Quantum Gates

Formula

In the basis $|0\rangle,|1\rangle$ ,

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},

H=\frac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}, \quad S=\begin{pmatrix}1&0\\0&i\end{pmatrix}, \quad T=\begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}.

Rotations are

R_k(\theta)=e^{-i\theta\sigma_k/2} =\cos\frac\theta2I-i\sin\frac\theta2\sigma_k.

For control $A$ and target $B$ ,

C(U)=|0\rangle\langle0|_A\otimes I_B +|1\rangle\langle1|_A\otimes U_B.

In the ordered basis $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ ,

CNOT= \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}, \qquad CZ=\operatorname{diag}(1,1,1,-1),

SWAP= \begin{pmatrix} 1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1 \end{pmatrix}.

Assumptions and Conventions

State columns are acted on from the left; the rightmost matrix acts first.
The two-qubit basis order above treats the left label as the first tensor factor.
$U\otimes I$ acts on the first factor and $I\otimes U$ on the second.
The displayed objects are ideal unitaries, not hardware noise models.

Symbols

Symbol	Meaning
$X,Y,Z$	Pauli gates
$H$	Hadamard gate
$S,T$	phase and eighth-root phase gates
$R_k(\theta)$	Bloch-sphere rotation about axis $k$
$C(U)$	coherently controlled unitary

Validity and Warnings

Do not read a matrix product left-to-right in time.
State the basis order before using a multi-qubit matrix.
Keep the half-angle in $R_k(\theta)$ .
Equality up to a global phase gives the same isolated unitary channel but can fail after coherent control because the phase becomes relative.
Clifford gates alone are not universal for arbitrary quantum computation.
Measurement, reset, postselection, and noisy hardware operations are not unitaries on the visible register in general.

Canonical Treatment

Circuit conventions, controlled-phase subtleties, Bell-state preparation, universality, physical-operation cautions, verification workflow, exercises, and references are at Gates, Circuits, and Computation Models.

Detailed gate families are at Single-Qubit Gates and Multi-Qubit Gates.

The full projector-controlled, branch-phase, coherent-conditioning, and access-assumption audit is at Controlled Operations; this card remains a compact lookup aid.