Chapter 5: Architecture - Computer and Information Science - CUNY

5.1. BITS AND QUBITS 3
A bit is either in state |0〉 or in state |1〉, which was sufficient for the classical
world. Either electricity is running through a circuit or it is not. Either a switch
is on or it is off. Either a proposition is true or it is false. But either/or is not
sufficient in the quantum world. In that world, there are situations where we
are in one state and in the other simultaneously. In the realm of the quantum
world, there are systems where a switch is both on and off. One quantum
system can be in state |0〉 and in state |1〉. Hence we are led to the definition
of a qubit:
Definition 5.1.1 A quantum bit or a qubit is a way of describing a quantum
system of dimension two.
We shall represent a qubit as a two by one matrix with complex numbers
[ ]
0 c 0
, (5.3)
1 c 1
where |c 0 | 2 + |c 1 | 2 = 1. Notice that a classical bit is a special type of qubit.
|c 0 | 2 is to be interpreted as the probability that after measuring the qubit, it
will be found in state |0〉. |c 1 | 2 is to be interpreted as the probability that
after measuring the qubit it will be found in state |1〉. Whenever we measure a
qubit, it automatically becomes a bit. So we shall never “see” a general qubit.
Nevertheless, they do exist and are the main characters in our tale. We might
visualize this “collapsing” of a qubit to a bit as
[1, 0] T
[c 0 , c 1 ] T
[0, 1]. T (5.4)
Following the normalization procedures that we learned in chapter 4 on page
???, any element of C 2 can be converted into a qubit. For example, the vector
⎡ ⎤
V = ⎣ 5 + 3i ⎦ (5.5)
6i