List of quantum logic gates

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

class="wikitable"

!Name

!# qubits

!Operator symbol

!Matrix

!Circuit diagram

!Properties

!Refs

Identity,

no-op

|1 (any)

| $I,\;\mathbb{I}$ , {{not a typo|𝟙}}

| $\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$

|150px
or

150px

Global phase

|1 (any)

| $\mathrm{Ph}$ , $\mathrm{Phase}$ or $\mathrm e^{i\delta}I$

| $\mathrm{e}^{i\delta}\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$

|150px

Continuous parameters: $\delta$ (period $2\pi$ )
Exponential form: $\exp(i\delta I)$

The identity gate is the identity operation $I|\psi\rangle=|\psi\rangle$ , 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",{{cite web|url=https://qiskit.org/documentation/stubs/qiskit.circuit.library.IGate.html#qiskit.circuit.library.IGate|title=IGate|website=qiskit.org}} Qiskit online documentation. and a NOP.{{cite web|url=https://docs.microsoft.com/en-us/qsharp/api/qsharp/microsoft.quantum.intrinsic.i|title=I operation|website=docs.microsoft.com|date=28 July 2023 }} Q# online documentation.

The global phase gate introduces a global phase $e^{i\varphi}$ 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: $|e^{i\varphi}|=1$ for any $\varphi$ .

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

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

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

Clifford qubit gates

This table includes commonly used Clifford gates for qubits.{{cite book |author=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=Springer |year=2011 |isbn=978-1-84628-887-6}}{{cite journal | last=Feynman | first=Richard P. | title=Quantum mechanical computers | journal=Foundations of Physics | publisher=Springer Science and Business Media LLC | volume=16 | issue=6 | year=1986 | issn=0015-9018 | doi=10.1007/bf01886518 | pages=507–531| bibcode=1986FoPh...16..507F | s2cid=122076550 }}{{cite journal | last1=Barenco | first1=Adriano | last2=Bennett | first2=Charles H. | last3=Cleve | first3=Richard | last4=DiVincenzo | first4=David P. | last5=Margolus | first5=Norman | last6=Shor | first6=Peter | last7=Sleator | first7=Tycho | last8=Smolin | first8=John A. | last9=Weinfurter | first9=Harald | title=Elementary gates for quantum computation | journal=Physical Review A | publisher=American Physical Society (APS) | volume=52 | issue=5 | date=1995-11-01 | issn=1050-2947 | doi=10.1103/physreva.52.3457 | pages=3457–3467| pmid=9912645 |arxiv=quant-ph/9503016| bibcode=1995PhRvA..52.3457B | s2cid=8764584 }}

class="wikitable"

!Names

!# qubits

!Operator symbol

!Matrix

!Circuit diagram

!Some properties

!Refs

Pauli X,
NOT,
bit flip

| $X,\;\mathrm{NOT},\;\sigma_x$

| $\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$

150px

Hermitian
Pauli group
Traceless
Involutory

|{{Cite book |last=Nielsen |first=Michael A. |url=https://www.worldcat.org/oclc/665137861 |title=Quantum computation and quantum information |publisher=Cambridge University Press |others=Isaac L. Chuang |year=2010 |isbn=978-1-107-00217-3 |edition=10th anniversary |location=Cambridge |oclc=665137861}}

Pauli Y

| $Y,\;\sigma_y$

| $\begin{bmatrix} 0 & -i \\ i & 0\end{bmatrix}$

|150px

Hermitian
Pauli group
Traceless
Involutory

Pauli Z,
phase flip

| $Z,\;\sigma_z$

| $\begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}$

|150px

Hermitian
Pauli group
Traceless
Involutory

Phase gate S,
square root of Z

| $S,\;P,\;\sqrt{Z}$

| $\begin{bmatrix} 1 & 0 \\ 0 & i\end{bmatrix}$

|150px

Square root of X,
square root of NOT

| $\sqrt{X}$ , $V$ , $\sqrt{\mathrm{NOT}},\;\mathrm{SX}$

| $\frac{1}{2}\begin{bmatrix} 1+i & 1-i \\ 1-i & 1+i\end{bmatrix}$

|150px

|{{cite journal |last1=Hung |first1=W. N. N. |last2=Song |first2=Xiaoyu |last3=Yang |first3=Guowu |last4=Yang |first4=Jin |last5=Perkowski |first5=M. |date=September 2006 |title=Optimal synthesis of multiple output Boolean functions using a set of quantum gates by symbolic reachability analysis |journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems |volume=25 |issue=9 |pages=1652–1663 |doi=10.1109/tcad.2005.858352 |issn=0278-0070 |s2cid=14123321}}

Hadamard,
Walsh-Hadamard

| $H$

| $\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1\end{bmatrix}$

|150px

Hermitian
Traceless
Involutory

Controlled NOT,
controlled-X,
controlled-bit flip,
reversible exclusive OR,
Feynman

| $\mathrm{CNOT}$ , $\mathrm{XOR},\;\mathrm{CX}$

| $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{bmatrix}$
$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\0 & 0 & 1 &0 \\ 0 & 1 & 0 & 0\end{bmatrix}$

|150px

150px

Hermitian
Involutory

Implementation:

Cirac–Zoller gate

Anticontrolled-NOT,
anticontrolled-X,
zero control,
control-on-0-NOT,
reversible exclusive NOR

| $\overline{\mathrm C}\mathrm X$ , $\text{controlled[0]-NOT}$ , $\mathrm{XNOR}$

| $\begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$

|150px

Hermitian
Involutory

Controlled-Z,
controlled sign flip,
controlled phase flip

| $\mathrm{CZ}$ , $\mathrm{CPF}$ , $\mathrm{CSIGN}$ , $\mathrm{CPHASE}$

| $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1\end{bmatrix}$

|150px

Hermitian
Involutory
Symmetrical

Implementation:

Duan-Kimble gate

Double-controlled NOT

| $\mathrm{DCNOT}$

| $\begin{bmatrix} 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0\end{bmatrix}$

|150px

Special unitary

|{{Cite journal |last1=Collins |first1=Daniel |last2=Linden |first2=Noah |last3=Popescu |first3=Sandu |date=2001-08-07 |title=Nonlocal content of quantum operations |url=https://link.aps.org/doi/10.1103/PhysRevA.64.032302 |journal=Physical Review A |language=en |volume=64 |issue=3 |pages=032302 |doi=10.1103/PhysRevA.64.032302 |arxiv=quant-ph/0005102 |bibcode=2001PhRvA..64c2302C |s2cid=29769034 |issn=1050-2947}}

Swap

| $\mathrm{SWAP}$

| $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$

|150px
or
150px

Hermitian
Involutory
Symmetrical

Imaginary swap

| $\mbox{iSWAP}$

| $\begin{bmatrix} 1&0&0&0\\0&0&i&0\\0&i&0&0\\0&0&0&1\end{bmatrix}$

|120px
or
150px

Special unitary
Symmetrical

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

Note that if a Clifford gate A is not in the Pauli group, $\sqrt{A}$ or controlled-A are not in the Clifford gates.{{Citation needed|date=September 2022}}

The Clifford set is not a universal quantum gate set.

Non-Clifford qubit gates

= Relative phase gates =

class="wikitable"

!Names

!# qubits

!Operator symbol

!Matrix

!Circuit diagram

!Properties

!Refs

Phase shift

| $P(\varphi),\;R(\varphi),\;u_1(\varphi)$

| $\begin{bmatrix} 1 & 0 \\ 0 & \mathrm e^{i \varphi}\end{bmatrix}$

|150px

Continuous parameters: $\varphi$ (period $2\pi$ )

|{{Cite book |last=Pathak |first=Anirban |url=https://books.google.com/books?id=cEPSBQAAQBAJ |title=Elements of Quantum Computation and Quantum Communication |date=2013-06-20 |publisher=Taylor & Francis |isbn=978-1-4665-1792-9 |language=en}}{{Cite book |last1=Yanofsky |first1=Noson S. |url=https://books.google.com/books?id=U1chAwAAQBAJ |title=Quantum Computing for Computer Scientists |last2=Mannucci |first2=Mirco A. |date=2008-08-11 |publisher=Cambridge University Press |isbn=978-1-139-64390-0 |language=en}}

Phase gate T,
π/8 gate,
fourth root of Z

| $T,P(\pi/4)$ or $\sqrt[4]{Z}$

| $\begin{bmatrix} 1 & 0 \\ 0 & \mathrm e^{i\pi /4}\end{bmatrix}$

|150px

Controlled phase

| $\mathrm{CPhase}(\varphi),\mathrm{CR}(\varphi)$

| $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{i \varphi} \end{bmatrix}$

|150px

Continuous parameters: $\varphi$ (period $2\pi$ )
Symmetrical

Implementation:

Jaksch Gate{{cite journal |author=Jaksch |first=D. |last2=Cirac |first2=J. I. |last3=Zoller |first3=P. |last4=Rolston |first4=S. L. |last5=Côté |first5=R. |last6=Lukin |first6=M. D. |name-list-style=and |year=2000 |title=Fast Quantum Gates for Neutral Atoms |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.2208 |journal=Phys. Rev. Lett. |volume=85 |issue=10 |pages=2208–2211 |arxiv=quant-ph/0004038 |bibcode=2000PhRvL..85.2208J |doi=10.1103/PhysRevLett.85.2208 |pmid=10970499}}

Controlled phase S

| $\mathrm{CS},\text{controlled-}S$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & i
\end{bmatrix}$

|150px

Symmetrical

The phase shift is a family of single-qubit gates that map the basis states $P(\varphi)|0\rangle = |0\rangle$ and $P(\varphi)|1\rangle= e^{i\varphi}|1\rangle$ . The probability of measuring a $|0\rangle$ or $|1\rangle$ 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 $\varphi$ radians. A common example is the T gate where $\varphi = \frac{\pi}{4}$ (historically known as the $\pi /8$ gate), the phase gate. Note that some Clifford gates are special cases of the phase shift gate: $P(0)=I,\;P(\pi)=Z;P(\pi/2)=S.$

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. $P(\varphi)$ rotates the phase about {{nowrap| $|1\rangle$ )}}. Extending $P(\varphi)$ 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: $P(\beta) \cdot X \cdot P(\alpha) \cdot X = \begin{bmatrix} e^{i\alpha} & 0 \\ 0 & e^{i\beta} \end{bmatrix}$ . When $\alpha = -\beta$ this gate is the rotation operator $R_z(2\beta)$ gate and if $\alpha =\beta$ it is a global phase.{{efn| $\alpha = -\beta$ when $P(\alpha)=P(\beta)^\dagger$ , where $\dagger$ is the conjugate transpose (or Hermitian adjoint).}}{{efn|Also: $\left(P(\delta) \cdot Y\right)^2 = e^{i\delta} I$ }}

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

Arbitrary single-qubit phase shift gates $P(\varphi)$ are natively available for transmon quantum processors through timing of microwave control pulses.{{cite journal |author=Chatterjee |first=Dibyendu |last2=Roy |first2=Arijit |year=2015 |title=A transmon-based quantum half-adder scheme |url=https://paperity.org/p/73955611/a-transmon-based-quantum-half-adder-scheme |journal=Progress of Theoretical and Experimental Physics |volume=2015 |issue=9 |pages=7–8 |bibcode=2015PTEP.2015i3A02C |doi=10.1093/ptep/ptv122 |doi-access=free}} It can be explained in terms of change of frame.{{cite journal|first1=David C. |last1=McKay |first2=Christopher J. |last2=Wood |first3=Sarah |last3=Sheldon |first4=Jerry M. |last4=Chow |first5=Jay M. |last5=Gambetta|journal=Physical Review A|title=Efficient Z gates for quantum computing|date=31 August 2017|volume=96|issue=2|page=022330|doi=10.1093/ptep/ptv122|arxiv=1612.00858|bibcode=2015PTEP.2015i3A02C }}{{cite web|url=https://qiskit.org/documentation/stubs/qiskit.circuit.library.PhaseGate.html#qiskit.circuit.library.PhaseGate|title=qiskit.circuit.library.PhaseGate|publisher=IBM (qiskit documentation)}}

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

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

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

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

= Rotation operator gates =

{{Further|Bloch sphere#Rotations|Rotation operator (quantum mechanics)}}

class="wikitable"

!Names

!# qubits

!Operator symbol

!Exponential form

!Matrix

!Circuit diagram

!Properties

!Refs

Rotation about x-axis

| $R_x(\theta)$

| $\exp(-iX\theta/2)$

| ${\begin{bmatrix}\cos(\theta /2)&-i\sin(\theta /2)\\-i\sin(\theta /2)&\cos(\theta /2)\end{bmatrix}}$

|150px

Special unitary
Continuous parameters: $\theta$ (period $4\pi$ )

Rotation about y-axis

| $R_y(\theta)$

| $\exp(-iY\theta/2)$

| $\begin{bmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{bmatrix}$

|150px

Special unitary
Continuous parameters: $\theta$ (period $4\pi$ )

Rotation about z-axis

| $R_z(\theta)$

| $\exp(-iZ\theta/2)$

| $\begin{bmatrix} \exp(-i\theta/2) & 0 \\0 & \exp(i\theta/2)
\end{bmatrix}$

|150px

Special unitary
Continuous parameters: $\theta$ (period $4\pi$ )

The rotation operator gates $R_x(\theta),R_y(\theta)$ and $R_z(\theta)$ are the analog rotation matrices in three Cartesian axes of SO(3),{{efn|a SU(2) double cover. See also Hopf fibration.}} 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 $2 \times 2$ 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 {{math|4π}}. A rotation of {{math|2π}} (360 degrees) returns the same statevector with a different phase.{{cite book |author=Griffiths |first=D. J. |title=Introduction to Elementary Particles |publisher=John Wiley & Sons |year=2008 |isbn=978-3-527-40601-2 |edition=2nd |pages=127–128 |author-link=David J. Griffiths}}

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

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

It is possible to work out the adjoint action of rotations on the Pauli vector, namely rotation effectively by double the angle {{mvar|a}} to apply Rodrigues' rotation formula:

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

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

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

Global phase and phase shift can be transformed into each other's with the Z-rotation operator: $R_z(\gamma) \operatorname{Ph}\left(\frac{\gamma}{2}\right) = P(\gamma)$ .{{r|Barenco|pages=11}}{{r|Williams|pages=77–83}}

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

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

= Two-qubit interaction gates =

class="wikitable"

!Names

!# qubits

!Operator symbol

!Exponential form

!Matrix

!Circuit diagram

!Properties

!Refs

XX interaction

| $R_{xx}(\phi)$ , $\text{XX} (\phi)$

| $\exp\left(-i \frac{\phi}{2} X\otimes X\right)$

| $\begin{bmatrix}
\cos\left(\frac{\phi}{2}\right) & 0 & 0 & -i \sin\left(\frac{\phi}{2}\right) \\
0 &\cos\left(\frac{\phi}{2}\right) & -i \sin\left(\frac{\phi}{2}\right) & 0 \\
0 & -i \sin\left(\frac{\phi}{2}\right) & \cos\left(\frac{\phi}{2}\right) & 0 \\
-i \sin\left(\frac{\phi}{2}\right) & 0 & 0 & \cos\left(\frac{\phi}{2}\right) \\
\end{bmatrix}$

Special unitary
Continuous parameters: $\theta$ (period $4\pi$ )

Implementation:

Mølmer–Sørensen gate

|{{Citation needed|date=September 2022}}

YY interaction

| $R_{yy}(\phi)$ , $\text{YY} (\phi)$

| $\exp\left(-i \frac{\phi}{2} Y\otimes Y\right)$

| $\begin{bmatrix}
\cos\left(\frac{\phi}{2}\right) & 0 & 0 & i\sin\left(\frac{\phi}{2}\right) \\
0 & \cos\left(\frac{\phi}{2}\right) & -i\sin\left(\frac{\phi}{2}\right) & 0 \\
0 & -i\sin\left(\frac{\phi}{2}\right) & \cos\left(\frac{\phi}{2}\right) & 0 \\
i\sin\left(\frac{\phi}{2}\right) & 0 & 0 & \cos\left(\frac{\phi}{2}\right) \\
\end{bmatrix}$

Special unitary
Continuous parameters: $\theta$ (period $4\pi$ )

Implementation:

Mølmer–Sørensen gate

|{{Citation needed|date=September 2022}}

ZZ interaction

| ${\displaystyle R_{zz}(\phi )}$ , $\text{ZZ} (\phi)$

| ${\displaystyle \exp \left(-i{\frac {\phi }{2}}Z\otimes Z\right)}$

| $\begin{bmatrix}
e^{-i \phi/2} & 0 & 0 & 0 \\
0 & e^{i \phi/2} & 0 & 0 \\
0 & 0 & e^{i \phi/2} & 0 \\
0 & 0 & 0 & e^{-i \phi/2} \\
\end{bmatrix}$

Special unitary
Continuous parameters: $\theta$ (period $4\pi$ )

|{{Citation needed|date=September 2022}}

XY,
XX plus YY

| ${\displaystyle R_{xy}(\phi )}$ , $\text{XY} (\phi)$

| ${\displaystyle \exp \left[-i\frac {\phi }{4}(X\otimes X+Y\otimes Y)\right]}$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos(\phi/2) & - i\sin(\phi/2) & 0 \\
0 & -i\sin(\phi/2) & \cos(\phi/2) & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$

Special unitary
Continuous parameters: $\theta$ (period $4\pi$ )

|{{Citation needed|date=September 2022}}

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.{{cite web |title=Monroe Conference |url=http://online.kitp.ucsb.edu/online/mbl_c15/monroe/pdf/Monroe_MBL15Conf_KITP.pdf |website=online.kitp.ucsb.edu}}{{cite web |title=Demonstration of a small programmable quantum computer with atomic qubits |url=http://iontrap.umd.edu/wp-content/uploads/2012/12/nature18648.pdf |access-date=2019-02-10}}

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

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

: $\mbox{CNOT} =e^{-i\frac{\pi}{4}}R_{y_1}(-\pi/2)R_{x_1}(-\pi/2)R_{x_2}(-\pi/2)R_{xx}(\pi/2)R_{y_1}(\pi/2).$

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

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,{{cite journal | last1=Foxen | first1=B. | last2=Neill | first2=C. | last3=Dunsworth | first3=A. | last4=Roushan | first4=P. | last5=Chiaro | first5=B. | last6=Megrant | first6=A. | last7=Kelly | first7=J. | last8=Chen | first8=Zijun | last9=Satzinger | first9=K. | last10=Barends | first10=R. | last11=Arute | first11=F. | last12=Arya | first12=K. | last13=Babbush | first13=R. | last14=Bacon | first14=D. | last15=Bardin | first15=J. C. | last16=Boixo | first16=S. | last17=Buell | first17=D. | last18=Burkett | first18=B. | last19=Chen | first19=Yu | last20=Collins | first20=R. | last21=Farhi | first21=E. | last22=Fowler | first22=A. | last23=Gidney | first23=C. | last24=Giustina | first24=M. | last25=Graff | first25=R. | last26=Harrigan | first26=M. | last27=Huang | first27=T. | last28=Isakov | first28=S. V. | last29=Jeffrey | first29=E. | last30=Jiang | first30=Z. | last31=Kafri | first31=D. | last32=Kechedzhi | first32=K. | last33=Klimov | first33=P. | last34=Korotkov | first34=A. | last35=Kostritsa | first35=F. | last36=Landhuis | first36=D. | last37=Lucero | first37=E. | last38=McClean | first38=J. | last39=McEwen | first39=M. | last40=Mi | first40=X. | last41=Mohseni | first41=M. | last42=Mutus | first42=J. Y. | last43=Naaman | first43=O. | last44=Neeley | first44=M. | last45=Niu | first45=M. | last46=Petukhov | first46=A. | last47=Quintana | first47=C. | last48=Rubin | first48=N. | last49=Sank | first49=D. | last50=Smelyanskiy | first50=V. | last51=Vainsencher | first51=A. | last52=White | first52=T. C. | last53=Yao | first53=Z. | last54=Yeh | first54=P. | last55=Zalcman | first55=A. | last56=Neven | first56=H. | last57=Martinis | first57=J. M. | title=Demonstrating a Continuous Set of Two-qubit Gates for Near-term Quantum Algorithms | journal=Physical Review Letters | volume=125 | issue=12 | date=2020-09-15 | issn=0031-9007 | doi=10.1103/PhysRevLett.125.120504 | page=120504| pmid=33016760 | arxiv=2001.08343 | bibcode=2020PhRvL.125l0504F }} or using microwave drives in fixed-frequency qubits with fixed coupling.{{cite journal |last1=Nguyen |first1=L.B. |last2=Kim |first2=Y. |last3=Hashim |first3=A. |last4=Goss |first4=N.|last5=Marinelli |first5=B.|last6=Bhandari |first6=B.|last7=Das |first7=D.|last8=Naik |first8=R.K.|last9=Kreikebaum |first9=J.M.|last10=Jordan |first10=A.|last11=Santiago |first11=D.I.|last12=Siddiqi |first12=I. |title=Programmable Heisenberg interactions between Floquet qubits

|journal=Nature Physics |date=16 January 2024 |volume=20 |issue=1 |pages=240–246 |doi=10.1038/s41567-023-02326-7 |bibcode=2024NatPh..20..240N |doi-access=free |arxiv=2211.10383}}

= Non-Clifford swap gates =

class="wikitable"

!Names

!# qubits

!Operator symbol

!Matrix

!Circuit diagram

!Properties

!Refs

Square root swap

| $\sqrt{\mathrm{SWAP}}$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \frac{1}{{2}} (1+i) & \frac{1}{{2}} (1-i) & 0 \\
0 & \frac{1}{{2}} (1-i) & \frac{1}{{2}} (1+i) & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$

|150px

Square root imaginary swap

| $\sqrt{\mbox{iSWAP}}$

| $\begin{bmatrix} 1&0&0&0\\0&\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\0&\frac{i}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\0&0&0&1\end{bmatrix}$

Special unitary

|{{Cite book |last1=Stancil |first1=Daniel D. |url=https://books.google.com/books?id=awxsEAAAQBAJ |title=Principles of Superconducting Quantum Computers |last2=Byrd |first2=Gregory T. |date=2022-04-19 |publisher=John Wiley & Sons |isbn=978-1-119-75074-1 |language=en}}

Swap (raised to a power)

| $\mbox{SWAP}^\alpha$

| $\begin{bmatrix} 1&0&0&0\\0&\frac{1+e^{i \pi \alpha}}{2}&\frac{1-e^{i \pi \alpha}}{2}&0\\0&\frac{1-e^{i \pi \alpha}}{2}&\frac{1+e^{i \pi \alpha}}{2}&0\\0&0&0&1\end{bmatrix}$

|150px

Continuous parameters: $\alpha$ (period $2$ )

Fredkin,

controlled swap

| $\mathrm{CSWAP}$ , $\mathrm{FREDKIN}$

| $\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{bmatrix}$

|150px
or
150px

Hermitian
Involutory
Functionally complete reversible gate for Boolean algebra

The {{radic|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 {{radic|SWAP}} and single qubit gates. More than one application of the {{radic|SWAP}} is required to produce a Bell state from product states. The {{radic|SWAP}} gate arises naturally in systems that exploit exchange interaction.{{Citation|last1=Nemirovsky|first1=Jonathan|last2=Sagi|first2=Yoav|title=Fast universal two-qubit gate for neutral fermionic atoms in optical tweezers|year=2021|journal=Physical Review Research|volume=3|issue=1|page=013113|doi=10.1103/PhysRevResearch.3.013113|arxiv=2008.09819|bibcode=2021PhRvR...3a3113N|doi-access=free}}

For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap{{Cite journal |last1=Rasmussen |first1=S. E. |last2=Zinner |first2=N. T. |date=2020-07-17 |title=Simple implementation of high fidelity controlled- i swap gates and quantum circuit exponentiation of non-Hermitian gates |journal=Physical Review Research |language=en |volume=2 |issue=3 |pages=033097 |arxiv=2002.11728 |bibcode=2020PhRvR...2c3097R |doi=10.1103/PhysRevResearch.2.033097 |issn=2643-1564 |doi-access=free}} or iSWAP.{{Cite journal |last1=Schuch |first1=Norbert |last2=Siewert |first2=Jens |date=2003-03-10 |title=Natural two-qubit gate for quantum computation using the XY interaction |url=https://link.aps.org/doi/10.1103/PhysRevA.67.032301 |journal=Physical Review A |language=en |volume=67 |issue=3 |pages=032301 |arxiv=quant-ph/0209035 |bibcode=2003PhRvA..67c2301S |doi=10.1103/PhysRevA.67.032301 |issn=1050-2947 |s2cid=50823541}}{{Cite journal |last1=Dallaire-Demers |first1=Pierre-Luc |last2=Wilhelm |first2=Frank K. |date=2016-12-05 |title=Quantum gates and architecture for the quantum simulation of the Fermi-Hubbard model |url=https://link.aps.org/doi/10.1103/PhysRevA.94.062304 |journal=Physical Review A |language=en |volume=94 |issue=6 |pages=062304 |arxiv=1606.00208 |bibcode=2016PhRvA..94f2304D |doi=10.1103/PhysRevA.94.062304 |issn=2469-9926 |s2cid=118408193}} Note that $i\mbox{SWAP}=R_{xx}(-\pi/2)R_{yy}(-\pi/2)$ and $\sqrt{i\mbox{SWAP}}=R_{xx}(-\pi/4)R_{yy}(-\pi/4)$ , or more generally $\sqrt[n]{i\mbox{SWAP}}=R_{xx}(-\pi/2n)R_{yy}(-\pi/2n)$ for all real n except 0.

SWAP^α arises naturally in spintronic quantum computers.

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.{{anchor|Deutsch|Deutsch gate}}

= Other named gates =

class="wikitable"

!Names

!# qubits

!Operator symbol

!Matrix

!Circuit diagram

!Properties

!Named after

!Refs

General single qubit rotation

| $U(\theta,\phi,\lambda)$

| ${\begin{bmatrix}\cos(\theta /2)&-e^{i\lambda}\sin(\theta /2)\\e^{i\phi}\sin(\theta /2)&e^{i(\lambda+\phi)}\cos(\theta /2)\end{bmatrix}}$

Implements an arbitrary single-qubit rotation
Continuous parameters: $\theta,\phi,\lambda$ (period $2\pi$ )

|OpenQASM U gate{{Efn|The matrix shown here is from openQASM 3.0, which differs from $U(\theta,\phi,\lambda)$ from a global phase (OpenQASM 2.0 U gate is in SU(2) ) .}}

|{{Cite journal |last1=Cross |first1=Andrew |last2=Javadi-Abhari |first2=Ali |last3=Alexander |first3=Thomas |last4=De Beaudrap |first4=Niel |last5=Bishop |first5=Lev S. |last6=Heidel |first6=Steven |last7=Ryan |first7=Colm A. |last8=Sivarajah |first8=Prasahnt |last9=Smolin |first9=John |last10=Gambetta |first10=Jay M. |last11=Johnson |first11=Blake R. |date=2022 |title=OpenQASM 3: A Broader and Deeper Quantum Assembly Language |journal=ACM Transactions on Quantum Computing |volume=3 |issue=3 |pages=1–50 |doi=10.1145/3505636 |s2cid=233476587 |issn=2643-6809|doi-access=free |arxiv=2104.14722 }}

| $\mathrm{BARENCO}(\alpha,\phi,\theta)$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & e^{i \alpha}\cos\theta & -\mathrm {i} e^{\mathrm{i} (\alpha-\phi)}\sin\theta \\
0 & 0 & -\mathrm {i} e^{\mathrm{i} (\alpha+\phi)}\sin\theta &e^{i \alpha}\cos\theta
\end{bmatrix}$

Implements a controlled arbitrary qubit rotation
Universal quantum gate
Continuous parameters: $\alpha,\phi,\theta$ (period $2\pi$ )

|Adriano Barenco

Berkeley B

| $B$

| $\begin{bmatrix}
\cos(\pi/8) & 0 & 0 & i \sin(\pi/8) \\
0 & \cos(3\pi/8) & i \sin(3\pi/8) & 0 \\
0 & i \sin(3\pi/8) & \cos(3\pi/8) & 0 \\
i \sin(\pi/8) & 0 & 0 & \cos(\pi/8) \\
\end{bmatrix}$

Special unitary
Exponential form:

: $\exp\left[i \frac\pi 8 (2X\otimes X + Y\otimes Y) \right]$

|University of California Berkeley{{Cite journal |last1=Zhang |first1=Jun |last2=Vala |first2=Jiri |last3=Sastry |first3=Shankar |last4=Whaley |first4=K. Birgitta |date=2004-07-07 |title=Minimum Construction of Two-Qubit Quantum Operations |url=https://link.aps.org/doi/10.1103/PhysRevLett.93.020502 |journal=Physical Review Letters |language=en |volume=93 |issue=2 |pages=020502 |doi=10.1103/PhysRevLett.93.020502 |pmid=15323888 |arxiv=quant-ph/0312193 |bibcode=2004PhRvL..93b0502Z |s2cid=9632700 |issn=0031-9007}}

Controlled-V,

controlled square root NOT

| $\mathrm{CSX},\text{controlled-}\sqrt{X},$ $\text{controlled-}V$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & e^{i \pi /4} & e^{-i \pi /4} \\
0 & 0 & e^{-i \pi /4} & e^{i \pi /4}
\end{bmatrix}$

Core entangling, canonical decomposition

| $N(a,b,c)$ , $\mathrm{can}(a,b,c)$

| $\begin{bmatrix}
e^{ic}\cos(a-b) & 0 & 0 & i e^{ic}\sin(a-b) \\
0 & e^{-ic}\cos(a+b) & i e^{-ic}\sin(a+b) & 0 \\
0 & i e^{-ic}\sin(a+b) & e^{-ic}\cos(a+b) & 0 \\
i e^{ic}\sin(a-b) & 0 & 0 & e^{ic}\cos(a-b) \\
\end{bmatrix}$

Special unitary
Universal quantum gate
Exponential form

: $\exp\left[i (a X\otimes X + b Y\otimes Y + c Z\otimes Z) \right]$

Continuous parameters: $a,b,c$ (period $2\pi$ )

Dagwood Bumstead

| $\text{DB}$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos(3\pi/8) & - i\sin(3\pi/8) & 0 \\
0 & -i\sin(3\pi/8) & \cos(3\pi/8) & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$

Special unitary
Exponential form:

: $\exp\left[-i \frac{3\pi}{16} (X\otimes X + Y\otimes Y) \right]$

|Comicbook Dagwood Bumstead

|{{Cite journal |last1=Peterson |first1=Eric C. |last2=Crooks |first2=Gavin E. |last3=Smith |first3=Robert S. |date=2020-03-26 |title=Fixed-Depth Two-Qubit Circuits and the Monodromy Polytope |url=https://quantum-journal.org/papers/q-2020-03-26-247/ |journal=Quantum |language=en-GB |volume=4 |pages=247 |doi=10.22331/q-2020-03-26-247|s2cid=214690323 |doi-access=free |arxiv=1904.10541 }}{{Cite journal |last=AbuGhanem |first=M. |date=2021-01-01 |title=Two-qubit Entangling Gate for Superconducting Quantum Computers |url=https://papers.ssrn.com/abstract=4188257 |language=en |location=Rochester, NY|doi=10.2139/ssrn.4188257 |ssrn=4188257 |s2cid=252264545 }}

Echoed cross resonance

| $\text{ECR}$

| $\frac1{\sqrt2}\begin{bmatrix}
0 & 0 & 1 & i \\
0 & 0 & i & 1 \\
1 & -i & 0 & 0 \\
-i & 1 & 0 & 0 \\
\end{bmatrix}$

Special unitary

|{{Cite journal |last1=Córcoles |first1=A. D. |last2=Magesan |first2=Easwar |last3=Srinivasan |first3=Srikanth J. |last4=Cross |first4=Andrew W. |last5=Steffen |first5=M. |last6=Gambetta |first6=Jay M. |last7=Chow |first7=Jerry M. |date=2015-04-29 |title=Demonstration of a quantum error detection code using a square lattice of four superconducting qubits |journal=Nature Communications |language=en |volume=6 |issue=1 |pages=6979 |doi=10.1038/ncomms7979 |pmid=25923200 |pmc=4421819 |arxiv=1410.6419 |bibcode=2015NatCo...6.6979C |issn=2041-1723}}

Fermionic simulation

| $U_\text{fSim}(\theta,\phi)$ , $\text{fSim}(\theta,\phi)$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos(\theta) & - i\sin(\theta) & 0 \\
0 & -i\sin(\theta) & \cos(\theta) & 0 \\
0 & 0 & 0 & e^{i\phi} \\
\end{bmatrix}$

Special unitary
Continuous parameters: $\theta,\phi$ (period $2\pi$ )

|{{Cite journal |last1=Kyriienko |first1=Oleksandr |last2=Elfving |first2=Vincent E. |date=2021-11-15 |title=Generalized quantum circuit differentiation rules |url=https://link.aps.org/doi/10.1103/PhysRevA.104.052417 |journal=Physical Review A |language=en |volume=104 |issue=5 |pages=052417 |doi=10.1103/PhysRevA.104.052417 |arxiv=2108.01218 |bibcode=2021PhRvA.104e2417K |hdl=10871/127818 |s2cid=236881494 |issn=2469-9926|hdl-access=free }}

Givens

| $G(\theta)$ , $\text{Givens}(\theta)$

| $\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos(\theta) & - \sin(\theta) & 0 \\
0 & \sin(\theta) & \cos(\theta) & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$

Special unitary
Exponential form:

: $\exp\left[-i \frac \theta 2 (Y\otimes X - X\otimes Y) \right]$

Continuous parameters: $\theta,\phi$ (period $2\pi$ )

|Givens rotations

|{{Cite journal |last1=Arrazola |first1=Juan Miguel |last2=Matteo |first2=Olivia Di |last3=Quesada |first3=Nicolás |last4=Jahangiri |first4=Soran |last5=Delgado |first5=Alain |last6=Killoran |first6=Nathan |date=2022-06-20 |title=Universal quantum circuits for quantum chemistry |url=https://quantum-journal.org/papers/q-2022-06-20-742/ |journal=Quantum |language=en-GB |volume=6 |pages=742 |doi=10.22331/q-2022-06-20-742|arxiv=2106.13839 |bibcode=2022Quant...6..742A |s2cid=235658488 |doi-access=free }}

Magic

| $\mathcal{M}$

| $\frac1{\sqrt{2}}\begin{bmatrix}
1 & i & 0 & 0 \\
0 & 0 & i & 1 \\
0 & 0 &i & -1 \\
1 & -i & 0 & 0 \\
\end{bmatrix}$

Sycamore

| $\text{syc}$ , $\text{fSim}(\pi/2,\pi/6)$

| $\begin{bmatrix} 1&0&0&0\\0&0&-i&0\\0&-i&0&0\\0&0&0&\mathrm e^{-i \pi/6}\end{bmatrix}$

|Google's Sycamore processor

|{{Cite journal |last1=Arute |first1=Frank |last2=Arya |first2=Kunal |last3=Babbush |first3=Ryan |last4=Bacon |first4=Dave |last5=Bardin |first5=Joseph C. |last6=Barends |first6=Rami |last7=Biswas |first7=Rupak |last8=Boixo |first8=Sergio |last9=Brandao |first9=Fernando G. S. L. |last10=Buell |first10=David A. |last11=Burkett |first11=Brian |last12=Chen |first12=Yu |last13=Chen |first13=Zijun |last14=Chiaro |first14=Ben |last15=Collins |first15=Roberto |date=2019 |title=Quantum supremacy using a programmable superconducting processor |journal=Nature |language=en |volume=574 |issue=7779 |pages=505–510 |doi=10.1038/s41586-019-1666-5 |pmid=31645734 |arxiv=1910.11333 |bibcode=2019Natur.574..505A |s2cid=204836822 |issn=1476-4687|doi-access=free }}

CZ-SWAP

| $\text{CZS}(\theta,\phi,\gamma)$ ,

| $\begin{bmatrix} 1&0&0&0\\0&-e^{i\gamma}\sin^2(\theta/2)+\cos^2(\theta/2)&\frac{1}{2}(1+e^{i\gamma})e^{-i\phi}\sin(\theta)&0\\0&\frac{1}{2}(1+e^{i\gamma})e^{i\phi}\sin(\theta)&-e^{i\gamma}\cos^2(\theta/2)+\sin^2(\theta/2)&0\\0&0&0&\mathrm -e^{i\gamma}\end{bmatrix}$

Continuous parameters: $\theta,\phi,\gamma$
Submatrix of a controlled-CZS (CCZS)

|{{Cite journal |last1=Gu |first1=Xiu |last2=Fernández-Pendás |first2=Jorge |last3=Vikstål |first3=Pontus |last4=Abad |first4=Tahereh |last5=Warren |first5=Christopher |last6=Bengtsson |first6=Andreas |last7=Tancredi |first7=Giovanna |last8=Shumeiko |first8=Vitaly |last9=Bylander |first9=Jonas |last10=Johansson |first10=Göran |last11=Frisk Kockum |first11=Anton |date=2021 |title=Fast Multiqubit Gates through Simultaneous Two-Qubit Gates |journal=PRX Quantum |language=en |volume=2 |issue=4|pages=040348 |doi=10.1103/PRXQuantum.2.040348|arxiv=2108.11358 |issn=2691-3399|doi-access=free |bibcode=2021PRXQ....2d0348G }}

Deutsch

| $D_\theta$ , $D(\theta)$

| $\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & i \cos\theta & \sin \theta \\
0 & 0 & 0 & 0 & 0 & 0 & \sin\theta & i \cos\theta \\
\end{bmatrix}$

Continuous parameters: $\theta,\phi$ (period $2\pi$ )
Universal quantum gate

|David Deutsch

Margolus,
simplified Toffoli

| $M$ , $\text{RCCX}$

| $\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
\end{bmatrix}$

|150px

Hermitian
Involutory
Special unitary
Functionally complete reversible gate for Boolean algebra

|Norman Margolus

|{{Cite journal |last=Maslov |first=Dmitri |date=2016-02-10 |title=Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization |journal=Physical Review A |language=en |volume=93 |issue=2 |pages=022311 |doi=10.1103/PhysRevA.93.022311 |arxiv=1508.03273 |bibcode=2016PhRvA..93b2311M |s2cid=5226873 |issn=2469-9926|doi-access=free }}{{Cite journal |last1=Song |first1=Guang |last2=Klappenecker |first2=Andreas |date=2003-12-31 |title=The simplified Toffoli gate implementation by Margolus is optimal |arxiv=quant-ph/0312225 |bibcode=2003quant.ph.12225S }}

Peres

| $\mathrm{PG}$ , $\mathrm{Peres}$

| $\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0& 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
\end{bmatrix}$

|150px

Functionally complete reversible gate for Boolean algebra

|Asher Peres

|{{Cite book |last1=Thapliyal |first1=Himanshu |last2=Ranganathan |first2=Nagarajan |title=2009 IEEE Computer Society Annual Symposium on VLSI |chapter=Design of Efficient Reversible Binary Subtractors Based on a New Reversible Gate |date=2009 |chapter-url=https://ieeexplore.ieee.org/document/5076412 |pages=229–234 |doi=10.1109/ISVLSI.2009.49|isbn=978-1-4244-4408-3 |s2cid=16182781 }}

Toffoli,
controlled-controlled NOT

| $\mathrm{CCNOT},\mathrm{CCX}, \mathrm{Toff}$

| $\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
\end{bmatrix}$

|150px

Hermitian
Involutory
Functionally complete reversible gate for Boolean algebra

|Tommaso Toffoli

Fermionic-Fredkin,

Controlled-fermionic SWAP

| $\mathrm{fFredkin}$ ,

$\mathrm{CCZS}(\pi/2,0,0)$ , $\mathrm{CfSWAP}$

{{Cite journal |last1=Warren |first1=Christopher |last2=Fernández-Pendás |first2=Jorge |last3=Ahmed |first3=Shahnawaz |last4=Abad |first4=Tahereh |last5=Bengtsson |first5=Andreas |last6=Biznárová |first6=Janka |last7=Debnath |first7=Kamanasish |last8=Gu |first8=Xiu |last9=Križan |first9=Christian |last10=Osman |first10=Amr |last11=Fadavi Roudsari |first11=Anita |last12=Delsing |first12=Per |last13=Johansson |first13=Göran |last14=Frisk Kockum |first14=Anton |last15=Tancredi |first15=Giovanna |last16=Bylander |first16=Jonas |date=2023 |title=Extensive characterization and implementation of a family of three-qubit gates at the coherence limit |journal=npj Quantum Information |language=en |volume=9 |issue=1|pages=44 |doi=10.1038/s41534-023-00711-x|arxiv=2207.02938 |issn=2056-6387|doi-access=free |bibcode=2023npjQI...9...44W }}

Notes

References

Category:Quantum computing

Category:Quantum gates