WOLFRAM|DEMONSTRATIONS PROJECT

Quantum Computational Basis Vectors

​
number of qubits
n
1
2
3
4
th
i
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
n
2
basis states, defined by
|
ϵ
1
ϵ
2
…ϵ
n
〉, where
ϵ
i
=0
or
1
,
0≤i≤
n
2
-1
.