WOLFRAM|DEMONSTRATIONS PROJECT

Quantum Computational Basis Vectors

​
number of qubits
n
1
2
3
4
i
th
component
i
2
|10〉 = 
0
0
1
0

〈10| = (
0
0
1
0
)

A quantum computer is based on the notion of a quantum bit (or qubit). A qubit has two fundamental vector states denoted by
|0〉=

1
0

and
|1〉=
0
1

. These states represent basis vectors in the complex two-dimensional vector space

2
(equivalent to the Hilbert space

2
). A quantum computer manipulates these states by unitary matrix transformations. Two qubits are defined in the four-dimensional complex vector space

4
=
2
⊗
2
associated with four basis vectors

00〉=0〉⊗0〉=
1
0
0
0
,01〉=0〉⊗1〉=
0
1
0
0
,10〉=1〉⊗0〉=
0
0
1
0
,11〉=1〉⊗1〉=
0
0
0
1
.

More generally,
n
quantum bits have
2
n
basis states, defined by
|ϵ
1
ϵ
2
…ϵ
n
〉, where
ϵ
i
=0
or
1
,
0≤i≤2
n
-1
.