Chapter 5: Architecture - Computer and Information Science - CUNY

5.3. REVERSIBLE GATES 21
|x〉
|x〉
•
(5.59)
|y〉
•
|y〉
|z〉
|z ⊕ (x ∧ y)〉
This is similar to the controlled-NOT gate but with two controlling bits.
The bottom bit flips only when the top two bits are in state |1〉. We can write
this operation as taking state |x, y, z〉 to |x, y, (x ∧ y) ⊕ z〉.
Exercise 5.3.2 Show that the Toffoli gate is its own inverse.
The matrix that corresponds to this gate is
000 001 010 011 100 101 110 111
⎡
⎤
000 1 0 0 0 0 0 0 0
001 0 1 0 0 0 0 0 0
010 0 0 1 0 0 0 0 0
011 0 0 0 1 0 0 0 0
100 0 0 0 0 1 0 0 0
101 0 0 0 0 0 1 0 0
⎢
⎥
110 ⎣ 0 0 0 0 0 0 0 1 ⎦
111 0 0 0 0 0 0 1 0
(5.60)
The NOT gate has no controlling bit, the controlled-not gate has one controlling
bit, and the Toffoli gate has two controlling bits. Can we go on with
this? Yes. A gate with three controlling bits can be constructed from three
Toffoli gates as follows.