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# CCXGate

class CCXGate(label=None, ctrl_state=None)

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CCX gate, also known as Toffoli gate.

Can be applied to a QuantumCircuit with the ccx() and toffoli() methods.

Circuit symbol:

q_0: ──■──

q_1: ──■──
┌─┴─┐
q_2: ┤ X ├
└───┘

Matrix representation:

$\begin{split}CCX q_0, q_1, q_2 = I \otimes I \otimes |0 \rangle \langle 0| + CX \otimes |1 \rangle \langle 1| = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{pmatrix}\end{split}$
Note

In Qiskit’s convention, higher qubit indices are more significant (little endian convention). In many textbooks, controlled gates are presented with the assumption of more significant qubits as control, which in our case would be q_2 and q_1. Thus a textbook matrix for this gate will be:

┌───┐
q_0: ┤ X ├
└─┬─┘
q_1: ──■──

q_2: ──■──
$\begin{split}CCX\ q_2, q_1, q_0 = |0 \rangle \langle 0| \otimes I \otimes I + |1 \rangle \langle 1| \otimes CX = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix}\end{split}$

Create new CCX gate.

## Methods Defined Here

### control

CCXGate.control(num_ctrl_qubits=1, label=None, ctrl_state=None)

Controlled version of this gate.

Parameters

• num_ctrl_qubits (int) – number of control qubits.
• label (str or None) – An optional label for the gate [Default: None]
• ctrl_state (int or str or None) – control state expressed as integer, string (e.g. ‘110’), or None. If None, use all 1s.

Returns

controlled version of this gate.

Return type

ControlledGate

### inverse

CCXGate.inverse()

Return an inverted CCX gate (also a CCX).

## Attributes

### condition_bits

Get Clbits in condition.

Return type

List[Clbit]

### ctrl_state

Return the control state of the gate as a decimal integer.

Return type

int

### decompositions

Get the decompositions of the instruction from the SessionEquivalenceLibrary.

### definition

Return definition in terms of other basic gates. If the gate has open controls, as determined from self.ctrl_state, the returned definition is conjugated with X without changing the internal _definition.

Return type

List

### duration

Get the duration.

### label

Return instruction label

Return type

str

### name

Get name of gate. If the gate has open controls the gate name will become:

<original_name_o<ctrl_state>

where <original_name> is the gate name for the default case of closed control qubits and <ctrl_state> is the integer value of the control state for the gate.

Return type

str

### num_clbits

Return the number of clbits.

### num_ctrl_qubits

Get number of control qubits.

Returns

The number of control qubits for the gate.

Return type

int

### num_qubits

Return the number of qubits.

### params

Get parameters from base_gate.

Returns

List of gate parameters.

Return type

list

Raises

CircuitError – Controlled gate does not define a base gate

### unit

Get the time unit of duration.