List of quantum logic gates

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In gate-based quantum computing, various sets of quantum 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 or conjugate transpose (adjoint) versions of some of these gates may not be listed.

Identity gate and global phase

Name	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Refs
Identity, no-op	1 (any)	${\textstyle I,\;\mathbb {I} }$, 𝟙	${\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}}$	or	Hermitian Identity element	[1]
Global phase	1 (any)	${\textstyle \mathrm {Ph} }$, ${\textstyle \mathrm {Phase} }$ or ${\textstyle \mathrm {e} ^{i\delta }I}$	${\displaystyle \mathrm {e} ^{i\delta }{\begin{bmatrix}1&0\\0&1\end{bmatrix}}}$		Continuous parameters: ${\displaystyle \delta }$ (period ${\displaystyle 2\pi }$) Exponential form: ${\displaystyle \exp(i\delta I)}$	[1]

The identity gate is the identity operation ${\displaystyle I|\psi \rangle =|\psi \rangle }$, 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 a NOP. ^{[3] [1]}

The global phase gate introduces a global phase ${\displaystyle e^{i\varphi }}$ to the whole qubit quantum state. A quantum state is uniquely defined up to a phase. Because of the Born rule, a phase factor has no effect on a measurement outcome: ${\displaystyle |e^{i\varphi }|=1}$ for any ${\displaystyle \varphi }$.

Because ${\displaystyle e^{i\delta }|\psi \rangle \otimes |\phi \rangle =e^{i\delta }(|\psi \rangle \otimes |\phi \rangle ),}$ when the global phase gate is applied to a single qubit in a quantum register, the entire register's global phase is changed.

Also, ${\displaystyle \mathrm {Ph} (0)=I.}$

These gates can be extended to any number of qubits or qudits.

Clifford qubit gates

This table includes commonly used Clifford gates for qubits. ^{[1] [4] [5]}

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Some properties	Refs
Pauli X, NOT, bit flip	1	${\textstyle X,\;\mathrm {NOT} ,\;\sigma _{x}}$	${\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}$	or	Hermitian Pauli group Traceless Involutory	[1] [6]
Pauli Y	1	${\textstyle Y,\;\sigma _{y}}$	${\displaystyle {\begin{bmatrix}0&-i\\i&0\end{bmatrix}}}$		Hermitian Pauli group Traceless Involutory	[1] [6]
Pauli Z, phase flip	1	${\textstyle Z,\;\sigma _{z}}$	${\displaystyle {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$		Hermitian Pauli group Traceless Involutory	[1] [6]
Phase gate S, square root of Z	1	${\textstyle S,\;P,\;{\sqrt {Z}}}$	${\displaystyle {\begin{bmatrix}1&0\\0&i\end{bmatrix}}}$			[1] [6]
Square root of X, square root of NOT	1	${\textstyle {\sqrt {X}}}$, ${\textstyle V}$, ${\textstyle {\sqrt {\mathrm {NOT} }},\;\mathrm {SX} }$	${\displaystyle {\frac {1}{2}}{\begin{bmatrix}1+i&1-i\\1-i&1+i\end{bmatrix}}}$			[1] [7]
Hadamard, Walsh-Hadamard	1	${\textstyle H}$	${\displaystyle {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}}$		Hermitian Traceless Involutory	[1] [6]
Controlled NOT, controlled-X, controlled-bit flip, reversible exclusive OR, Feynman	2	${\textstyle \mathrm {CNOT} }$, ${\textstyle \mathrm {XOR} ,\;\mathrm {CX} }$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}1&0&0&0\\0&0&0&1\\0&0&1&0\\0&1&0&0\end{bmatrix}}}$		Hermitian Involutory Implementation: Cirac–Zoller gate	[1] [6]
Anticontrolled-NOT, anticontrolled-X, zero control, control-on-0-NOT, reversible exclusive NOR	2	${\textstyle {\overline {\mathrm {C} }}\mathrm {X} }$, ${\textstyle {\text{controlled[0]-NOT}}}$, ${\textstyle \mathrm {XNOR} }$	${\displaystyle {\begin{bmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}}$		Hermitian Involutory	[1]
Controlled-Z, controlled sign flip, controlled phase flip	2	${\textstyle \mathrm {CZ} }$, ${\textstyle \mathrm {CPF} }$, ${\textstyle \mathrm {CSIGN} }$, ${\textstyle \mathrm {CPHASE} }$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{bmatrix}}}$		Hermitian Involutory Symmetrical Implementation: Duan-Kimble gate	[1] [6]
Double-controlled NOT	2	${\textstyle \mathrm {DCNOT} }$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0\end{bmatrix}}}$		Special unitary	[8]
Swap	2	${\textstyle \mathrm {SWAP} }$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}}$	or	Hermitian Involutory Symmetrical	[1] [6]
Imaginary swap	2	${\displaystyle {\mbox{iSWAP}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&0&i&0\\0&i&0&0\\0&0&0&1\end{bmatrix}}}$	or	Special unitary Symmetrical	[1]

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

Note that if a Clifford gate A 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.

Non-Clifford qubit gates

Relative phase 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 Implementation: Jaksch Gate [12]	[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. ^{[a] [b]}

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. ^[13] It can be explained in terms of change 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 ${\displaystyle \varphi }$ 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]

Rotation operator gates

Further information: Bloch sphere § Rotations, and Rotation operator (quantum mechanics)

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), ^[c] along the x, y or z-axes of 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. ^[16]

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 other's 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.

Two-qubit interaction gates

Names	# qubits	Operator symbol	Exponential form	Matrix	Circuit diagram	Properties	Refs
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	[ citation needed ]
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	[ citation needed ]
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 ${\displaystyle 4\pi }$)	[ citation needed ]
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 ${\displaystyle 4\pi }$)	[ citation needed ]

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. ^{[17] [18]}

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: ${\displaystyle {\text{SWAP}}=e^{i{\frac {\pi }{4}}}R_{xx}(\pi /2)R_{yy}(\pi /2)R_{zz}(\pi /2)}$.

In superconducting circuits, the family of gates resulting from Heisenberg interactions is sometimes called the fSim 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]

Non-Clifford swap gates

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. ^{[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 ${\displaystyle i{\mbox{SWAP}}=R_{xx}(-\pi /2)R_{yy}(-\pi /2)}$ 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.

Other named gates

Names	# qubits	Operator symbol	Matrix	Circuit diagram	Properties	Named after	Refs
General single qubit rotation	1	${\displaystyle U(\theta ,\phi ,\lambda )}$	${\displaystyle {\begin{bmatrix}\cos(\theta /2)&-e^{i\lambda }\sin(\theta /2)\\e^{i\phi }\sin(\theta /2)&e^{i(\lambda +\phi )}\cos(\theta /2)\end{bmatrix}}}$		Implements an arbitrary single-qubit rotation Continuous parameters: ${\displaystyle \theta ,\phi ,\lambda }$ (period ${\displaystyle 2\pi }$)	OpenQASM U gate [d]	[11] [25]
Barenco	2	${\displaystyle \mathrm {BARENCO} (\alpha ,\phi ,\theta )}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&e^{i\alpha }\cos \theta &-\mathrm {i} e^{\mathrm {i} (\alpha -\phi )}\sin \theta \\0&0&-\mathrm {i} e^{\mathrm {i} (\alpha +\phi )}\sin \theta &e^{i\alpha }\cos \theta \end{bmatrix}}}$		Implements a controlled arbitrary qubit rotation Universal quantum gate Continuous parameters: ${\displaystyle \alpha ,\phi ,\theta }$ (period ${\displaystyle 2\pi }$)	Adriano Barenco	[1]
Berkeley B	2	${\displaystyle B}$	${\displaystyle {\begin{bmatrix}\cos(\pi /8)&0&0&i\sin(\pi /8)\\0&\cos(3\pi /8)&i\sin(3\pi /8)&0\\0&i\sin(3\pi /8)&\cos(3\pi /8)&0\\i\sin(\pi /8)&0&0&\cos(\pi /8)\\\end{bmatrix}}}$		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 {CSX} ,{\text{controlled-}}{\sqrt {X}},}$${\displaystyle {\text{controlled-}}V}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&e^{i\pi /4}&e^{-i\pi /4}\\0&0&e^{-i\pi /4}&e^{i\pi /4}\end{bmatrix}}}$				[9]
Core entangling, canonical decomposition	2	${\displaystyle N(a,b,c)}$, ${\displaystyle \mathrm {can} (a,b,c)}$	${\displaystyle {\begin{bmatrix}e^{ic}\cos(a-b)&0&0&ie^{ic}\sin(a-b)\\0&e^{-ic}\cos(a+b)&ie^{-ic}\sin(a+b)&0\\0&ie^{-ic}\sin(a+b)&e^{-ic}\cos(a+b)&0\\ie^{ic}\sin(a-b)&0&0&e^{ic}\cos(a-b)\\\end{bmatrix}}}$		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	${\displaystyle {\text{DB}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(3\pi /8)&-i\sin(3\pi /8)&0\\0&-i\sin(3\pi /8)&\cos(3\pi /8)&0\\0&0&0&1\\\end{bmatrix}}}$		Special unitary Exponential form: ${\displaystyle \exp \left[-i{\frac {3\pi }{16}}(X\otimes X+Y\otimes Y)\right]}$	Comicbook Dagwood Bumstead [27]	[28] [27]
Echoed cross resonance	2	${\displaystyle {\text{ECR}}}$	${\displaystyle {\frac {1}{\sqrt {2}}}{\begin{bmatrix}0&0&1&i\\0&0&i&1\\1&-i&0&0\\-i&1&0&0\\\end{bmatrix}}}$		Special unitary		[29]
Fermionic simulation	2	${\displaystyle U_{\text{fSim}}(\theta ,\phi )}$, ${\displaystyle {\text{fSim}}(\theta ,\phi )}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-i\sin(\theta )&0\\0&-i\sin(\theta )&\cos(\theta )&0\\0&0&0&e^{i\phi }\\\end{bmatrix}}}$		Special unitary Continuous parameters: ${\displaystyle \theta ,\phi }$ (period ${\displaystyle 2\pi }$)		[30] [19] [20]
Givens	2	${\displaystyle G(\theta )}$, ${\displaystyle {\text{Givens}}(\theta )}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\\\end{bmatrix}}}$		Special unitary Exponential form: ${\displaystyle \exp \left[-i{\frac {\theta }{2}}(Y\otimes X-X\otimes Y)\right]}$ Continuous parameters: ${\displaystyle \theta ,\phi }$ (period ${\displaystyle 2\pi }$)	Givens rotations	[31]
Magic	2	${\displaystyle {\mathcal {M}}}$	${\displaystyle {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&i&0&0\\0&0&i&1\\0&0&i&-1\\1&-i&0&0\\\end{bmatrix}}}$				[1]
Sycamore	2	${\displaystyle {\text{syc}}}$, ${\displaystyle {\text{fSim}}(\pi /2,\pi /6)}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&0&-i&0\\0&-i&0&0\\0&0&0&\mathrm {e} ^{-i\pi /6}\end{bmatrix}}}$			Google's Sycamore processor	[32]
CZ-SWAP	2	${\displaystyle {\text{CZS}}(\theta ,\phi ,\gamma )}$,	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&-e^{i\gamma }\sin ^{2}(\theta /2)+\cos ^{2}(\theta /2)&{\frac {1}{2}}(1+e^{i\gamma })e^{-i\phi }\sin(\theta )&0\\0&{\frac {1}{2}}(1+e^{i\gamma })e^{i\phi }\sin(\theta )&-e^{i\gamma }\cos ^{2}(\theta /2)+\sin ^{2}(\theta /2)&0\\0&0&0&\mathrm {-} e^{i\gamma }\end{bmatrix}}}$		Continuous parameters: ${\displaystyle \theta ,\phi ,\gamma }$ Submatrix of a controlled-CZS (CCZS)		[33]
Deutsch	3	${\displaystyle D_{\theta }}$, ${\displaystyle D(\theta )}$	${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&i\cos \theta &\sin \theta \\0&0&0&0&0&0&\sin \theta &i\cos \theta \\\end{bmatrix}}}$		Continuous parameters: ${\displaystyle \theta ,\phi }$ (period ${\displaystyle 2\pi }$) Universal quantum gate	David Deutsch	[1]
Margolus, simplified Toffoli	3	${\displaystyle M}$, ${\displaystyle {\text{RCCX}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&-1&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\\\end{bmatrix}}}$		Hermitian Involutory Special unitary Functionally complete reversible gate for Boolean algebra	Norman Margolus	[34] [35]
Peres	3	${\displaystyle \mathrm {PG} }$,${\displaystyle \mathrm {Peres} }$	${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\\end{bmatrix}}}$		Functionally complete reversible gate for Boolean algebra	Asher Peres	[36]
Toffoli, controlled-controlled NOT	3	${\displaystyle \mathrm {CCNOT} ,\mathrm {CCX} ,\mathrm {Toff} }$	${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\\\end{bmatrix}}}$		Hermitian Involutory Functionally complete reversible gate for Boolean algebra	Tommaso Toffoli	[1] [6]
Fermionic-Fredkin, Controlled-fermionic SWAP	3	${\displaystyle \mathrm {fFredkin} }$, ${\displaystyle \mathrm {CCZS} (\pi /2,0,0)}$, ${\displaystyle \mathrm {CfSWAP} }$	${\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&-1\\\end{bmatrix}}}$				[33] [37]

Notes

1. ${\displaystyle \alpha =-\beta }$ when ${\displaystyle P(\alpha )=P(\beta )^{\dagger }}$, where ${\displaystyle \dagger }$ is the conjugate transpose (or Hermitian adjoint).
2. Also: ${\displaystyle \left(P(\delta )\cdot Y\right)^{2}=e^{i\delta }I}$
3. a SU(2) double cover. See also Hopf fibration.
4. The matrix shown here is from openQASM 3.0, which differs from ${\displaystyle U(\theta ,\phi ,\lambda )}$ from a global phase (OpenQASM 2.0 U gate is in SU(2) ) .

References

1. Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN   978-1-84628-887-6.
2. "IGate". qiskit.org. Qiskit online documentation.
3. "I operation". docs.microsoft.com. 28 July 2023. Q# online documentation.
4. Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. 16 (6). Springer Science and Business Media LLC: 507–531. Bibcode:1986FoPh...16..507F. doi:10.1007/bf01886518. ISSN   0015-9018. S2CID   122076550.
5. Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation". Physical Review A. 52 (5). American Physical Society (APS): 3457–3467. arXiv: quant-ph/9503016 . Bibcode:1995PhRvA..52.3457B. doi:10.1103/physreva.52.3457. ISSN   1050-2947. PMID   9912645. S2CID   8764584.
6. Nielsen, Michael A. (2010). Quantum computation and quantum information. Isaac L. Chuang (10th anniversary ed.). Cambridge: Cambridge University Press. ISBN   978-1-107-00217-3. OCLC   665137861.
7. Hung, W.N.N.; Xiaoyu Song; Guowu Yang; Jin Yang; Perkowski, M. (September 2006). "Optimal synthesis of multiple output Boolean functions using a set of quantum gates by symbolic reachability analysis". IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 25 (9): 1652–1663. doi:10.1109/tcad.2005.858352. ISSN   0278-0070. S2CID   14123321.
8. Collins, Daniel; Linden, Noah; Popescu, Sandu (2001-08-07). "Nonlocal content of quantum operations". Physical Review A. 64 (3): 032302. arXiv: quant-ph/0005102 . Bibcode:2001PhRvA..64c2302C. doi:10.1103/PhysRevA.64.032302. ISSN   1050-2947. S2CID   29769034.
9. Pathak, Anirban (2013-06-20). Elements of Quantum Computation and Quantum Communication. Taylor & Francis. ISBN   978-1-4665-1792-9.
10. Yanofsky, Noson S.; Mannucci, Mirco A. (2008-08-11). Quantum Computing for Computer Scientists. Cambridge University Press. ISBN   978-1-139-64390-0.
11. Stancil, Daniel D.; Byrd, Gregory T. (2022-04-19). Principles of Superconducting Quantum Computers. John Wiley & Sons. ISBN   978-1-119-75074-1.
12. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin (2000). "Fast Quantum Gates for Neutral Atoms". Phys. Rev. Lett. 85 (10): 2208–2211. arXiv: quant-ph/0004038 . Bibcode:2000PhRvL..85.2208J. doi:10.1103/PhysRevLett.85.2208. PMID   10970499.
13. Dibyendu Chatterjee, Arijit Roy (2015). "A transmon-based quantum half-adder scheme". Progress of Theoretical and Experimental Physics . 2015 (9): 7–8. Bibcode:2015PTEP.2015i3A02C. doi: 10.1093/ptep/ptv122 .
14. McKay, David C.; Wood, Christopher J.; Sheldon, Sarah; Chow, Jerry M.; Gambetta, Jay M. (31 August 2017). "Efficient Z gates for quantum computing". Physical Review A . 96 (2): 022330. arXiv: 1612.00858 . Bibcode:2015PTEP.2015i3A02C. doi:10.1093/ptep/ptv122.
15. "qiskit.circuit.library.PhaseGate". IBM (qiskit documentation).
16. Griffiths, D. J. (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. pp. 127–128. ISBN   978-3-527-40601-2.
17. "Monroe Conference" (PDF). online.kitp.ucsb.edu.
18. "Demonstration of a small programmable quantum computer with atomic qubits" (PDF). Retrieved 2019-02-10.
19. Foxen, B.; Neill, C.; Dunsworth, A.; Roushan, P.; Chiaro, B.; Megrant, A.; Kelly, J.; Chen, Zijun; Satzinger, K.; Barends, R.; Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J. C.; Boixo, S.; Buell, D.; Burkett, B.; Chen, Yu; Collins, R.; Farhi, E.; Fowler, A.; Gidney, C.; Giustina, M.; Graff, R.; Harrigan, M.; Huang, T.; Isakov, S. V.; Jeffrey, E.; Jiang, Z.; Kafri, D.; Kechedzhi, K.; Klimov, P.; Korotkov, A.; Kostritsa, F.; Landhuis, D.; Lucero, E.; McClean, J.; McEwen, M.; Mi, X.; Mohseni, M.; Mutus, J. Y.; Naaman, O.; Neeley, M.; Niu, M.; Petukhov, A.; Quintana, C.; Rubin, N.; Sank, D.; Smelyanskiy, V.; Vainsencher, A.; White, T. C.; Yao, Z.; Yeh, P.; Zalcman, A.; Neven, H.; Martinis, J. M. (2020-09-15). "Demonstrating a Continuous Set of Two-qubit Gates for Near-term Quantum Algorithms". Physical Review Letters. 125 (12): 120504. arXiv: 2001.08343 . Bibcode:2020PhRvL.125l0504F. doi:10.1103/PhysRevLett.125.120504. ISSN   0031-9007. PMID   33016760.
20. Nguyen, L.B.; Kim, Y.; Hashim, A.; Goss, N.; Marinelli, B.; Bhandari, B.; Das, D.; Naik, R.K.; Kreikebaum, J.M.; Jordan, A.; Santiago, D.I.; Siddiqi, I. (16 January 2024). "Programmable Heisenberg interactions between Floquet qubits". Nature Physics. 20 (1): 240–246. arXiv: 2211.10383 . Bibcode:2024NatPh..20..240N. doi: 10.1038/s41567-023-02326-7 .
21. Nemirovsky, Jonathan; Sagi, Yoav (2021), "Fast universal two-qubit gate for neutral fermionic atoms in optical tweezers", Physical Review Research, 3 (1): 013113, arXiv: 2008.09819 , Bibcode:2021PhRvR...3a3113N, doi: 10.1103/PhysRevResearch.3.013113
22. Rasmussen, S. E.; Zinner, N. T. (2020-07-17). "Simple implementation of high fidelity controlled- i swap gates and quantum circuit exponentiation of non-Hermitian gates". Physical Review Research. 2 (3): 033097. arXiv: 2002.11728 . Bibcode:2020PhRvR...2c3097R. doi: 10.1103/PhysRevResearch.2.033097 . ISSN   2643-1564.
23. Schuch, Norbert; Siewert, Jens (2003-03-10). "Natural two-qubit gate for quantum computation using the XY interaction". Physical Review A. 67 (3): 032301. arXiv: quant-ph/0209035 . Bibcode:2003PhRvA..67c2301S. doi:10.1103/PhysRevA.67.032301. ISSN   1050-2947. S2CID   50823541.
24. Dallaire-Demers, Pierre-Luc; Wilhelm, Frank K. (2016-12-05). "Quantum gates and architecture for the quantum simulation of the Fermi-Hubbard model". Physical Review A. 94 (6): 062304. arXiv: 1606.00208 . Bibcode:2016PhRvA..94f2304D. doi:10.1103/PhysRevA.94.062304. ISSN   2469-9926. S2CID   118408193.
25. Cross, Andrew; Javadi-Abhari, Ali; Alexander, Thomas; De Beaudrap, Niel; Bishop, Lev S.; Heidel, Steven; Ryan, Colm A.; Sivarajah, Prasahnt; Smolin, John; Gambetta, Jay M.; Johnson, Blake R. (2022). "OpenQASM 3: A Broader and Deeper Quantum Assembly Language". ACM Transactions on Quantum Computing. 3 (3): 1–50. arXiv: 2104.14722 . doi: 10.1145/3505636 . ISSN   2643-6809. S2CID   233476587.
26. Zhang, Jun; Vala, Jiri; Sastry, Shankar; Whaley, K. Birgitta (2004-07-07). "Minimum Construction of Two-Qubit Quantum Operations". Physical Review Letters. 93 (2): 020502. arXiv: quant-ph/0312193 . Bibcode:2004PhRvL..93b0502Z. doi:10.1103/PhysRevLett.93.020502. ISSN   0031-9007. PMID   15323888. S2CID   9632700.
27. AbuGhanem, M. (2021-01-01). "Two-qubit Entangling Gate for Superconducting Quantum Computers". Rochester, NY. doi:10.2139/ssrn.4188257. S2CID   252264545. SSRN   4188257.{{cite journal}}: Cite journal requires |journal= (help)
28. Peterson, Eric C.; Crooks, Gavin E.; Smith, Robert S. (2020-03-26). "Fixed-Depth Two-Qubit Circuits and the Monodromy Polytope". Quantum. 4: 247. arXiv: 1904.10541 . doi: 10.22331/q-2020-03-26-247 . S2CID   214690323.
29. Córcoles, A. D.; Magesan, Easwar; Srinivasan, Srikanth J.; Cross, Andrew W.; Steffen, M.; Gambetta, Jay M.; Chow, Jerry M. (2015-04-29). "Demonstration of a quantum error detection code using a square lattice of four superconducting qubits". Nature Communications. 6 (1): 6979. arXiv: 1410.6419 . Bibcode:2015NatCo...6.6979C. doi:10.1038/ncomms7979. ISSN   2041-1723. PMC   4421819 . PMID   25923200.
30. Kyriienko, Oleksandr; Elfving, Vincent E. (2021-11-15). "Generalized quantum circuit differentiation rules". Physical Review A. 104 (5): 052417. arXiv: 2108.01218 . Bibcode:2021PhRvA.104e2417K. doi:10.1103/PhysRevA.104.052417. hdl: 10871/127818 . ISSN   2469-9926. S2CID   236881494.
31. Arrazola, Juan Miguel; Matteo, Olivia Di; Quesada, Nicolás; Jahangiri, Soran; Delgado, Alain; Killoran, Nathan (2022-06-20). "Universal quantum circuits for quantum chemistry". Quantum. 6: 742. arXiv: 2106.13839 . Bibcode:2022Quant...6..742A. doi: 10.22331/q-2022-06-20-742 . S2CID   235658488.
32. Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C.; Barends, Rami; Biswas, Rupak; Boixo, Sergio; Brandao, Fernando G. S. L.; Buell, David A.; Burkett, Brian; Chen, Yu; Chen, Zijun; Chiaro, Ben; Collins, Roberto (2019). "Quantum supremacy using a programmable superconducting processor". Nature. 574 (7779): 505–510. arXiv: 1910.11333 . Bibcode:2019Natur.574..505A. doi: 10.1038/s41586-019-1666-5 . ISSN   1476-4687. PMID   31645734. S2CID   204836822.
33. Gu, Xiu; Fernández-Pendás, Jorge; Vikstål, Pontus; Abad, Tahereh; Warren, Christopher; Bengtsson, Andreas; Tancredi, Giovanna; Shumeiko, Vitaly; Bylander, Jonas; Johansson, Göran; Frisk Kockum, Anton (2021). "Fast Multiqubit Gates through Simultaneous Two-Qubit Gates". PRX Quantum. 2 (4): 040348. arXiv: 2108.11358 . Bibcode:2021PRXQ....2d0348G. doi: 10.1103/PRXQuantum.2.040348 . ISSN   2691-3399.
34. Maslov, Dmitri (2016-02-10). "Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization". Physical Review A. 93 (2): 022311. arXiv: 1508.03273 . Bibcode:2016PhRvA..93b2311M. doi: 10.1103/PhysRevA.93.022311 . ISSN   2469-9926. S2CID   5226873.
35. Song, Guang; Klappenecker, Andreas (2003-12-31). "The simplified Toffoli gate implementation by Margolus is optimal". arXiv: quant-ph/0312225 . Bibcode:2003quant.ph.12225S.{{cite journal}}: Cite journal requires |journal= (help)
36. Thapliyal, Himanshu; Ranganathan, Nagarajan (2009). "Design of Efficient Reversible Binary Subtractors Based on a New Reversible Gate". 2009 IEEE Computer Society Annual Symposium on VLSI. pp. 229–234. doi:10.1109/ISVLSI.2009.49. ISBN   978-1-4244-4408-3. S2CID   16182781.
37. Warren, Christopher; Fernández-Pendás, Jorge; Ahmed, Shahnawaz; Abad, Tahereh; Bengtsson, Andreas; Biznárová, Janka; Debnath, Kamanasish; Gu, Xiu; Križan, Christian; Osman, Amr; Fadavi Roudsari, Anita; Delsing, Per; Johansson, Göran; Frisk Kockum, Anton; Tancredi, Giovanna; Bylander, Jonas (2023). "Extensive characterization and implementation of a family of three-qubit gates at the coherence limit". npj Quantum Information. 9 (1): 44. arXiv: 2207.02938 . Bibcode:2023npjQI...9...44W. doi: 10.1038/s41534-023-00711-x . ISSN   2056-6387.