CCX gate

The CCX gate, the Toffoli, flips the target qubit when both control qubits are $\vert 1 \rangle$ and leaves it unchanged otherwise. It acts on three qubits and takes no parameters. The Circuit panel draws a dot on each control wire joined to an X box on the target wire.

Kets such as $\vert 0 \rangle$ name the basis states and rotations are described on the Bloch sphere; the qubits lesson introduces both.

Call

mcx takes a list of any number of controls. IonQ backends accept up to seven; more than that fails at submission with TooManyControls, listed in the Error reference.

Matrix

The matrix uses the basis order $\vert \text{control}_1\;\text{control}_2\;\text{target} \rangle$ from $\vert 000 \rangle$ to $\vert 111 \rangle$; CCX is the identity with its last two rows swapped.

Effect on basis states

The controls are never changed in the computational basis. With the target held at $\vert 0 \rangle$, CCX writes the AND of the two controls into it, which is what makes it a universal gate for classical logic inside a quantum circuit.

Inverse

CCX is its own inverse. Two CCX gates in a row on the same qubits cancel.

Usage

from qiskit import QuantumCircuit

qc = QuantumCircuit(3, 3)
qc.x(0)
qc.x(1)
qc.ccx(0, 1, 2)
qc.measure([0, 1, 2], [0, 1, 2])

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(3, 3);
circuit.x(0);
circuit.x(1);
circuit.ccx(0, 1, 2);
circuit.measure([0, 1, 2], [0, 1, 2]);

Every shot reads 111: both controls are 1, so the target flips.

On IonQ hardware

CCX is not a native gate and costs more than a CX. The textbook decomposition uses six CX gates, against one for a CX. Qollab does not submit those six: a ccx goes to IonQ as a single X gate with two controls, the form IonQ recommends because a circuit expressed in fewer gates leaves its optimiser more room, and IonQ's compiler decides the two-qubit gates from there without publishing how many. See How your circuit is compiled.

Related

• CX gate
• X gate
• Gate reference
• Error reference