Chapter 5: Architecture - Computer and Information Science - CUNY

5.2. CLASSICAL GATES 9
This matrix satisfies
We can write this as
⎡ ⎤
0
⎡ ⎤
⎡ ⎤
⎣ 1 1 1 0
0
⎦
= ⎣ 0 ⎦ . (5.29)
0 0 0 1 ⎢
⎣
0⎥
⎦
1
1
AND|11〉 = |1〉. (5.30)
In contrast, consider another 4 by 1 matrix:
⎡ ⎤
0
⎡ ⎤
⎡ ⎤
⎣ 1 1 1 0
1
⎦
= ⎣ 1 ⎦ . (5.31)
0 0 0 1 ⎢
⎣
0⎥
⎦
0
0
We can write this as
Exercise 5.2.1 Calculate AND|10〉.
AND|01〉 = |0〉. (5.32)
What would happen if we put an arbitrary 4 by 1 matrix to the right of
AND?
⎡ ⎤
3.5
⎡ ⎤
⎡ ⎤
⎣ 1 1 1 0
2
⎦
= ⎣ 5.5 ⎦ (5.33)
0 0 0 1 ⎢
⎣
0 ⎥
⎦
−4.1
−4.1
This is clearly nonsense. We are only permitted to multiply these classical gates
with vectors that represent classical states, i.e., column matrices with a single
1 entry and all the other entries must be 0. In the classical world, the bits are
only one state and are described by such vectors. Only later, when we delve
into quantum gates will we have more room (and more fun).
The OR gate
can be represented by the matrix
⎡
OR = ⎣ 1 0 0 ⎤
0 ⎦ . (5.34)
0 1 1 1