Identity (I) gate

Definition

$$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$

$I$ is the $2\times 2$ identity matrix: the unique unitary that acts trivially on every state, and the identity element of $U(2)$ under matrix multiplication.

Action on basis states

$$I|0\rangle = |0\rangle, \qquad I|1\rangle = |1\rangle$$

and by linearity $I(\alpha|0\rangle+\beta|1\rangle) = \alpha|0\rangle+\beta|1\rangle$ for any superposition — no amplitude or phase is touched.

Key identities

• $I$ is the identity element for gate composition: $UI = IU = U$ for every single-qubit unitary $U$.
• Every involutory gate in this catalog squares to $I$: $X^2=Y^2=Z^2=H^2=I$, and every phase-ladder gate returns to $I$ after enough applications: $S^4=I$, $T^8=I$.
• $I = R_Z(0) = P(0)$: the identity is the $\theta=0$ (or $\lambda=0$) special case of every parametrized rotation and phase gate in this catalog.
• On real hardware, an explicit identity or delay instruction is used to pad circuit depth for timing alignment; unlike a mathematical no-op, it still accumulates decoherence, which is why idle qubits are a genuine error source in circuit scheduling.