# Identity Gate (I)

**Identity** (I or $I$) is the trivial single-qubit gate that leaves the quantum state unchanged. The quantum analog of the "do nothing" operation. Every quantum state is an eigenvector of the identity gate with eigenvalue 1.

Matrix:

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

Action: leaves all states unchanged.

$$I|\psi\rangle = |\psi\rangle$$

## Basis Action

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

## Properties

- **Self-inverse**: $I^2 = I$
- **Hermitian**: $I^\dagger = I$
- **Trivial gate**: no physical operation required
- **Eigenvalues**: all states are eigenstates with eigenvalue 1
- **Commutes with everything**: $[I, U] = 0$ for any unitary $U$

## Uses

- **Circuit padding**: align circuit depth when needed
- **Identity channels**: theoretical reference in error analysis
- **Placeholder**: represents "no operation" in quantum circuits
- **Mathematical proofs**: proving properties of other gates (e.g., $U U^\dagger = I$)

## Decomposition

- **Trivial**: identity gate is its own simplest form; no decomposition needed

## Implementation

- **All platforms**: identity operation is a no-op; requires no actual gate time or pulse
- **Cost**: negligible (zero gate time)
- **Hardware**: typically implemented by doing nothing during the time slot reserved for an operation