Chapter 5: Architecture - Computer and Information Science - CUNY

16 CHAPTER 5. ARCHITECTURE
First, let’s calculate the tensor product:
⎡ ⎤
0 0 0 1
⎡ ⎤ ⎡ ⎤
NOT ⊗ NOT = ⎣ 0 1 ⎦ ⊗ ⎣ 0 1
0 0 1 0
⎦ =
. (5.52)
1 0 1 0 ⎢
⎣
0 1 0 0⎥
⎦
1 0 0 0
This DeMorgan’s law corresponds to the following identity of matrices:
⎡ ⎤
0 0 0 1
⎡ ⎤ ⎡ ⎤
⎡ ⎤
⎣ 0 1 ⎦ ⋆ ⎣ 1 1 1 0
0 0 1 0
⎦ ⋆
= ⎣ 1 0 0 0 ⎦ . (5.53)
1 0 0 0 0 1 ⎢
⎣
0 1 0 0⎥
⎦
0 1 1 1
1 0 0 0
Exercise 5.2.6 Multiply out these matrices and confirm the identity.
There is one last piece of notation that must be included. Usually at the end
of a computation a measurement will be performed. A measurement of qubit(s)
shall be denoted with
(5.54)
Exercise 5.2.7 Formulate the other DeMorgan’s law ¬(¬P ∨ ¬Q) = P ∧ Q in
terms of matrices
Exercise 5.2.8 Write the matrix that would correspond to a one-bit adder. A
one-bit adder adds the bits x, y and c ( a carry-bit from an earlier adder) and
outputs the bits z and c ′ (a carry-bit for the next adder). There are three inputs
and two outputs, so the matrix will be of dimension 2 2 by 2 3 . Hint: Mark
the columns as 000, 001, 010, . . . , 110, 111 where column, say, 101 corresponds
to x = 1, y = 0, c = 1. Mark the rows as 00, 01, 10, 11 where row, say, 10,
corresponds to z = 1, c ′ = 0. When x = 1, y = 0, c = 1, the output should be
z = 0 and c ′ = 1. So place a 1 in the row marked 01 and a 0 in all other rows.