14, 2026 at 11:38] (current) – external edit 127.0.0.1
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+# Pauli gate
+**Pauli gates** are one of the three quantum gates $(X, Y, Z)$, which correspond to Pauli matrices $(\sigma_x, \sigma_y, \sigma_z)$, with the inclusion of the identity gate $I$.
+For a single qubit, they take the following matrix form
+$$I = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\quad
+X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\quad
+Y = \begin{pmatrix}0 & i \\ -i & 0 \end{pmatrix}\quad
+Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$$
+They are involutory, meaning they square up to identity matrix
+$$I^2 = X^2 = Y^2 = Z^2 = -iXYZ = I$$
+Multiplying two gates produces the third gate with an induced global phase shift of $\pi/2$ radians (because $e^{i\pi / 2} = i$)
+$$XY = iZ \qquad YZ = iX \qquad ZX = iY$$
+The Pauli gates $(X, Y, Z)$ anticommute, meaning multiplying them in revrse order produces a minus sign (equivalent to a global phase shift of $\pi$ radians, since $e^{i\pi} = -1$).
+$$XY = -YX\qquad YZ = -ZY\qquad ZX = -XZ$$