Quantum Gates

This table fixes common quantum-information gate conventions. It is a matrix lookup, not a circuit-design guide.

Convention

One-qubit matrices use the computational basis $\lvert0\rangle,\lvert1\rangle$. Two-qubit matrices use the basis order

$$\lvert00\rangle,\ \lvert01\rangle,\ \lvert10\rangle,\ \lvert11\rangle.$$

Gates act on state column vectors from the left.

One-Qubit Gates

Gate	Matrix	Action
$I$	$\begin{pmatrix}1&0\\0&1\end{pmatrix}$	identity
$X$	$\begin{pmatrix}0&1\\1&0\end{pmatrix}$	bit flip
$Y$	$\begin{pmatrix}0&-i\\i&0\end{pmatrix}$	bit and phase flip
$Z$	$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$	phase flip
$H$	$\frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$	Hadamard basis change
$S$	$\begin{pmatrix}1&0\\0&i\end{pmatrix}$	phase gate, $S^2=Z$
$T$	$\begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}$	$\pi/8$ phase gate

Two-Qubit Gates

Gate	Matrix or Action	Note
CNOT	$\lvert a,b\rangle\mapsto\lvert a,b\oplus a\rangle$	first qubit controls second
CZ	$\operatorname{diag}(1,1,1,-1)$	symmetric controlled phase
SWAP	$\lvert a,b\rangle\mapsto\lvert b,a\rangle$	exchanges tensor factors

In the stated basis, CNOT is

$$\mathrm{CNOT} = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}.$$

Common Mistakes

• Reversing qubit order between matrix convention and circuit diagram.
• Treating a global phase as a distinct physical gate on a closed state.
• Forgetting that $Y$ differs by phases from a simple bit flip.
• Comparing CNOT matrices from sources that use opposite control-target ordering.

Canonical Links

• Quantum Gates Formula Card
• Circuit Model
• Single-Qubit Gates
• Multi-Qubit Gates
• Stabilizer Identities
• Tensor Product Exercises
• Quantum Information Problem Map

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.