Exact & formalGatesSingle-qubit gate2026-07-17

Identity (I) gate

The single-qubit no-op gate that leaves every state exactly unchanged, used as a placeholder in circuit diagrams, a timing/idle slot on real hardware, and the base case for gate-composition identities.

Stabilizer

identitysingle qubitcliffordno-op

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I rarely appears as an algorithmic step, but it matters as the formal identity element of the single-qubit unitary group: every gate-composition identity in this catalog (e.g. S·S† = I, X² = I) is stated relative to it.

Circuit & simulation

q[0]

P(|0⟩) unchanged100%

P(|1⟩) unchanged0%

What this takes and returns

Takes1 qubitWhat joins here

Takes a 1-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 1 qubits can feed it.

15 entries meet this end. Named below.

Returns1 qubitWhat joins here

Returns a 1-qubit register a next stage can take. This is the end that joins.

13 entries meet this end. Named below.

A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →

Fits before this 15

Fits after this 13

How it works

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.

Implementation

FrameworkNative

identity.qasm

OPENQASM 3.0;
include "stdgates.inc";
qubit q;
id q;

Quantum vs classical

Classical baseline

Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.

Quantum claim

The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.

How to compare

Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.

Declared gaps

Nobody has reviewed this record for gaps yet.

Literature & references

Quantum Computation and Quantum Information: 10th Anniversary Edition2010 · Michael A. Nielsen and Isaac L. Chuang

Standard reference for gate matrices, the Clifford/non-Clifford distinction, and controlled-gate constructions used throughout this record.

doi.org/10.1017/cbo9780511976667 ↗

OpenQASM 3: A broader and deeper quantum assembly language2021 · Andrew W. Cross, Ali Javadi-Abhari, Thomas Alexander, Niel de Beaudrap, Lev S. Bishop, Steven Heidel, Colm A. Ryan, Prasahnt Sivarajah, John Smolin, Jay M. Gambetta, Blake R. Johnson

Defines the standard gate library (id, sdg, tdg, cy, crz, rxx, rzz, ccx, cx…) used natively by the code snippets on this record.

arxiv.org/abs/2104.14722 ↗