Ingate-based quantum computing, various sets ofquantum logic gates are commonly used to express quantum operations. The following tables list several unitary quantum logic gates, together with their common name, how they are represented, and some of their properties.Controlled orconjugate transpose (adjoint) versions of some of these gates may not be listed.

Name	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Refs
Identity, no-op	1 (any)	$I,\mathbb{I}$, 𝟙		or	Hermitian Identity element	[1]
Global phase	1 (any)	$\mathrm{P}\mathrm{h}$,$\mathrm{P}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}$ or${\mathrm{e}}^{i\delta }I$			Continuous parameters:${\displaystyle \delta }$ (period${\displaystyle 2\pi }$) Exponential form:${\displaystyle \exp (i\delta I)}$	[1]

The identity gate is theidentity operation${\displaystyle I|\psi ⟩=|\psi ⟩}$, most of the times this gate is not indicated in circuit diagrams, but it is useful when describing mathematical results. It has been described as being a "wait cycle",^[2] and aNOP.^[3][1]

The global phase gate introduces a global phase${\displaystyle e^{i\phi }}$ to the whole qubit quantum state. A quantum state is uniquely defined up to a phase. Because of theBorn rule, aphase factor has no effect on ameasurement outcome:${\displaystyle |e^{i\phi }|=1}$ for any${\displaystyle \phi }$. Because${\displaystyle e^{i\delta }|\psi ⟩\otimes |\varphi ⟩=e^{i\delta }(|\psi ⟩\otimes |\varphi ⟩),}$ when the global phase gate is applied to a single qubit in aquantum register, the entire register's global phase is changed. Also,${\displaystyle \mathrm{P}\mathrm{h}(0)=I.}$

These gates can be extended to any number ofqubits orqudits.

This table includes commonly usedClifford gates for qubits.^[1][4][5]

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Some properties	Refs
PauliX, NOT, bit flip	1	$X,\mathrm{N}\mathrm{O}\mathrm{T},{\sigma }_x$		or	Hermitian Pauli group Traceless Involutory	[1][6]
PauliY	1	$Y,{\sigma }_y$			Hermitian Pauli group Traceless Involutory	[1][6]
PauliZ, phase flip	1	$Z,{\sigma }_z$			Hermitian Pauli group Traceless Involutory	[1][6]
Phase gateS, square root ofZ	1	$S,P,\sqrt{Z}$				[1][6]
Square root ofX, square root of NOT	1	$\sqrt{X}$,$V$,$\sqrt{\mathrm{N}\mathrm{O}\mathrm{T}},\mathrm{S}\mathrm{X}$				[1][7]
Hadamard, Walsh-Hadamard	1	$H$			Hermitian Traceless Involutory	[1][6]
Controlled NOT, controlled-X, controlled-bit flip, reversibleexclusive OR, Feynman	2	$\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}$,$\mathrm{X}\mathrm{O}\mathrm{R},\mathrm{C}\mathrm{X}$			Hermitian Involutory Implementation: Cirac–Zoller gate	[1][6]
Anticontrolled-NOT, anticontrolled-X, open-controlled-CNOT, zero control, control-on-0-NOT, reversibleexclusive NOR	2	$\overline{\mathrm{C}}\mathrm{X}$,$\text{controlled[0]-NOT}$,$\mathrm{X}\mathrm{N}\mathrm{O}\mathrm{R}$			Hermitian Involutory	[1]
Controlled-Z, controlled sign flip, controlled phase flip	2	$\mathrm{C}\mathrm{Z}$,$\mathrm{C}\mathrm{P}\mathrm{F}$,$\mathrm{C}\mathrm{S}\mathrm{I}\mathrm{G}\mathrm{N}$,$\mathrm{C}\mathrm{P}\mathrm{H}\mathrm{A}\mathrm{S}\mathrm{E}$			Hermitian Involutory Symmetrical Implementation: Duan-Kimble gate	[1][6]
Double-controlled NOT	2	$\mathrm{D}\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}$			Special unitary	[8]
Swap	2	$\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P}$		or	Hermitian Involutory Symmetrical	[1][6]
Imaginary swap	2	${\displaystyle \text{iSWAP}}$		or	Special unitary Symmetrical	[1]

Other Clifford gates, including higher dimensional ones are not included here but by definition can be generated using$H,S$ and$\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}$.

Note that if a Clifford gateA is not in the Pauli group,${\displaystyle \sqrt{A}}$ or controlled-A are not in the Clifford gates.^{[citation needed]}

The Clifford set is not a universal quantum gate set.

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Refs
Phase shift	1	$P(\phi ),R(\phi ),u_1(\phi )$			Continuous parameters:${\displaystyle \phi }$ (period${\displaystyle 2\pi }$)	[9][10][11]
Phase gateT, π/8 gate, fourth root ofZ	1	$T,P(\pi /4)$ or$\sqrt[4]{Z}$				[1][6]
Controlled phase	2	$\mathrm{C}\mathrm{P}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}(\phi ),\mathrm{C}\mathrm{R}(\phi )$			Continuous parameters:${\displaystyle \phi }$ (period${\displaystyle 2\pi }$) Symmetrical Implementation: Jaksch Gate[12]	[11]
Controlled phase S	2	${\displaystyle \mathrm{C}\mathrm{S},\text{controlled-}S}$			Symmetrical	[6]

The phase shift is a family of single-qubit gates that map the basis states${\displaystyle P(\phi )|0⟩=|0⟩}$ and${\displaystyle P(\phi )|1⟩=e^{i\phi }|1⟩}$. The probability of measuring a${\displaystyle |0⟩}$ or${\displaystyle |1⟩}$ is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle (a line of latitude), or a rotation along the z-axis on theBloch sphere by${\displaystyle \phi }$ radians. A common example is theT gate where$\phi =\frac{\pi }{4}$ (historically known as the${\displaystyle \pi /8}$ gate), the phase gate. Note that some Clifford gates are special cases of the phase shift gate:${\displaystyle P(0)=I,P(\pi )=Z;P(\pi /2)=S.}$

The argument to the phase shift gate is inU(1), and the gate performs a phase rotation in U(1) along the specified basis state (e.g.${\displaystyle P(\phi )}$ rotates the phase about${\displaystyle |1⟩}$). Extending${\displaystyle P(\phi )}$ to a rotation about a generic phase of both basis states of a 2-level quantum system (aqubit) can be done with aseries circuit:. When${\displaystyle \alpha =-\beta }$ this gate is therotation operator${\displaystyle R_z(2\beta )}$ gate and if${\displaystyle \alpha =\beta }$ it is a global phase.^[a][b]

TheT gate's historic name of${\displaystyle \pi /8}$ gate comes from the identity${\displaystyle R_z(\pi /4)\mathrm{Ph}\left(\frac{\pi }{8}\right)=P(\pi /4)}$, where.

Arbitrary single-qubit phase shift gates${\displaystyle P(\phi )}$ are natively available fortransmon quantum processors through timing of microwave control pulses.^[13] It can be explained in terms ofchange of frame.^[14][15]

As with any single qubit gate one can build a controlled version of the phase shift gate. With respect to the computational basis, the 2-qubit controlled phase shift gate is: shifts the phase with${\displaystyle \phi }$ only if it acts on the state${\displaystyle |11⟩}$:

The controlled-Z (or CZ) gate is the special case where${\displaystyle \phi =\pi }$.

The controlled-S gate is the case of the controlled-${\displaystyle P(\phi )}$ when${\displaystyle \phi =\pi /2}$ and is a commonly used gate.^[6]

Names	# qubits	Operator symbol	Exponential form	Matrix	Circuit diagram	Properties	Refs
Rotation aboutx-axis	1	$R_x(\theta )$	${\displaystyle \exp (-iX\theta /2)}$			Special unitary Continuous parameters:${\displaystyle \theta }$ (period${\displaystyle 4\pi }$)	[1][6]
Rotation about y-axis	1	$R_y(\theta )$	${\displaystyle \exp (-iY\theta /2)}$			Special unitary Continuous parameters:${\displaystyle \theta }$ (period${\displaystyle 4\pi }$)	[1][6]
Rotation about z-axis	1	$R_z(\theta )$	${\displaystyle \exp (-iZ\theta /2)}$			Special unitary Continuous parameters:${\displaystyle \theta }$ (period${\displaystyle 4\pi }$)	[1][6]

The rotation operator gates${\displaystyle R_x(\theta ),R_y(\theta )}$ and${\displaystyle R_z(\theta )}$ are the analogrotation matrices in threeCartesian axes ofSO(3),^[c] along the x, y or z-axes of theBloch sphere projection.

AsPauli matrices are related to thegenerator of rotations, these rotation operators can be written asmatrix exponentials with Pauli matrices in the argument. Any${\displaystyle 2\times 2}$unitary matrix inSU(2) can be written as a product (i.e. series circuit) of three rotation gates or less. Note that for two-level systems such as qubits andspinors, these rotations have a period of4π. A rotation of2π (360 degrees) returns the same statevector with a differentphase.^[16]

We also have${\displaystyle R_b(-\theta )=R_b(\theta {)}^{†}}$ and${\displaystyle R_b(0)=I}$ for all${\displaystyle b\in \{x,y,z\}.}$

The rotation matrices are related to the Pauli matrices in the following way:${\displaystyle R_x(\pi )=-iX,R_y(\pi )=-iY,R_z(\pi )=-iZ.}$

It is possible to work out the adjoint action of rotations on thePauli vector, namely rotation effectively by double the anglea to applyRodrigues' rotation formula:

${\displaystyle R_n(-a)\overrightarrow{\sigma }{R}_n(a)=e^{i\frac{a}{2}\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}\overrightarrow{\sigma }{e}^{-i\frac{a}{2}\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}=\overrightarrow{\sigma }\cos (a)+\hat{n}\times \overrightarrow{\sigma }\sin (a)+\hat{n}\hat{n}\cdot \overrightarrow{\sigma }(1-\cos (a)).}$

Taking thedot product of any unit vector with the above formula generates the expression of any single qubit gate when sandwiched within adjoint rotation gates. For example, it can be shown that${\displaystyle R_y(-\pi /2)X{R}_y(\pi /2)=\hat{x}\cdot (\hat{y}\times \overrightarrow{\sigma })=Z}$. Also, using the anticommuting relation we have${\displaystyle R_y(-\pi /2)X{R}_y(\pi /2)=X{R}_y(+\pi /2)R_y(\pi /2)=X(-iY)=Z}$.

Rotation operators have interesting identities. For example,${\displaystyle R_y(\pi /2)Z=H}$ and${\displaystyle X{R}_y(\pi /2)=H.}$ Also, using the anticommuting relations we have${\displaystyle Z{R}_y(-\pi /2)=H}$ and${\displaystyle R_y(-\pi /2)X=H.}$

Global phase and phase shift can be transformed into each other's with the Z-rotation operator:${\displaystyle R_z(\gamma )\mathrm{Ph}\left(\frac{\gamma }{2}\right)=P(\gamma )}$.^{[5]: 11 [1]: 77–83}

The${\displaystyle \sqrt{X}}$ gate represents a rotation ofπ/2 about thex axis at the Bloch sphere${\displaystyle \sqrt{X}=e^{i\pi /4}{R}_x(\pi /2)}$.

Similar rotation operator gates exist forSU(3) usingGell-Mann matrices. They are the rotation operators used withqutrits.

Names	# qubits	Operator symbol	Exponential form	Matrix	Circuit diagram	Properties	Refs
XX interaction	2	${\displaystyle R_{xx}(\varphi )}$,${\displaystyle \text{XX}(\varphi )}$	${\displaystyle \exp \left(-i\frac{\varphi }{2}X\otimes X\right)}$			Special unitary Continuous parameters:${\displaystyle \theta }$ (period${\displaystyle 4\pi }$) Implementation: Mølmer–Sørensen gate	[citation needed]
YY interaction	2	${\displaystyle R_{yy}(\varphi )}$,${\displaystyle \text{YY}(\varphi )}$	${\displaystyle \exp \left(-i\frac{\varphi }{2}Y\otimes Y\right)}$			Special unitary Continuous parameters:${\displaystyle \theta }$ (period${\displaystyle 4\pi }$) Implementation: Mølmer–Sørensen gate	[citation needed]
ZZ interaction	2	${\displaystyle R_{zz}(\varphi )}$,${\displaystyle \text{ZZ}(\varphi )}$	${\displaystyle {\displaystyle \exp \left(-i\frac{\varphi }{2}Z\otimes Z\right)}}$			Special unitary Continuous parameters:${\displaystyle \theta }$ (period${\displaystyle 4\pi }$)	[citation needed]
XY, XX plus YY	2	${\displaystyle R_{xy}(\varphi )}$,${\displaystyle \text{XY}(\varphi )}$	${\displaystyle {\displaystyle \exp \left[-i\frac{\varphi }{4}(X\otimes X+Y\otimes Y)\right]}}$			Special unitary Continuous parameters:${\displaystyle \theta }$ (period${\displaystyle 4\pi }$)	[citation needed]

The qubit-qubit Ising coupling or Heisenberg interaction gatesR_xx,R_yy andR_zz are 2-qubit gates that are implemented natively in sometrapped-ion quantum computers, using for example theMølmer–Sørensen gate procedure.^[17][18]

Note that these gates can be expressed in sinusoidal form also, for example${\displaystyle R_{xx}(\varphi )=\exp \left(-i\frac{\varphi }{2}X\otimes X\right)=\cos \left(\frac{\varphi }{2}\right)I\otimes I-i\sin \left(\frac{\varphi }{2}\right)X\otimes X}$.

The CNOT gate can be further decomposed as products of rotation operator gates and exactly a single two-qubit interaction gate, for example

${\displaystyle \text{CNOT}=e^{-i\frac{\pi }{4}}{R}_{y_1}(-\pi /2)R_{x_1}(-\pi /2)R_{x_2}(-\pi /2)R_{xx}(\pi /2)R_{y_1}(\pi /2).}$

The SWAP gate can be constructed from other gates, for example using thetwo-qubit interaction gates:${\displaystyle \text{SWAP}=e^{i\frac{\pi }{4}}{R}_{xx}(\pi /2)R_{yy}(\pi /2)R_{zz}(\pi /2)}$.

In superconducting circuits, the family of gates resulting from Heisenberg interactions is sometimes called thefSim gate set. They can be realized using flux-tunable qubits with flux-tunable coupling,^[19] or using microwave drives in fixed-frequency qubits with fixed coupling.^[20]

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Refs
Square root swap	2	${\displaystyle \sqrt{\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P}}}$				[1]
Square root imaginary swap	2	${\displaystyle \sqrt{\text{iSWAP}}}$			Special unitary	[11]
Swap (raised to a power)	2	${\displaystyle {\text{SWAP}}^{\alpha }}$			Continuous parameters:${\displaystyle \alpha }$ (period${\displaystyle 2}$)	[1]
Fredkin, controlled swap	3	${\displaystyle \mathrm{C}\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P}}$,${\displaystyle \mathrm{F}\mathrm{R}\mathrm{E}\mathrm{D}\mathrm{K}\mathrm{I}\mathrm{N}}$		or	Hermitian Involutory Functionally complete reversible gate for Boolean algebra	[1][6]

The√SWAP gate performs half-way of a two-qubit swap (see Clifford gates). It is universal such that any many-qubit gate can be constructed from only√SWAP and single qubit gates. More than one application of the√SWAP is required to produce aBell state from product states. The√SWAP gate arises naturally in systems that exploitexchange interaction.^[21][1]

For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap^[22] or iSWAP.^[23][24] Note that${\displaystyle i\text{SWAP}=R_{xx}(-\pi /2)R_{yy}(-\pi /2)}$ and${\displaystyle \sqrt{i\text{SWAP}}=R_{xx}(-\pi /4)R_{yy}(-\pi /4)}$, or more generally${\displaystyle \sqrt[n]{i\text{SWAP}}=R_{xx}(-\pi /2n)R_{yy}(-\pi /2n)}$ for all realn except 0.

SWAP^α arises naturally in spintronic quantum computers.^[1]

TheFredkin gate (also CSWAP or CS gate), named afterEdward Fredkin, is a 3-bit gate that performs acontrolledswap. It isuniversal for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in thebilliard ball model means the same number of balls are output as input.

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Named after	Refs
General single qubit rotation	1	${\displaystyle U(\theta ,\varphi ,\lambda )}$			Implements an arbitrary single-qubit rotation Continuous parameters:${\displaystyle \theta ,\varphi ,\lambda }$ (period${\displaystyle 2\pi }$)	OpenQASM U gate[d]	[11][25]
Barenco	2	${\displaystyle \mathrm{B}\mathrm{A}\mathrm{R}\mathrm{E}\mathrm{N}\mathrm{C}\mathrm{O}(\alpha ,\varphi ,\theta )}$			Implements a controlled arbitrary qubit rotation Universal quantum gate Continuous parameters:${\displaystyle \alpha ,\varphi ,\theta }$ (period${\displaystyle 2\pi }$)	Adriano Barenco	[1]
BerkeleyB	2	${\displaystyle B}$			Special unitary Exponential form: ${\displaystyle \exp \left[i\frac{\pi }{8}(2X\otimes X+Y\otimes Y)\right]}$	University of California Berkeley[26]	[1]
Controlled-V, controlled square root NOT	2	${\displaystyle \mathrm{C}\mathrm{S}\mathrm{X},\text{controlled-}\sqrt{X},}$${\displaystyle \text{controlled-}V}$					[9]
Core entangling, canonical decomposition	2	${\displaystyle N(a,b,c)}$,${\displaystyle \mathrm{c}\mathrm{a}\mathrm{n}(a,b,c)}$			Special unitary Universal quantum gate Exponential form ${\displaystyle \exp \left[i(aX\otimes X+bY\otimes Y+cZ\otimes Z)\right]}$ Continuous parameters:${\displaystyle a,b,c}$ (period${\displaystyle 2\pi }$)		[1]
Dagwood Bumstead	2	${\displaystyle \text{DB}}$			Special unitary Exponential form: ${\displaystyle \exp \left[-i\frac{3\pi }{16}(X\otimes X+Y\otimes Y)\right]}$	ComicbookDagwood Bumstead[27]	[28][27]
Echoed cross resonance	2	${\displaystyle \text{ECR}}$			Special unitary		[29]
Fermionic simulation	2	${\displaystyle U_{\text{fSim}}(\theta ,\varphi )}$,${\displaystyle \text{fSim}(\theta ,\varphi )}$			Special unitary Continuous parameters:${\displaystyle \theta ,\varphi }$ (period${\displaystyle 2\pi }$)		[30][19][20]
Givens	2	${\displaystyle G(\theta )}$,${\displaystyle \text{Givens}(\theta )}$			Special unitary Exponential form: ${\displaystyle \exp \left[-i\frac{\theta }{2}(Y\otimes X-X\otimes Y)\right]}$ Continuous parameters:${\displaystyle \theta ,\varphi }$ (period${\displaystyle 2\pi }$)	Givens rotations	[31]
Magic	2	${\displaystyle \mathcal{M}}$					[1]
Sycamore	2	${\displaystyle \text{syc}}$,${\displaystyle \text{fSim}(\pi /2,\pi /6)}$				Google'sSycamore processor	[32]
CZ-SWAP	2	${\displaystyle \text{CZS}(\theta ,\varphi ,\gamma )}$,			Continuous parameters:${\displaystyle \theta ,\varphi ,\gamma }$ Submatrix of a controlled-CZS (CCZS)		[33]
Deutsch	3	${\displaystyle D_{\theta }}$,${\displaystyle D(\theta )}$			Continuous parameters:${\displaystyle \theta ,\varphi }$ (period${\displaystyle 2\pi }$) Universal quantum gate	David Deutsch	[1]
Margolus, simplified Toffoli	3	${\displaystyle M}$,${\displaystyle \text{RCCX}}$			Hermitian Involutory Special unitary Functionally complete reversible gate for Boolean algebra	Norman Margolus	[34][35]
Peres	3	${\displaystyle \mathrm{P}\mathrm{G}}$,${\displaystyle \mathrm{P}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{s}}$			Functionally complete reversible gate for Boolean algebra	Asher Peres	[36]
Toffoli, controlled-controlled NOT	3	${\displaystyle \mathrm{C}\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T},\mathrm{C}\mathrm{C}\mathrm{X},\mathrm{T}\mathrm{o}\mathrm{f}\mathrm{f}}$			Hermitian Involutory Functionally complete reversible gate for Boolean algebra	Tommaso Toffoli	[1][6]
Fermionic-Fredkin, Controlled-fermionic SWAP	3	${\displaystyle \mathrm{f}\mathrm{F}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{k}\mathrm{i}\mathrm{n}}$, ${\displaystyle \mathrm{C}\mathrm{C}\mathrm{Z}\mathrm{S}(\pi /2,0,0)}$,${\displaystyle \mathrm{C}\mathrm{f}\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P}}$					[33] [37]