Introduction to Quantum Physics and Information Processing

126 Introduction to Quantum Physics and Information Processing

Box 7.1: Useful Representations of Single Qubit Gates

The Pauli matrices, along with the 2 × 2 identity matrix, are said to form

a basis for the space of 2 × 2 matrices. So any single qubit gate A can be

expressed as a linear combination

A = a

0

+ ~a · ~σ.

1. Since A is unitary, it can be expressed up to an overall phase as

U = u

0

+ i~u · ~σ, (7.11)

for real u

0

, u

1

, u

2

, u

3

s.t. u

2

0

+ ~u · ~u = 1.

2. This can be re-expressed as

U = e

iα

e

iβ ˆn·~σ

, (7.12)

where α is a phase, ˆn is a unit vector parallel to ~u and β is an angle,

which turns out to be half the angle of rotation of the initial state ab ...