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.

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

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

• 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 $U$

• 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., $U U^\dagger = I$)

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

• 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