Complex Vector State
Set of Vectors
Example: set of vectors of length 4.
a typical element of it will look like:
To simply put C means a matrix of n
in it.
Operations
That is V + W ∈ C
4
n
⎡
7 + 3i
4.2 − 8.1i
⎣
−3i
Properties followed by addition operator:
commutative V
Associative (V
+ W = W + V
C
⎤
⎦
4
+
+ W ) + X = V + (W + X)
Additive Inverse, ie. V + Z = 0
⎢⎥
Notes from Quantum Computing for Computer Scientists book by Noson S. Yanofsky
and Micro A. Mannucci
Primary example of a complex vector space is set of vectors of a fixed length with
complex enteries.
These vectors describe the states of a quantum systems and quantum computers.
= C × C × C × C
⎡
⎣
⎡
⎣
here Z
6 − 4i
7 + 3i
4.2 − 8.1i
−3i
× 1
−7i
6
−4i
= −V
⎤
⎦
( 1D array )having complex numbers as
All operations that we can perform on Real vector space can be performed on
complex vector space. taking example of
Addition
Consider:
6 − 4i 16 + 2.3i
⎤
⎦
=
⎡
⎣
22 − 1.7i
7 − 4i
10.2 − 8.1i
−7i
⎤
⎦
,
as
Scalar multplication
some other Scaler multplication properties:
1.V = V
c 1 . (c 2 . V ) = (c 1
c. (V + W ) = c. V + c. W
C
m × n
× c 2 ). V
(c 1 + c 2 ). V = c 1 . V + c 2 . V
3
⎡
−2
⎣
5
3
⎤
⎦
(3 + 2i).
+ 5
3
⎡
⎣
⎦ ⎣
0
1
4
⎤
⎦
⎢⎥
⎡
⎣
4
− 4
6 + 3i
⎦ ⎣
⎡
⎣
0
5 + 1i
4
, the set of all m-by-n matrices, with complex enteries in it.
Basis and Dimension
−6
1
0
⎤
0
⎦
⎤
⎦
⎦
=
A set B = {V 0, V 1, . . . , V n − 1} ⊆ V of vectors is called a basis of a (complex) vector
space V if both
every, V
B
∈ V can be written as a liner combination of vectors from B and
is linearly independent,each of the vectors in the set {V
written as a combination of the others in the set.
Example:
we can say [45.3, −2.9, 31.1] is a liner combination of
T
⎡
⎣
−2
5
⎤ ⎡
,
0
1
⎤ ⎡
,
−6
⎤
1
+ 2.1
The dimension of a (complex) vector space is the number of elements in a basis of
the vector space.
For example, if V is a complex vector space with a basis B = {v , v , … , v }, then
the dimension of V is n.
R
C
n
n
has dimension n as a real vector space.
has dimension n as a complex vector space
⎡
⎣
, and
12 + 12i
3
⎡ ⎤
⎣ ⎦
3
1
1
⎡ ⎤
⎣ ⎦
1
0
13 + 13i
12 + 8i
1
=
⎤
⎦
⎡
⎣
0,
45.3
−2.9
31.1
V 1 , … , V n−1 }
⎤
⎦
1 2
cannot be
n
Set of Vectors
Example: set of vectors of length 4.
a typical element of it will look like:
To simply put C means a matrix of n
in it.
Operations
That is V + W ∈ C
4
n
⎡
7 + 3i
4.2 − 8.1i
⎣
−3i
Properties followed by addition operator:
commutative V
Associative (V
+ W = W + V
C
⎤
⎦
4
+
+ W ) + X = V + (W + X)
Additive Inverse, ie. V + Z = 0
⎢⎥
Notes from Quantum Computing for Computer Scientists book by Noson S. Yanofsky
and Micro A. Mannucci
Primary example of a complex vector space is set of vectors of a fixed length with
complex enteries.
These vectors describe the states of a quantum systems and quantum computers.
= C × C × C × C
⎡
⎣
⎡
⎣
here Z
6 − 4i
7 + 3i
4.2 − 8.1i
−3i
× 1
−7i
6
−4i
= −V
⎤
⎦
( 1D array )having complex numbers as
All operations that we can perform on Real vector space can be performed on
complex vector space. taking example of
Addition
Consider:
6 − 4i 16 + 2.3i
⎤
⎦
=
⎡
⎣
22 − 1.7i
7 − 4i
10.2 − 8.1i
−7i
⎤
⎦
,
as
Scalar multplication
some other Scaler multplication properties:
1.V = V
c 1 . (c 2 . V ) = (c 1
c. (V + W ) = c. V + c. W
C
m × n
× c 2 ). V
(c 1 + c 2 ). V = c 1 . V + c 2 . V
3
⎡
−2
⎣
5
3
⎤
⎦
(3 + 2i).
+ 5
3
⎡
⎣
⎦ ⎣
0
1
4
⎤
⎦
⎢⎥
⎡
⎣
4
− 4
6 + 3i
⎦ ⎣
⎡
⎣
0
5 + 1i
4
, the set of all m-by-n matrices, with complex enteries in it.
Basis and Dimension
−6
1
0
⎤
0
⎦
⎤
⎦
⎦
=
A set B = {V 0, V 1, . . . , V n − 1} ⊆ V of vectors is called a basis of a (complex) vector
space V if both
every, V
B
∈ V can be written as a liner combination of vectors from B and
is linearly independent,each of the vectors in the set {V
written as a combination of the others in the set.
Example:
we can say [45.3, −2.9, 31.1] is a liner combination of
T
⎡
⎣
−2
5
⎤ ⎡
,
0
1
⎤ ⎡
,
−6
⎤
1
+ 2.1
The dimension of a (complex) vector space is the number of elements in a basis of
the vector space.
For example, if V is a complex vector space with a basis B = {v , v , … , v }, then
the dimension of V is n.
R
C
n
n
has dimension n as a real vector space.
has dimension n as a complex vector space
⎡
⎣
, and
12 + 12i
3
⎡ ⎤
⎣ ⎦
3
1
1
⎡ ⎤
⎣ ⎦
1
0
13 + 13i
12 + 8i
1
=
⎤
⎦
⎡
⎣
0,
45.3
−2.9
31.1
V 1 , … , V n−1 }
⎤
⎦
1 2
cannot be
n
