Types of Quantum Gates

Note

Installation Required: This functionality requires MATLAB Support Package for Quantum Computing.

This topic provides a list of functions that you can use to create quantum gates in MATLAB^®. Quantum gates are the building blocks of quantum circuits, and they enable you to program algorithms for a quantum computer. Quantum gates are reversible and have unitary matrix representations.

Creation Functions for `SimpleGate` Objects

Gates on One Target Qubit

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
`hGate`	Hadamard gate	1	$\frac{1}{\sqrt{2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}]$	Traceless Involutory
`idGate`	Identity gate	1	$[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$	Identity Involutory
`xGate`	Pauli X gate	1	$[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$	Traceless Involutory Pauli group
`yGate`	Pauli Y gate	1	$[\begin{matrix} 0 & - i \\ i & 0 \end{matrix}]$	Traceless Involutory Pauli group
`zGate`	Pauli Z gate	1	$[\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$	Traceless Involutory Pauli group

Rotation Gates

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
`rxGate`	x-axis rotation gate	1	$[\begin{matrix} \cos (\frac{θ}{2}) & - i \sin (\frac{θ}{2}) \\ - i \sin (\frac{θ}{2}) & \cos (\frac{θ}{2}) \end{matrix}]$	Special unitary (determinant is 1) Continuous parameter $θ$ with period $4 π$
`ryGate`	y-axis rotation gate	1	$[\begin{matrix} \cos (\frac{θ}{2}) & - \sin (\frac{θ}{2}) \\ \sin (\frac{θ}{2}) & \cos (\frac{θ}{2}) \end{matrix}]$	Special unitary (determinant is 1) Continuous parameter $θ$ with period $4 π$
`rzGate`	z-axis rotation gate	1	$[\begin{matrix} \exp (- i \frac{θ}{2}) & 0 \\ 0 & \exp (i \frac{θ}{2}) \end{matrix}]$	Special unitary (determinant is 1) Continuous parameter $θ$ with period $4 π$
`r1Gate`	z-axis rotation gate with global phase	1	$[\begin{matrix} 1 & 0 \\ 0 & \exp (i θ) \end{matrix}]$	Continuous parameter $θ$ with period $2 π$
`sGate`	S gate	1	$[\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}]$
`siGate`	Inverse S gate	1	$[\begin{matrix} 1 & 0 \\ 0 & - i \end{matrix}]$
`tGate`	T gate	1	$[\begin{matrix} 1 & 0 \\ 0 & \frac{1 + i}{\sqrt{2}} \end{matrix}]$
`tiGate`	Inverse T gate	1	$[\begin{matrix} 1 & 0 \\ 0 & \frac{1 - i}{\sqrt{2}} \end{matrix}]$

Gates with One Control Qubit and One Target Qubit

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
`chGate`	Controlled Hadamard gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & 0 & \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{matrix}]$	Involutory
`cxGate` or `cnotGate`	Controlled X or CNOT gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}]$	Involutory
`cyGate`	Controlled Y gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - i \\ 0 & 0 & i & 0 \end{matrix}]$	Involutory
`czGate`	Controlled Z gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}]$	Involutory Swapping control and target qubits does not change gate operation

Gate That Swap States of Two Qubits

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
`swapGate`	Swap gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$	Involutory Swapping the two target qubits does not change gate operation

Controlled Rotation Gates

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
`crxGate`	Controlled x-axis rotation gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos (\frac{θ}{2}) & - i \sin (\frac{θ}{2}) \\ 0 & 0 & - i \sin (\frac{θ}{2}) & \cos (\frac{θ}{2}) \end{matrix}]$	Continuous parameter $θ$ with period $4 π$
`cryGate`	Controlled y-axis rotation gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos (\frac{θ}{2}) & - \sin (\frac{θ}{2}) \\ 0 & 0 & \sin (\frac{θ}{2}) & \cos (\frac{θ}{2}) \end{matrix}]$	Continuous parameter $θ$ with period $4 π$
`crzGate`	Controlled z-axis rotation gate	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \exp (- i \frac{θ}{2}) & 0 \\ 0 & 0 & 0 & \exp (i \frac{θ}{2}) \end{matrix}]$	Continuous parameter $θ$ with period $4 π$
`cr1Gate`	Controlled z-axis rotation gate with global phase	2	$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \exp (i θ) \end{matrix}]$	Continuous parameter $θ$ with period $2 π$ Swapping control and target qubits does not change gate operation

Controlled Controlled X Gate

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
`ccxGate`	Controlled controlled X gate (CCNOT or Toffoli gate)	3	$[\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}]$	Involutory Swapping the two control qubits does not change gate operation

Ising Coupling Gates

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
`rxxGate`	Ising XX coupling gate	2	$[\begin{matrix} \cos (\frac{θ}{2}) & 0 & 0 & - i \sin (\frac{θ}{2}) \\ 0 & \cos (\frac{θ}{2}) & - i \sin (\frac{θ}{2}) & 0 \\ 0 & - i \sin (\frac{θ}{2}) & \cos (\frac{θ}{2}) & 0 \\ - i \sin (\frac{θ}{2}) & 0 & 0 & \cos (\frac{θ}{2}) \end{matrix}]$	Special unitary (determinant is 1) Continuous parameter $θ$ with period $4 π$ Swapping the two target qubits does not change gate operation
`ryyGate`	Ising YY coupling gate	2	$[\begin{matrix} \cos (\frac{θ}{2}) & 0 & 0 & i \sin (\frac{θ}{2}) \\ 0 & \cos (\frac{θ}{2}) & - i \sin (\frac{θ}{2}) & 0 \\ 0 & - i \sin (\frac{θ}{2}) & \cos (\frac{θ}{2}) & 0 \\ i \sin (\frac{θ}{2}) & 0 & 0 & \cos (\frac{θ}{2}) \end{matrix}]$	Special unitary (determinant is 1) Continuous parameter $θ$ with period $4 π$ Swapping the two target qubits does not change gate operation
`rzzGate`	Ising ZZ coupling gate	2	$[\begin{matrix} \exp (- i \frac{θ}{2}) & 0 & 0 & 0 \\ 0 & \exp (i \frac{θ}{2}) & 0 & 0 \\ 0 & 0 & \exp (i \frac{θ}{2}) & 0 \\ 0 & 0 & 0 & \exp (- i \frac{θ}{2}) \end{matrix}]$	Special unitary (determinant is 1) Continuous parameter $θ$ with period $4 π$ Swapping the two target qubits does not change gate operation

Creation Functions for `CompositeGate` Objects

Composite and Specialized Gates

Creation Function	Gate Name	No. of Qubits	Gate Symbol	Matrix Representation
`compositeGate`	Composite gate	Varies	Example: Quantum circuit with two composite gates named `"bell"`. The equivalent internal gates of each composite gate are a Hadamard gate and a controlled X gate.
`compositeGate`	Composite gate	Varies		$\frac{1}{2} [\begin{matrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & - 1 & 0 & 1 & 0 & - 1 \\ 1 & 0 & - 1 & 0 & 1 & 0 & - 1 & 0 \\ 1 & 0 & 1 & 0 & - 1 & 0 & - 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & - 1 & 0 & - 1 \\ 0 & - 1 & 0 & 1 & 0 & 1 & 0 & - 1 \\ - 1 & 0 & 1 & 0 & 1 & 0 & - 1 & 0 \end{matrix}]$
`qftGate`	Quantum Fourier transform (QFT) gate	Varies	Example: Quantum Fourier transform gate on three qubits. The equivalent internal gates are Hadamard gates, R1 gates, and a swap gate.
`qftGate`	Quantum Fourier transform (QFT) gate	Varies		$\begin{array}{l} \frac{1}{\sqrt{8}} [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & ω & ω^{2} & ω^{3} & ω^{4} & ω^{5} & ω^{6} & ω^{7} \\ 1 & ω^{2} & ω^{4} & ω^{6} & 1 & ω^{2} & ω^{4} & ω^{6} \\ 1 & ω^{3} & ω^{6} & ω & ω^{4} & ω^{7} & ω^{2} & ω^{5} \\ 1 & ω^{4} & 1 & ω^{4} & 1 & ω^{4} & 1 & ω^{4} \\ 1 & ω^{5} & ω^{2} & ω^{7} & ω^{4} & ω & ω^{6} & ω^{3} \\ 1 & ω^{6} & ω^{4} & ω^{2} & 1 & ω^{6} & ω^{4} & ω^{2} \\ 1 & ω^{7} & ω^{6} & ω^{5} & ω^{4} & ω^{3} & ω^{2} & ω \end{matrix}] \\ where ω = \exp (\frac{2 π i}{8}) \end{array}$
`initGate`	Initialization gate	Varies	Example: Initialization gate on three target qubits. The equivalent internal gates are inverse uniform controlled y-axis rotation gates.
`initGate`	Initialization gate	Varies		`initGate` applies a matrix U such that $\| ψ 〉 = U^{*} \| 0 〉$ , where $\| ψ 〉$ is the vector representation of the input state.
`unitaryGate`	Unitary matrix gate	Varies	Example: Unitary matrix gate on three target qubits. The equivalent internal gates are four unitary matrix gates alternating between two uniform controlled z-axis rotation gates and one uniform controlled y-axis rotation gate.
`unitaryGate`	Unitary matrix gate	Varies		`unitaryGate` applies the unitary matrix to the target qubits (up to global phase).
`mcxGate`	Multi-controlled X gate	Varies	Example: Multi-controlled X gate with three control qubits, one target qubit, and no ancilla qubit. The equivalent internal gates are Hadamard gates, controlled R1 gates, and controlled X gates.
`mcxGate`	Multi-controlled X gate	Varies		$[\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ⋱ & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}]$

Uniformly Controlled Rotation Gates

Creation Function	Gate Name	No. of Qubits	Gate Symbol	Matrix Representation
`ucrxGate`	Uniformly controlled x-axis rotation gate	Varies	Example: Uniformly controlled x-axis rotation gate with one control qubit and one target qubit using rotation angle vector $[\begin{matrix} θ_{1} & θ_{2} \end{matrix}]$ . The equivalent internal gates are x-axis rotation gates and controlled Z gates.
`ucrxGate`	Uniformly controlled x-axis rotation gate	Varies		$[\begin{matrix} \cos (\frac{θ_{1}}{2}) & - i \sin (\frac{θ_{1}}{2}) & 0 & 0 \\ - i \sin (\frac{θ_{1}}{2}) & \cos (\frac{θ_{1}}{2}) & 0 & 0 \\ 0 & 0 & \cos (\frac{θ_{2}}{2}) & - i \sin (\frac{θ_{2}}{2}) \\ 0 & 0 & - i \sin (\frac{θ_{2}}{2}) & \cos (\frac{θ_{2}}{2}) \end{matrix}]$
`ucryGate`	Uniformly controlled y-axis rotation gate	Varies	Example: Uniformly controlled y-axis rotation gate with one control qubit and one target qubit using rotation angle vector $[\begin{matrix} θ_{1} & θ_{2} \end{matrix}]$ . The equivalent internal gates are y-axis rotation gates and controlled X gates.
`ucryGate`	Uniformly controlled y-axis rotation gate	Varies		$[\begin{matrix} \cos (\frac{θ_{1}}{2}) & - \sin (\frac{θ_{1}}{2}) & 0 & 0 \\ \sin (\frac{θ_{1}}{2}) & \cos (\frac{θ_{1}}{2}) & 0 & 0 \\ 0 & 0 & \cos (\frac{θ_{2}}{2}) & - \sin (\frac{θ_{2}}{2}) \\ 0 & 0 & \sin (\frac{θ_{2}}{2}) & \cos (\frac{θ_{2}}{2}) \end{matrix}]$
`ucrzGate`	Uniformly controlled z-axis rotation gate	Varies	Example: Uniformly controlled z-axis rotation gate with one control qubit and one target qubit using rotation angle vector $[\begin{matrix} θ_{1} & θ_{2} \end{matrix}]$ . The equivalent internal gates are z-axis rotation gates and controlled X gates.
`ucrzGate`	Uniformly controlled z-axis rotation gate	Varies		$[\begin{matrix} \exp (- i \frac{θ_{1}}{2}) & 0 & 0 & 0 \\ 0 & \exp (i \frac{θ_{1}}{2}) & 0 & 0 \\ 0 & 0 & \exp (- i \frac{θ_{2}}{2}) & 0 \\ 0 & 0 & 0 & \exp (i \frac{θ_{2}}{2}) \end{matrix}]$