0198506961.pdf

13.1 Qubits and their properties 283

13.1 Qubits and their properties

Fig. 13.2The two hyperfine levels in

thegroundstateofanion. Gener-

ally, experiments use an ion with to-

tal angular momentumJ=1/2inthe

lowest electronic configuration, so that

F=I± 1 /2. The qubits| 0 〉and| 1 〉

correspond to two particular Zeeman

states in the two levels, e.g.|F, M〉and

|F+1,M′〉, respectively.

A classical computer uses bits with two values 0 or 1 to represent binary
numbers, but a quantum computer stores information as quantum bits
or qubits (pronounced Q-bits). Each qubit has two states, labelled| 0 〉
and| 1 〉in the Dirac ket notion for quantum states. Most theoretical
discussions of quantum computing consider the qubit as a spin-1/2 ob-
ject,sothetwostatescorrespondtospindown|ms=− 1 / 2 〉and spin up
|ms=+1/ 2 〉. However, for a trapped ion the two states usually corre-
spond to two hyperfine levels of the ground configuration, as illustrated
in Fig. 13.2. In the following discussion,| 1 〉represents the ion in the
upper hyperfine level and| 0 〉is used for the lower hyperfine level, but
all of the arguments apply equally well to spin-1/2 particles since the
principles of quantum computing clearly do not depend on the things
used as qubits. Ions, and other physical qubits, give a compact way of
storing information, e.g.| 1101 〉represents the binary number 1101 in
Fig. 13.1, but the quantum features of this new way of encoding infor-
mation only become apparent when we consider the properties of more
than one qubit in Section 13.1.1. Even though a single qubit gener-
ally exists in a superposition of the two states, a qubit does not carry
more classical information than a classical bit, as shown by the following
argument. The superposition of the two states

ψqubit=a| 0 〉+b| 1 〉 (13.1)

obeys the normalisation condition|a|^2 +|b|^2 = 1. We write this super-
position in the general form

ψqubit=

{

cos

(

θ

2

)

| 0 〉+eiφsin

(

θ

2

)

| 1 〉

}

eiφ

′

. (13.2)

The overall phase factor has little significance and the possible states cor-
respond to vectors of unit length with direction specified by two anglesθ
andφ. These are the position vectors of points lying on the surface of a
sphere, as in Fig. 13.3. The state| 0 〉lies at the north pole of thisBloch
sphereand| 1 〉at the south pole; all other position vectors are superpo-
sitions of these two basis states. Since these position vectors correspond
to a simple classical object such as a pointer in three-dimensional space,
it follows that the information encoded by each qubit can be modelled
in a classical way.
The analogy between a qubit and a three-dimensional pointer seems to
imply that a qubit stores more information than a bit with two possible
values 0 or 1, e.g. like one of the hands of an analogue clock that gives us
information about the time by its orientation in two-dimensional space.
Generally, however, this is not true because we cannot determine the ori-
entation of a quantum object as precisely as the hands on a clock. Mea-
surements can only distinguish quantum states, with a high probability,
if the states are very different from each other, occupying well-separated
positions on opposite sides of the Bloch sphere. A measurement on a