List of quantum logic gates

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Refs
Phase shift	1	${\textstyle P(\varphi ),\;R(\varphi ),\;u_{1}(\varphi )}$	${\displaystyle {\begin{bmatrix}1&0\\0&\mathrm {e} ^{i\varphi }\end{bmatrix}}}$		Continuous parameters: ${\displaystyle \varphi }$ (period ${\displaystyle 2\pi }$)	[9][10][11]
Phase gate T, π/8 gate, fourth root of Z	1	${\textstyle T,P(\pi /4)}$ or ${\textstyle {\sqrt[{4}]{Z}}}$	${\displaystyle {\begin{bmatrix}1&0\\0&\mathrm {e} ^{i\pi /4}\end{bmatrix}}}$			[1][6]
Controlled phase	2	${\textstyle \mathrm {CPhase} (\varphi ),\mathrm {CR} (\varphi )}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&e^{i\varphi }\end{bmatrix}}}$		Continuous parameters: ${\displaystyle \varphi }$ (period ${\displaystyle 2\pi }$) Symmetrical	[11]
Controlled phase S	2	${\displaystyle \mathrm {CS} ,{\text{controlled-}}S}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&i\end{bmatrix}}}$		Symmetrical	[6]

The phase shift is a family of single-qubit gates that map the basis states ${\displaystyle P(\varphi )|0\rangle =|0\rangle }$ and ${\displaystyle P(\varphi )|1\rangle =e^{i\varphi }|1\rangle }$. The probability of measuring a ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$ 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 the Bloch sphere by ${\displaystyle \varphi }$ radians. A common example is the T gate where ${\textstyle \varphi ={\frac {\pi }{4}}}$ (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 in U(1), and the gate performs a phase rotation in U(1) along the specified basis state (e.g. ${\displaystyle P(\varphi )}$ rotates the phase about ${\displaystyle |1\rangle }$). Extending ${\displaystyle P(\varphi )}$ to a rotation about a generic phase of both basis states of a 2-level quantum system (a qubit) can be done with a series circuit: ${\displaystyle P(\beta )\cdot X\cdot P(\alpha )\cdot X={\begin{bmatrix}e^{i\alpha }&0\\0&e^{i\beta }\end{bmatrix}}}$. When ${\displaystyle \alpha =-\beta }$ this gate is the rotation operator ${\displaystyle R_{z}(2\beta )}$ gate and if ${\displaystyle \alpha =\beta }$ it is a global phase.[lower-alpha 1][lower-alpha 2]

The T gate's historic name of ${\displaystyle \pi /8}$ gate comes from the identity ${\displaystyle R_{z}(\pi /4)\operatorname {Ph} \left({\frac {\pi }{8}}\right)=P(\pi /4)}$, where ${\displaystyle R_{z}(\pi /4)={\begin{bmatrix}e^{-i\pi /8}&0\\0&e^{i\pi /8}\end{bmatrix}}}$.

Arbitrary single-qubit phase shift gates ${\displaystyle P(\varphi )}$ are natively available for transmon quantum processors through timing of microwave control pulses.[12] It can be explained in terms of change of frame.[13][14]

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 ${\displaystyle \varphi }$ only if it acts on the state ${\displaystyle |11\rangle }$:

${\displaystyle |a,b\rangle \mapsto {\begin{cases}e^{i\varphi }|a,b\rangle &{\mbox{for }}a=b=1\\|a,b\rangle &{\mbox{otherwise.}}\end{cases}}}$

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

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

Names	# qubits	Operator symbol	Exponential form	Matrix	Circuit diagram	Properties	Refs
Rotation about x-axis	1	${\textstyle R_{x}(\theta )}$	${\displaystyle \exp(-iX\theta /2)}$	${\displaystyle {\begin{bmatrix}\cos(\theta /2)&-i\sin(\theta /2)\\-i\sin(\theta /2)&\cos(\theta /2)\end{bmatrix}}}$		Special unitary Continuous parameters: ${\displaystyle \theta }$ (period ${\displaystyle 4\pi }$)	[1][6]
Rotation about y-axis	1	${\textstyle R_{y}(\theta )}$	${\displaystyle \exp(-iY\theta /2)}$	${\displaystyle {\begin{bmatrix}\cos(\theta /2)&-\sin(\theta /2)\\\sin(\theta /2)&\cos(\theta /2)\end{bmatrix}}}$		Special unitary Continuous parameters: ${\displaystyle \theta }$ (period ${\displaystyle 4\pi }$)	[1][6]
Rotation about z-axis	1	${\textstyle R_{z}(\theta )}$	${\displaystyle \exp(-iZ\theta /2)}$	${\displaystyle {\begin{bmatrix}\exp(-i\theta /2)&0\\0&\exp(i\theta /2)\end{bmatrix}}}$		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 analog rotation matrices in three Cartesian axes of SO(3), the axes on the Bloch sphere projection.

As Pauli matrices are related to the generator of rotations, these rotation operators can be written as matrix exponentials with Pauli matrices in the argument. Any ${\displaystyle 2\times 2}$ unitary matrix in SU(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 and spinors, these rotations have a period of 4π. A rotation of 2π (360 degrees) returns the same statevector with a different phase.[15]

We also have ${\displaystyle R_{b}(-\theta )=R_{b}(\theta )^{\dagger }}$ 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's possible to work out the adjoint action of rotations on the Pauli vector, namely rotation effectively by double the angle a to apply Rodrigues' rotation formula:

${\displaystyle R_{n}(-a){\vec {\sigma }}R_{n}(a)=e^{i{\frac {a}{2}}\left({\hat {n}}\cdot {\vec {\sigma }}\right)}~{\vec {\sigma }}~e^{-i{\frac {a}{2}}\left({\hat {n}}\cdot {\vec {\sigma }}\right)}={\vec {\sigma }}\cos(a)+{\hat {n}}\times {\vec {\sigma }}~\sin(a)+{\hat {n}}~{\hat {n}}\cdot {\vec {\sigma }}~(1-\cos(a))~.}$

Taking the dot 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 ${\displaystyle R_{y}(-\pi /2)XR_{y}(\pi /2)={\hat {x}}\cdot ({\hat {y}}\times {\vec {\sigma }})=Z}$. Also, using the anticommuting relation we have ${\displaystyle R_{y}(-\pi /2)XR_{y}(\pi /2)=XR_{y}(+\pi /2)R_{y}(\pi /2)=X(-iY)=Z}$.

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

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

The ${\displaystyle {\sqrt {X}}}$ gate represents a rotation of π/2 about the x axis at the Bloch sphere ${\displaystyle {\sqrt {X}}=e^{i\pi /4}R_{x}(\pi /2)}$.

Similar rotation operator gates exist for SU(3) using Gell-Mann matrices. They are the rotation operators used with qutrits.

Names	# qubits	Operator symbol	Exponential form	Matrix	Circuit diagram	Properties	Res
XX interaction	2	${\displaystyle R_{xx}(\phi )}$, ${\displaystyle {\text{XX}}(\phi )}$	${\displaystyle \exp \left(-i{\frac {\phi }{2}}X\otimes X\right)}$	${\displaystyle {\begin{bmatrix}\cos \left({\frac {\phi }{2}}\right)&0&0&-i\sin \left({\frac {\phi }{2}}\right)\\0&\cos \left({\frac {\phi }{2}}\right)&-i\sin \left({\frac {\phi }{2}}\right)&0\\0&-i\sin \left({\frac {\phi }{2}}\right)&\cos \left({\frac {\phi }{2}}\right)&0\\-i\sin \left({\frac {\phi }{2}}\right)&0&0&\cos \left({\frac {\phi }{2}}\right)\\\end{bmatrix}}}$		Special unitary Continuous parameters: ${\displaystyle \theta }$ (period ${\displaystyle 4\pi }$) Implementation: Mølmer–Sørensen gate
YY interaction	2	${\displaystyle R_{yy}(\phi )}$, ${\displaystyle {\text{YY}}(\phi )}$	${\displaystyle \exp \left(-i{\frac {\phi }{2}}Y\otimes Y\right)}$	${\displaystyle {\begin{bmatrix}\cos \left({\frac {\phi }{2}}\right)&0&0&i\sin \left({\frac {\phi }{2}}\right)\\0&\cos \left({\frac {\phi }{2}}\right)&-i\sin \left({\frac {\phi }{2}}\right)&0\\0&-i\sin \left({\frac {\phi }{2}}\right)&\cos \left({\frac {\phi }{2}}\right)&0\\i\sin \left({\frac {\phi }{2}}\right)&0&0&\cos \left({\frac {\phi }{2}}\right)\\\end{bmatrix}}}$		Special unitary Continuous parameters: ${\displaystyle \theta }$ (period ${\displaystyle 4\pi }$) Implementation: Mølmer–Sørensen gate
ZZ interaction	2	${\textstyle {\displaystyle R_{zz}(\phi )}}$, ${\displaystyle {\text{ZZ}}(\phi )}$	${\displaystyle {\displaystyle \exp \left(-i{\frac {\phi }{2}}Z\otimes Z\right)}}$	${\displaystyle {\begin{bmatrix}e^{-i\phi /2}&0&0&0\\0&e^{i\phi /2}&0&0\\0&0&e^{i\phi /2}&0\\0&0&0&e^{-i\phi /2}\\\end{bmatrix}}}$		Special unitary Continuous parameters: ${\displaystyle \theta }$ (period ${\displaystyle 4\pi }$)
XY, XX plus YY	2	${\textstyle {\displaystyle R_{xy}(\phi )}}$, ${\displaystyle {\text{XY}}(\phi )}$	${\displaystyle {\displaystyle \exp \left[-i{\frac {\phi }{4}}(X\otimes X+Y\otimes Y)\right]}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\phi /2)&-i\sin(\phi /2)&0\\0&-i\sin(\phi /2)&\cos(\phi /2)&0\\0&0&0&1\\\end{bmatrix}}}$		Special unitary Continuous parameters: ${\displaystyle \theta }$ (period ${\displaystyle 4\pi }$)

The qubit-qubit Ising coupling or Heisenberg interaction gates R_xx, R_yy and R_zz are 2-qubit gates that are implemented natively in some trapped-ion quantum computers, using for example the Mølmer–Sørensen gate procedure.[16][17]

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

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

${\displaystyle {\mbox{CNOT}}=e^{-i{\frac {\pi }{4}}}R_{y_{1}}(-\pi /2)R_{x_{1}}(-\pi /2)R_{x_{2}}(-\pi /2)R_{xx}(\pi /2)R_{y_{1}}(\pi /2).}$

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

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Refs
Square root swap	2	${\displaystyle {\sqrt {\mathrm {SWAP} }}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&{\frac {1}{2}}(1+i)&{\frac {1}{2}}(1-i)&0\\0&{\frac {1}{2}}(1-i)&{\frac {1}{2}}(1+i)&0\\0&0&0&1\\\end{bmatrix}}}$			[1]
Square root imaginary swap	2	${\displaystyle {\sqrt {\mbox{iSWAP}}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&{\frac {1}{\sqrt {2}}}&{\frac {i}{\sqrt {2}}}&0\\0&{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\0&0&0&1\end{bmatrix}}}$		Special unitary	[11]
Swap (raised to a power)	2	${\displaystyle {\mbox{SWAP}}^{\alpha }}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&{\frac {1+e^{i\pi \alpha }}{2}}&{\frac {1-e^{i\pi \alpha }}{2}}&0\\0&{\frac {1-e^{i\pi \alpha }}{2}}&{\frac {1+e^{i\pi \alpha }}{2}}&0\\0&0&0&1\end{bmatrix}}}$		Continuous parameters: ${\displaystyle \alpha }$ (period ${\displaystyle 2}$)	[1]
Fredkin, controlled swap	3	${\displaystyle \mathrm {CSWAP} }$, ${\displaystyle \mathrm {FREDKIN} }$	${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\\end{bmatrix}}}$	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 a Bell state from product states. The √SWAP gate arises naturally in systems that exploit exchange interaction.[18][1]

For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap[19] or iSWAP.[20][21] Note that ${\displaystyle i{\mbox{SWAP}}=R_{xx}(-\pi /2)R_{yy}(-\pi /2)}$ and ${\displaystyle {\sqrt {i{\mbox{SWAP}}}}=R_{xx}(-\pi /4)R_{yy}(-\pi /4)}$, or more generally ${\displaystyle {\sqrt[{n}]{i{\mbox{SWAP}}}}=R_{xx}(-\pi /2n)R_{yy}(-\pi /2n)}$ for all real n except 0.

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

The Fredkin gate (also CSWAP or CS gate), named after Edward Fredkin, is a 3-bit gate that performs a controlled swap. It is universal for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the billiard ball model means the same number of balls are output as input.