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	$$\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$$	Traceless Involutory
idGate	Identity gate	1	$$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$	Identity Involutory
xGate	Pauli X gate	1	$$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$	Traceless Involutory Pauli group
yGate	Pauli Y gate	1	$$\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$$	Traceless Involutory Pauli group
zGate	Pauli Z gate	1	$$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$	Traceless Involutory Pauli group

Rotation Gates

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

Gates with One Control Qubit and One Target Qubit

Creation Function	Gate Name	No. of Qubits	Matrix Representation	Properties
chGate	Controlled Hadamard gate	2	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right]$$	Involutory
cxGate or cnotGate	Controlled X or CNOT gate	2	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$$	Involutory
cyGate	Controlled Y gate	2	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -i\\ 0& 0& i& 0\end{array}\right]$$	Involutory
czGate	Controlled Z gate	2	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]$$	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	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$	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	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \cos \left(\frac{\theta }{2}\right)& -i\sin \left(\frac{\theta }{2}\right)\\ 0& 0& -i\sin \left(\frac{\theta }{2}\right)& \cos \left(\frac{\theta }{2}\right)\end{array}\right]$$	Continuous parameter $$\theta$$ with period $$\text{4}\pi$$
cryGate	Controlled y-axis rotation gate	2	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \cos \left(\frac{\theta }{2}\right)& -\sin \left(\frac{\theta }{2}\right)\\ 0& 0& \sin \left(\frac{\theta }{2}\right)& \cos \left(\frac{\theta }{2}\right)\end{array}\right]$$	Continuous parameter $$\theta$$ with period $$\text{4}\pi$$
crzGate	Controlled z-axis rotation gate	2	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \exp \left(-i\frac{\theta }{2}\right)& 0\\ 0& 0& 0& \exp \left(i\frac{\theta }{2}\right)\end{array}\right]$$	Continuous parameter $$\theta$$ with period $$\text{4}\pi$$
cr1Gate	Controlled z-axis rotation gate with global phase	2	$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \exp \left(i\theta \right)\end{array}\right]$$	Continuous parameter $$\theta$$ with period $$2\pi$$ 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	$$\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 1& 0\end{array}\right]$$	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	$$\left[\begin{array}{cccc}\cos \left(\frac{\theta }{2}\right)& 0& 0& -i\sin \left(\frac{\theta }{2}\right)\\ 0& \cos \left(\frac{\theta }{2}\right)& -i\sin \left(\frac{\theta }{2}\right)& 0\\ 0& -i\sin \left(\frac{\theta }{2}\right)& \cos \left(\frac{\theta }{2}\right)& 0\\ -i\sin \left(\frac{\theta }{2}\right)& 0& 0& \cos \left(\frac{\theta }{2}\right)\end{array}\right]$$	Special unitary (determinant is 1) Continuous parameter $$\theta$$ with period $$\text{4}\pi$$ Swapping the two target qubits does not change gate operation
ryyGate	Ising YY coupling gate	2	$$\left[\begin{array}{cccc}\cos \left(\frac{\theta }{2}\right)& 0& 0& i\sin \left(\frac{\theta }{2}\right)\\ 0& \cos \left(\frac{\theta }{2}\right)& -i\sin \left(\frac{\theta }{2}\right)& 0\\ 0& -i\sin \left(\frac{\theta }{2}\right)& \cos \left(\frac{\theta }{2}\right)& 0\\ i\sin \left(\frac{\theta }{2}\right)& 0& 0& \cos \left(\frac{\theta }{2}\right)\end{array}\right]$$	Special unitary (determinant is 1) Continuous parameter $$\theta$$ with period $$\text{4}\pi$$ Swapping the two target qubits does not change gate operation
rzzGate	Ising ZZ coupling gate	2	$$\left[\begin{array}{cccc}\exp \left(-i\frac{\theta }{2}\right)& 0& 0& 0\\ 0& \exp \left(i\frac{\theta }{2}\right)& 0& 0\\ 0& 0& \exp \left(i\frac{\theta }{2}\right)& 0\\ 0& 0& 0& \exp \left(-i\frac{\theta }{2}\right)\end{array}\right]$$	Special unitary (determinant is 1) Continuous parameter $$\theta$$ with period $$\text{4}\pi$$ 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	Equivalent Internal Gates	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.
					$$\frac{1}{2}\left[\begin{array}{cccccccc}1& 0& 1& 0& 1& 0& 1& 0\\ 0& 1& 0& 1& 0& 1& 0& 1\\ 0& 1& 0& -1& 0& 1& 0& -1\\ 1& 0& -1& 0& 1& 0& -1& 0\\ 1& 0& 1& 0& -1& 0& -1& 0\\ 0& 1& 0& 1& 0& -1& 0& -1\\ 0& -1& 0& 1& 0& 1& 0& -1\\ -1& 0& 1& 0& 1& 0& -1& 0\end{array}\right]$$
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.
					$$\begin{array}{l}\frac{1}{\sqrt{8}}\left[\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 1& \omega & {\omega }^2& {\omega }^3& {\omega }^4& {\omega }^5& {\omega }^6& {\omega }^7\\ 1& {\omega }^2& {\omega }^4& {\omega }^6& 1& {\omega }^2& {\omega }^4& {\omega }^6\\ 1& {\omega }^3& {\omega }^6& \omega & {\omega }^4& {\omega }^7& {\omega }^2& {\omega }^5\\ 1& {\omega }^4& 1& {\omega }^4& 1& {\omega }^4& 1& {\omega }^4\\ 1& {\omega }^5& {\omega }^2& {\omega }^7& {\omega }^4& \omega & {\omega }^6& {\omega }^3\\ 1& {\omega }^6& {\omega }^4& {\omega }^2& 1& {\omega }^6& {\omega }^4& {\omega }^2\\ 1& {\omega }^7& {\omega }^6& {\omega }^5& {\omega }^4& {\omega }^3& {\omega }^2& \omega \end{array}\right]\\ \text{where}\omega =\exp \left(\frac{2\pi i}{8}\right)\end{array}$$
initGate	Initialization gate	Varies	Example: Initialization gate on three target qubits. The equivalent internal gates are inverse uniform controlled y-axis rotation gates.
					initGate applies a matrix U such that $$|\psi \rangle =U^{*}|0\rangle$$, where $$|\psi \rangle$$ 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 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.
					$$\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& \ddots & 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0\end{array}\right]$$

Uniformly Controlled Rotation Gates

Creation Function	Gate Name	No. of Qubits	Gate Symbol	Equivalent Internal Gates	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 $$\left[\begin{array}{cc}{\theta }_1& {\theta }_2\end{array}\right]$$. The equivalent internal gates are x-axis rotation gates and controlled Z gates.
					$$\left[\begin{array}{cccc}\cos \left(\frac{{\theta }_1}{2}\right)& -i\sin \left(\frac{{\theta }_1}{2}\right)& 0& 0\\ -i\sin \left(\frac{{\theta }_1}{2}\right)& \cos \left(\frac{{\theta }_1}{2}\right)& 0& 0\\ 0& 0& \cos \left(\frac{{\theta }_2}{2}\right)& -i\sin \left(\frac{{\theta }_2}{2}\right)\\ 0& 0& -i\sin \left(\frac{{\theta }_2}{2}\right)& \cos \left(\frac{{\theta }_2}{2}\right)\end{array}\right]$$
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 $$\left[\begin{array}{cc}{\theta }_1& {\theta }_2\end{array}\right]$$. The equivalent internal gates are y-axis rotation gates and controlled X gates.
					$$\left[\begin{array}{cccc}\cos \left(\frac{{\theta }_1}{2}\right)& -\sin \left(\frac{{\theta }_1}{2}\right)& 0& 0\\ \sin \left(\frac{{\theta }_1}{2}\right)& \cos \left(\frac{{\theta }_1}{2}\right)& 0& 0\\ 0& 0& \cos \left(\frac{{\theta }_2}{2}\right)& -\sin \left(\frac{{\theta }_2}{2}\right)\\ 0& 0& \sin \left(\frac{{\theta }_2}{2}\right)& \cos \left(\frac{{\theta }_2}{2}\right)\end{array}\right]$$
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 $$\left[\begin{array}{cc}{\theta }_1& {\theta }_2\end{array}\right]$$. The equivalent internal gates are z-axis rotation gates and controlled X gates.
					$$\left[\begin{array}{cccc}\exp \left(-i\frac{{\theta }_1}{2}\right)& 0& 0& 0\\ 0& \exp \left(i\frac{{\theta }_1}{2}\right)& 0& 0\\ 0& 0& \exp \left(-i\frac{{\theta }_2}{2}\right)& 0\\ 0& 0& 0& \exp \left(i\frac{{\theta }_2}{2}\right)\end{array}\right]$$