List of quantum gates

This is a listing of different quantum gates and their function.

Quantum gate with one input

Quantum gates to individual quantum refer

Symbol and function 1	designation	function	description
	identity	${\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$	Identity of the hyper-complex entrance and therefore no change in the quantum state
	Pauli-X-gate non-gate	${\displaystyle {\begin{pmatrix}0&1\\1&0\end{pmatrix}}=\left|1\right\rangle \left\langle 0\right|+\left|0\right\rangle \left\langle 1\right|}$	Mirroring the hyper-complex entrance on the X-axis Example: ${\displaystyle X\left|0\right\rangle =\left|1\right\rangle }$
	Pauli Y gate	${\displaystyle {\begin{pmatrix}0&-i\\i&0\end{pmatrix}}=i\left|1\right\rangle \left\langle 0\right|-i\left|0\right\rangle \left\langle 1\right|}$	Mirroring the hyper-complex entrance on the Y-axis Example: ${\displaystyle Y\left|0\right\rangle =i\left|1\right\rangle }$
	Pauli-Z-gate	${\displaystyle {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}=\left|0\right\rangle \left\langle 0\right|-\left|1\right\rangle \left\langle 1\right|}$	Mirroring the hyper-complex entrance on the Z-axis
	Hadamard Gate	${\displaystyle {\frac {1}{\sqrt {2}}}\cdot {\begin{pmatrix}1&1\\1&-1\end{pmatrix}}}$	Reflection of the hypercomplex input on the X + Z axis
	X rotation gate	${\displaystyle {\frac {1}{\sqrt {2}}}\cdot {\begin{pmatrix}1&-i\\-i&1\end{pmatrix}}}$	Rotates the complex input 90 ° (π / 2) around the X axis. Also known as a gate. ${\displaystyle {\sqrt {\mathrm {NOT} }}}$
	Y rotation gate	${\displaystyle {\frac {1}{\sqrt {2}}}\cdot {\begin{pmatrix}1&-1\\1&1\end{pmatrix}}}$	Rotates the hypercomplex input 90 ° (π / 2) around the Y axis
	(−X) rotation gate	${\displaystyle {\frac {1}{\sqrt {2}}}\cdot {\begin{pmatrix}1&i\\i&1\end{pmatrix}}}$	Rotates the complex input −90 ° (−π / 2) around the X axis
	(−Y) rotation gate	${\displaystyle {\frac {1}{\sqrt {2}}}\cdot {\begin{pmatrix}1&1\\-1&1\end{pmatrix}}}$	Rotates the hypercomplex input −90 ° (−π / 2) around the Y axis
	S-gate, phase gate	${\displaystyle {\begin{pmatrix}1&0\\0&i\end{pmatrix}}=\left|0\right\rangle \left\langle 0\right|+i\left|1\right\rangle \left\langle 1\right|}$	Rotates the phase 90 ° (π / 2) around the Z axis. Also known as a gate. ${\displaystyle {\sqrt {\mathrm {Z} }}}$
	T-gate, π / 8-gate phase (shifter) gate	${\displaystyle {\begin{pmatrix}1&0\\0&e^{\frac {i\pi }{4}}\end{pmatrix}}=\left|0\right\rangle \left\langle 0\right|+e^{\frac {i\pi }{4}}\left|1\right\rangle \left\langle 1\right|}$	Rotates the phase 45 ° (π / 4) around the Z axis. Also known as a gate. ${\displaystyle {\sqrt {\mathrm {S} }}}$
	General phase (shifter) gate 2,3 .	${\displaystyle {\begin{pmatrix}1&0\\0&e^{\frac {i\pi }{2^{k}}}\end{pmatrix}}=\left|0\right\rangle \left\langle 0\right|+e^{\frac {i\pi }{2^{k}}}\left|1\right\rangle \left\langle 1\right|}$	k is set arbitrarily Rotates the phase π / 2 k around the Z axis.
	Arbitrary unitary gate 3	${\displaystyle {\begin{pmatrix}a&b\\-b^{*}&a^{*}\end{pmatrix}}}$ With ${\displaystyle \left|a\right|^{2}+\left|b\right|^{2}=1}$	All properties are set arbitrarily
1 Using the example of three different input signals with different spins and their position after crossing the gate. The Z-axis (at the blue input) shows the real value, the X- (at the red input) and Y-axis (at the green input) the phase position. The input is marked with A, the output with A '. see also: Bloch sphere 2 output shown for the values ​​k = 0, k = 1 and k = 2 3 output depending on the parameters used

Quantum gate with two inputs

Quantum gates that relate to two quantum bits

symbol	designation	function	description
	Controlled-not gate (CNOT, XOR operation )	${\displaystyle \left|00\right\rangle \to \left|00\right\rangle }$ ${\displaystyle \left|01\right\rangle \to \left|01\right\rangle }$ ${\displaystyle \left|10\right\rangle \to \left|11\right\rangle }$ ${\displaystyle \left|11\right\rangle \to \left|10\right\rangle }$ Matrix display ${\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{pmatrix}}}$	The real value of the second qubit (B) is either retained (A = 0) or negated (A = 1) depending on the real value of the first qubit (A). ${\displaystyle B'\leftarrow A\oplus B=\left(\neg A\land B\right)\vee \left(\neg B\land A\right)}$ The value of the first qubit is retained. ${\displaystyle A'\leftarrow A\,}$
	Exchange node ("Swap")	${\displaystyle \left|00\right\rangle \to \left|00\right\rangle }$ ${\displaystyle \left|01\right\rangle \to \left|10\right\rangle }$ ${\displaystyle \left|10\right\rangle \to \left|01\right\rangle }$ ${\displaystyle \left|11\right\rangle \to \left|11\right\rangle }$ Matrix representation: ${\displaystyle {\begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}}}$	The two input qubits are swapped
	Root swap	${\displaystyle {\begin{pmatrix}1&0&0&0\\0&{\frac {1}{2}}(1+i)&{\frac {1}{2}}(1-i)&0\\0&{\frac {1}{2}}(1-i)&{\frac {1}{2}}(1+i)&0\\0&0&0&1\end{pmatrix}}}$	Universal gate that half swaps the input qubits
	Controlled Z-flip (CZ)	${\displaystyle {\left|11\right\rangle }\to -{\left|11\right\rangle }}$	Also known as controlled Z-gate, controlled phase flip (CPF) or controlled-SIGN (CSIGN)
	Controlled phase (C phase)	${\displaystyle {\left|11\right\rangle }\to e^{\frac {2i\pi }{2^{k}}}\cdot {\left|11\right\rangle }}$	${\displaystyle k}$ can be chosen arbitrarily.
	Controlled ${\displaystyle U}$	${\displaystyle \left|0\phi \right\rangle \to \left|0\phi \right\rangle }$ ${\displaystyle \left|1\phi \right\rangle \to \left|1\right\rangle {\begin{pmatrix}x_{00}&x_{10}\\x_{01}&x_{11}\end{pmatrix}}\cdot \left|\phi \right\rangle }$ Matrix representation: ${\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&x_{00}&x_{10}\\0&0&x_{01}&x_{11}\\\end{pmatrix}}}$ Dirac representation: ${\displaystyle \left|00\right\rangle \left\langle 00\right|}$+ + + + + ${\displaystyle \left|01\right\rangle \left\langle 01\right|}$ ${\displaystyle x_{00}\left|10\right\rangle \left\langle 10\right|}$ ${\displaystyle x_{01}\left|10\right\rangle \left\langle 11\right|}$ ${\displaystyle x_{10}\left|11\right\rangle \left\langle 10\right|}$ ${\displaystyle x_{11}\left|11\right\rangle \left\langle 11\right|}$	The second qubit is transformed according to the unitary mapping if the first qubit has the value "1" and remains otherwise unchanged. (C-NOT and C-phase are special cases of CU) ${\displaystyle U}$
	Any unitary transformation	${\displaystyle \left|\psi \right\rangle \to U\cdot \left|\psi \right\rangle }$	The independent variables of the complex unitary 4x4 matrix (16 real parameters) can be chosen as desired. In this way one can describe all interactions between the two qubits.

Quantum gate with three inputs

Quantum gates that relate to three quantum bits.

symbol	designation	function	description
	Toffoli Gate	${\displaystyle \left|111\right\rangle \leftrightarrow \left|110\right\rangle }$ ${\displaystyle \left|0yx\right\rangle \leftrightarrow \left|0yx\right\rangle }$ ${\displaystyle \left|10x\right\rangle \leftrightarrow \left|10x\right\rangle }$ ${\displaystyle {\begin{pmatrix}1&0\\1&1\end{pmatrix}}\cdot {\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}x\\x\oplus y\end{pmatrix}}}$ Matrix representation: ${\displaystyle {\begin{pmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&1\\0&0&0&0&0&0&1&0\end{pmatrix}}}$	The first two qubits (A and B) remain unchanged. ${\displaystyle A'\leftarrow A\,}$ ${\displaystyle B'\leftarrow B\,}$ The real value of the third qubit (C) is negated if the real value of the first two qubits is positive (i.e., logic 1). ${\displaystyle C'\leftarrow C\oplus \left(A\land B\right)\,}$ The Toffoli gate can carry out logical AND, XOR, NOT and FANOUT operations, which means that it can be used universally for classic calculations.
	Fredkin Gate	${\displaystyle \left|001\right\rangle \leftrightarrow \left|010\right\rangle }$ ${\displaystyle \left|010\right\rangle \leftrightarrow \left|001\right\rangle }$ ${\displaystyle \left|101\right\rangle \leftrightarrow \left|101\right\rangle }$ ${\displaystyle \left|110\right\rangle \leftrightarrow \left|110\right\rangle }$ ...	The Fredkin Gate swaps the second and third qubits when the real value of the first qubit is negative (i.e., logic 0).
	German gate	${\displaystyle |11x\rangle \rightarrow |11(1-x)\rangle \cdot \sin(\theta )+i\cdot |11x\rangle \cdot \cos(\theta )}$	The Deutsch gate is a universal three-qubit gate with which any interactions between the first two qubits and the third qubit can take place. The first two qubits are not changed. 1

See also

• Quantum gate
• Qubit