CNOT and Toffoli Gates in Multi-Qubit Setting

gate

CNOT

Toffoli

number of qubits

n

2

3

4

5

first control bit

1

2

3

second control bit

1

2

3

target bit

1

2

3

view

array

matrix

The most commonly used two-qubit gate is the controlled-not (CNOT) gate. In two-qubit space, CNOT is a matrix of the form

1	0	0	0
0	1	0	0
0	0	0	1
0	0	1	0

. In general it is a

n

2

×

n

2

matrix. The most important three-qubit gate is the universal Toffoli gate, or controlled-controlled-not (CCNOT) gate with two control bits. For the Toffoli gate, both control bits are operative; for CNOT, only the first control bit.