List of quantum gates

Overview

Many ideas in quantum computing rely on a small set of standard qubit states and gates. This page summarizes the one- and two-qubit basis states and the quantum gates that act on them.

The following list can also be downloaded as pdf.

One- and two-qubit basis states

One-qubit basis states (ket and vector notation)

$| 0 ⟩ = (\begin{matrix} 1 \\ 0 \end{matrix})$ , $| + ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ + | 1 ⟩) = (\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix})$

and

$| 1 ⟩ = (\begin{matrix} 0 \\ 1 \end{matrix})$ , $| - ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ - | 1 ⟩) = (\begin{matrix} \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} \end{matrix})$

Two-qubit basis states

$| 00 ⟩ = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) | 01 ⟩ = (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}) | 10 ⟩ = (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}) | 11 ⟩ = (\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix})$

Quantum gates

Identity Identity

Matrix symbol: $I$

Circuit symbol:

Matrix representation for one qubit

(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

Matrix representation for two qubits

$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

What it does

It leaves the qubit(s) unchanged

Example of the quantum gate being applied to one or two qubits

$(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 0 \end{matrix})$

$I \cdot | 0 ⟩ = | 0 ⟩$

Pauli X (NOT) Pauli X (NOT)

Matrix symbol: $X$

Circuit symbol:

Matrix representation for one qubit

$(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$

Matrix representation for two qubits

$(\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix})$

What it does

It flips the two qubit states. This gate is also called bit flip.

Example of the quantum gate being applied to one or two qubits

$(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 1 \end{matrix})$

$(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) = (\begin{matrix} 1 \\ 0 \end{matrix})$

$X \cdot | 0 ⟩ = | 1 ⟩ and X \cdot | 1 ⟩ = | 0 ⟩$

$(\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) \cdot | 00 ⟩ = | 11 ⟩$

Pauli Z Pauli Z

Matrix symbol: $Z$

Circuit symbol:

Matrix representation for one qubit

$(\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix})$

Matrix representation for two qubits

$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

What it does

Changes the sign of the $| 1 ⟩$ state resp. the $| 01 ⟩$ and $| 10 ⟩$ states; corresponds to a phase flip.

Example of the quantum gate being applied to one or two qubits

$(\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 0 \end{matrix})$

$(\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}) \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) = (\begin{matrix} 0 \\ -1 \end{matrix})$

$Z \cdot | 0 ⟩ = | 0 ⟩ and Z \cdot | 1 ⟩ = - | 1 ⟩$

Pauli Y Pauli Y

Matrix symbol: $Y$

Circuit symbol:

Matrix representation for one qubit

$(\begin{matrix} 0 & - i \\ i & 0 \end{matrix})$

What it does

Combination of $X$ and $Z$ gate – it flips the qubit state and the phase.

Example of the quantum gate being applied to one or two qubits

$(\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \end{matrix}) = i (\begin{matrix} 0 \\ 1 \end{matrix})$

$(\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) = - i (\begin{matrix} 1 \\ 0 \end{matrix})$

$Y \cdot | 0 ⟩ = i | 1 ⟩ and Y \cdot | 0 ⟩ = i | 1 ⟩$

Controlled NOT Controlled NOT

Matrix symbol: $C N O T$

Circuit symbol:

Matrix representation for one qubit

Matrix representation for two qubits

$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix})$

What it does

Flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is $| 1 ⟩$ .
It leaves the first qubit (the control qubit) unchanged.

Example of the quantum gate being applied to one or two qubits

$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) \cdot (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix})$

$C N O T \cdot | 00 ⟩ = | 00 ⟩$

$C N O T \cdot | 10 ⟩ = | 11 ⟩$

Hadamard Hadamard

Matrix symbol: $H$

Circuit symbol:

Matrix representation for one qubit

$\frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix})$

Matrix representation for two qubits

$\frac{1}{2} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & - 1 & 1 & - 1 \\ 1 & 1 & - 1 & - 1 \\ 1 & - 1 & - 1 & 1 \end{matrix})$

What it does

Creates an equal superposition state of a qubit.

Example of the quantum gate being applied to one or two qubits

$\frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}) \cdot (\begin{matrix} 1 \\ 0 \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} (\begin{matrix} 1 \\ 0 \end{matrix}) + (\begin{matrix} 0 \\ 1 \end{matrix}) \end{matrix})$

$H \cdot | 0 ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ + | 1 ⟩)$

$H \cdot | 1 ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ - | 1 ⟩)$

SWAP SWAP

Matrix symbol: $S W A P$

Circuit symbol:

Matrix representation for one qubit

Matrix representation for two qubits

$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

What it does

Swaps the states of two qubits.

Example of the quantum gate being applied to one or two qubits

$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) \cdot (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix})$

$S W A P \cdot | 10 ⟩ = | 01 ⟩$

$S W A P \cdot | 01 ⟩ = | 10 ⟩$

How to go from a 2x2-matrix (1 qubit) to a 4x4-matrix (2 qubits)

$X \otimes X = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) \otimes (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) = (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix})$

$H \otimes H = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}) \otimes \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}) = \frac{1}{2} (\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & - 1 & 1 & - 1 \\ 1 & 1 & - 1 & - 1 \\ 1 & - 1 & - 1 & 1 \end{matrix})$

$| 00 ⟩ = | 0 ⟩ \otimes | 0 ⟩ = (\begin{matrix} 1 \\ 0 \end{matrix}) \otimes (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}) | 11 ⟩ = | 1 ⟩ \otimes | 1 ⟩ = (\begin{matrix} 0 \\ 1 \end{matrix}) \otimes (\begin{matrix} 0 \\ 1 \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix})$

$| 01 ⟩ = | 0 ⟩ \otimes | 1 ⟩ = (\begin{matrix} 1 \\ 0 \end{matrix}) \otimes (\begin{matrix} 0 \\ 1 \end{matrix}) = (\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}) | 10 ⟩ = | 1 ⟩ \otimes | 0 ⟩ = (\begin{matrix} 0 \\ 1 \end{matrix}) \otimes (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix})$

References References

Images "Identity", "Pauli X (NOT)" and "Pauli Z" circuit symbol: By Geek3 - Own work, CC BY 3.0

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List of quantum gates

Overview

One- and two-qubit basis states

One-qubit basis states (ket and vector notation)

Two-qubit basis states

Quantum gates

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