Gate Library

A gate name specifies a unitary only after basis, tensor order, angle, and phase conventions have been fixed. This library records exact representatives for calculation. Single-Qubit Gates, Multi-Qubit Gates, and Controlled Operations explain their actions and synthesis.

Single-qubit matrices

Use column vectors in the basis $(\lvert0\rangle,\lvert1\rangle)$; in a matrix product the rightmost factor acts first. Angles are in radians. These are the conventional representatives used by Nielsen and Chuang.

Gate	Exact matrix	Useful relation
Identity	$I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$	$I^2=I$
Pauli X	$X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$	$X^2=I$
Pauli Y	$Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$	$Y=iXZ$
Pauli Z	$Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$	$Z^2=I$
Hadamard	$H=\frac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$	$H^2=I$
Phase S	$S=\operatorname{diag}(1,i)$	$S^2=Z$
Phase T	$T=\operatorname{diag}(1,e^{i\pi/4})$	$T^2=S$
General phase	$P(\phi)=\operatorname{diag}(1,e^{i\phi})$	$P(\phi)^\dagger=P(-\phi)$

For $j=x,y,z$, define $R_j(\theta)=e^{-i\theta\sigma_j/2}$. Writing $c=\cos(\theta/2)$ and $s=\sin(\theta/2)$ gives

$$R_x(\theta)=\begin{pmatrix}c&-is\\-is&c\end{pmatrix},$$ $$R_y(\theta)=\begin{pmatrix}c&-s\\s&c\end{pmatrix},$$ $$R_z(\theta)=\operatorname{diag}(e^{-i\theta/2},e^{i\theta/2}).$$

Thus $P(\phi)=e^{i\phi/2}R_z(\phi)$ and $R_j(\pi)=-i\sigma_j$. The phase factor disappears from an isolated unitary channel, but may become a relative phase when that unitary is controlled. In particular, $R_j(2\pi)=-I$ is not exact matrix equality to $I$.

Gates on two and three qubits

Use the two-qubit basis $(\lvert00\rangle,\lvert01\rangle,\lvert10\rangle,\lvert11\rangle)$, with the first bit the left tensor factor. Here $a,b,c\in\{0,1\}$ and $\oplus$ denotes addition modulo two.

Gate	Action and qualification
CNOT, first qubit controls second	$\lvert a,b\rangle\mapsto\lvert a,b\oplus a\rangle$ Reverse control direction is a different matrix.
CZ	$\lvert a,b\rangle\mapsto(-1)^{ab}\lvert a,b\rangle$ Symmetric under exchange of the qubits.
Controlled phase	$\operatorname{CP}(\phi)=\operatorname{diag}(1,1,1,e^{i\phi})$ Equal to CZ at $\phi=\pi$.
SWAP	$\lvert a,b\rangle\mapsto\lvert b,a\rangle$ Exchanges tensor factors without entangling product inputs.
iSWAP	$\lvert01\rangle\mapsto i\lvert10\rangle$, $\lvert10\rangle\mapsto i\lvert01\rangle$ Leaves $\lvert00\rangle$ and $\lvert11\rangle$ fixed; differs from SWAP by conditional phases.
Toffoli	$\lvert a,b,c\rangle\mapsto\lvert a,b,c\oplus ab\rangle$ Two controls and one target.
Fredkin	$\lvert0,b,c\rangle\mapsto\lvert0,b,c\rangle$, $\lvert1,b,c\rangle\mapsto\lvert1,c,b\rangle$ Controlled SWAP.

More generally,

$$C(U)=\lvert0\rangle\langle0\rvert\otimes I +\lvert1\rangle\langle1\rvert\otimes U.$$

The tensor order in this equation defines which subsystem is the control. A software bit-order convention must be translated to this order before comparing arrays.

Native interactions and a phase check

A native gate is an operation directly supported by a control implementation. Common mathematical families include $R_{ZZ}(\theta)=e^{-i\theta Z\otimes Z/2}$, $R_{XX}(\theta)=e^{-i\theta X\otimes X/2}$, and exchange-type rotations. Their calibrated phase offsets, signs, and local rotations are part of the definition; the family name alone does not specify the implemented gate. See Gate Decomposition and Pulse-Level Control.

For a useful diagnostic, compare $C(X)$ with $C(R_x(\pi))$ on $\lvert+0\rangle$. The former yields $(\lvert00\rangle+\lvert11\rangle)/\sqrt2$, whereas the latter yields $(\lvert00\rangle-i\lvert11\rangle)/\sqrt2$. These differ by a relative phase. Multiplying the control by $P(\pi/2)$ after the latter restores the former output. This is why dropping a one-qubit global phase before adding a control changes a circuit.

References

• Nielsen, Michael A., and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press, 2010, doi:10.1017/CBO9780511976667 — conventional gates, controlled operations, and universal circuit constructions.