Gate	Matrix	Basis state action
$CNOT$	$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$	$$\begin{align} CNOT\vert 00 \rangle &= \vert 00 \rangle \\ CNOT\vert 01 \rangle &= \vert 01 \rangle \\ CNOT\vert 10 \rangle &= \vert 11 \rangle \\ CNOT\vert 11 \rangle &= \vert 10 \rangle \end{align}$$
$CZ$	$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$	$$\begin{align} CZ\vert 00 \rangle &= \vert 00 \rangle \\ CZ\vert 01 \rangle &= \vert 01 \rangle \\ CZ\vert 10 \rangle &= \vert 10 \rangle \\ CZ\vert 11 \rangle &= -\vert 11 \rangle \end{align}$$
$CRZ$	$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{-i \frac{\theta}{2}} & 0 \\ 0 & 0 & 0 & e^{i \frac{\theta}{2}} \end{pmatrix}$	$$\begin{align} CZ\vert 00 \rangle &= \vert 00 \rangle \\ CZ\vert 01 \rangle &= \vert 01 \rangle \\ CZ\vert 10 \rangle &= e^{-i \frac{\theta}{2}} \vert 10 \rangle \\ CZ\vert 11 \rangle &= e^{i \frac{\theta}{2}} \vert 11 \rangle \end{align}$$
$CRX$	$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \left(\frac{\theta}{2} \right) & -i \sin \left(\frac{\theta}{2} \right) \\ 0 & 0 & -i \sin \left(\frac{\theta}{2} \right) & \cos \left(\frac{\theta}{2} \right) \end{pmatrix}$	$$\begin{align} CRX(\theta)\vert 00 \rangle &= \vert 00 \rangle \\ CRX(\theta)\vert 01 \rangle &= \vert 01 \rangle \\ CRX(\theta)\vert 10 \rangle &= \cos \frac{\theta}{2} \vert 10 \rangle - i \sin \frac{\theta}{2} \vert 11\rangle \\ CRX(\theta)\vert 11 \rangle &= - i \sin \frac{\theta}{2} \vert 10 \rangle + \cos \frac{\theta}{2} \vert 11\rangle \end{align}$$
$CRY$	$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \left(\frac{\theta}{2} \right) & - \sin \left(\frac{\theta}{2} \right) \\ 0 & 0 & \sin \left(\frac{\theta}{2} \right) & \cos \left(\frac{\theta}{2} \right) \end{pmatrix}$	$$\begin{align} CRY(\theta)\vert 00 \rangle &= \vert 00 \rangle \\ CRY(\theta)\vert 01 \rangle &= \vert 01 \rangle \\ CRY(\theta)\vert 10 \rangle &= \cos \frac{\theta}{2} \vert 10 \rangle + \sin \frac{\theta}{2} \vert 11\rangle \\ CRY(\theta)\vert 11 \rangle &= -\sin \frac{\theta}{2} \vert 10 \rangle + \cos \frac{\theta}{2} \vert 11\rangle \end{align} $$
$CU$	$\begin{pmatrix} I_2 & 0 \\ 0 & U \\ \end{pmatrix}$	$$\begin{align} CU\vert 00 \rangle &= \vert 00 \rangle \\ CU\vert 01 \rangle &= \vert 01 \rangle \\ CU\vert 10 \rangle &= \vert 1 \rangle \otimes U\vert 0 \rangle \\ CU\vert 11 \rangle &= \vert 1 \rangle \otimes U \vert 1 \rangle \end{align}$$
$SWAP$	$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$	$$\begin{align} SWAP\vert 00 \rangle &= \vert 00 \rangle \\ SWAP\vert 01 \rangle &= \vert 10 \rangle \\ SWAP\vert 10 \rangle &= \vert 01 \rangle \\ SWAP\vert 11 \rangle &= \vert 11 \rangle \end{align}$$
$TOF$	$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \end{pmatrix}$	$$\begin{align} TOF\vert 000 \rangle &= \vert 000 \rangle \\ TOF\vert 001 \rangle &= \vert 001 \rangle \\ \vdots &= \vdots \\ TOF \vert 101 \rangle &= \vert 101 \rangle \\ TOF \vert 110 \rangle &= \vert 111 \rangle \\ TOF \vert 111 \rangle &= \vert 110 \rangle \end{align}$$
$CCZ$	$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}$	$$\begin{align} CCZ\vert 000 \rangle &= \vert 000 \rangle \\ \vdots &= \vdots \\ CCZ \vert 110 \rangle &= \vert 110 \rangle \\ CCZ \vert 111 \rangle &= -\vert 111 \rangle \end{align}$$

Codercise I.14.1 — The Bell states

Codercise I.14.2 — Quantum Multiplexer

Multi-qubit gate reference

Single-qubit gate reference