Quantum Computing Lecture 2 Review of Linear Algebra Linear

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Linear Algebra
Quantum Computing
Lecture 2
The state space of a quantum system is described as a vector space.
Vector spaces are the object of study in Linear Algebra.
Anuj Dawar
In this lecture we review definitions from linear algebra that we
need in the rest of the course.
Review of Linear Algebra
We are mainly interested in vector spaces over the complex number
field – C.
We use the Dirac notation—|vi, |φi (read as ket) for vectors.
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Vector Spaces
A vector space over C is a set V with
• a commutative, associative addition operation + that has
– an identity 0: |vi + 0 = |vi
– inverses: |vi + (−|vi) = 0
• an operation of multiplication by a scalar α ∈ C such that:
– α(β|vi) = (αβ)|vi
– (α + β)|vi = α|vi + β|vi and α(|ui + |vi) = α|ui + α|vi
– 1|vi = |vi.
Cn


α1
 . 
n
. 
C is the vector space of n-tuples of complex numbers: 
 . .
αn

α1


β1


 .   .  
.   .  
with addition 
 . + . =
αn
βn



α1 + β 1

..

.

αn + β n


α1
zα1
 .   . 
.   . 
and scalar multiplication z 
 . = . 
αn
zαn
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Basis
Bases for Cn

A basis of a vector space V is a minimal collection of vectors
|v1 i, . . . , |vn i such that every vector |vi ∈ V can be expressed as a
linear combination of these:
|vi = α1 |v1 i + · · · + αn |vn i.
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


The standard basis for Cn is 


0
..
.
"
3
0
(written |0i, . . . , |n − 1i).
But other bases are possible:
n—the size of the basis—is uniquely determined by V and is called
the dimension of V.
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 
0
 
 
 
,
 
 
1
..
.
# "
4
,
0








,...




−i
#
0
0
..
.
1







is a basis for C2 .
We’ll be interested in orthonormal bases. That is bases of vectors
of unit length that are mutually orthogonal. Examples are |0i, |1i
and √12 (|0i + |1i), √12 (|0i − |1i).
Given a basis, every vector |vi can be represented as an n-tuple of
numbers.
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Linear Operators
A linear operator A from one vector space V to another W is a
function such that:
A(α|ui + β|vi) = α(A|ui) + β(A|vi)
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Matrices
Given a choice of bases |v1 i, . . . , |vn i and |w1 i, . . . , |wm i, let
A|vj i =
m
X
αij |wi i
i=1
Then, the matrix representation of A is given by the entries αij .
If V is of dimension n and W is of dimension m, then the operator
A can be represented as an m × n-matrix.
The matrix representation depends on the choice of bases for V
and W.
Multiplying this matrix by the representation of a vector |vi in the
basis |v1 i, . . . , |vn i gives the representation of A|vi in the basis
|w1 i, . . . , |wm i.
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Examples
Inner Products
◦
A 45 rotation of the real plane that takes
"
0
1
#
matrix
to
"
#
− √12
1
√
2
The operator 
0
i
of the real plane.
1
0
#
to
"
1
√
2
1
√
2
#
An inner product on V is an operation that associates to each pair
|ui, |vi of vectors a complex number
and
hu|vi.
is represented, in the standard basis by the
The operation satisfies


"
−i
0


√1
2
√1
2
− √12
√1
2

• hu|αv + βwi = αhu|vi + βhu|wi

• hu|vi = hv|ui∗ where the
∗
denotes the complex conjugate.
• hv|vi ≥ 0 (note: hv|vi is a real number) and hv|vi = 0 iff
|vi = 0.
 does not correspond to a transformation
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Inner Product on Cn
The standard inner product on Cn is obtained by taking, for
X
X
|ui =
ui |ii and |vi =
vi |ii
i
i
hu|vi =
X
u∗i vi
i
Note: hu| is a bra, which together with |vi forms the bra-ket hu|vi.
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Norms
The norm of a vector |vi (written || |vi||) is the non-negative, real
number:
p
|| |vi|| = hv|vi.
A unit vector is a vector with norm 1.
Two vectors |ui and |vi are orthogonal if hu|vi = 0.
An orthonormal basis for an inner product space V is a basis made
up of pairwise orthogonal, unit vectors.
the term Hilbert space is also used for an inner product space
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Outer Product
Eigenvalues
With a pair of vectors |ui ∈ U, |vi ∈ V we associate a linear
operator |uihv| : V → U, known as the outer product of |ui and |vi.
An eigenvector of a linear operator A : V → V is a non-zero vector
|vi such that
A|vi = λ|vi
(|uihv|)|v 0 i = hv|v 0 i|ui
for some complex number λ
λ is the eigenvalue corresponding to the eigenvector v.
|vihv| is the projection on the one-dimensional space generated by
|vi.
The eigenvalues of A are obtained as solutions of the characteristic
equation:
Any linear operator can be expressed as a linear combination of
outer products:
A=
X
det(A − λI) = 0
Each operator has at least one eigenvalue.
Aij |iihj|.
ij
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Diagonal Representation
A linear operator A is diagonalisable if
X
A=
λi |vi ihvi |
Adjoints
Associated with any linear operator A is its adjoint A† which
satisfies
hv|Awi = hA† v|wi
i
where the |vi i are an orthonormal set of eigenvectors of A with
corresponding eigenvalues λi .
In terms of matrices, A† = (A∗ )T
where ∗ denotes complex conjugation and T denotes transposition.
Equivalently, A can be written as a matrix


λ1




..


.


λn
in the basis |v1 i, . . . , |vn i of its eigenvectors.


1+i 1−i
−1
1
†

 =
1 − i −1
1+i
1


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Normal and Hermitian Operators
Unitary Operators
An operator A is said to be normal if
A linear operator A is unitary if
AA† = A† A
AA† = A† A = I
Fact: An operator is diagonalisable if, and only if, it is normal.
Unitary operators are normal and therefore diagonalisable.
A is said to be Hermitian if A = A†
Unitary operators are norm-preserving and invertible.
A normal operator is Hermitian if, and only if, it has real
eigenvalues.
hAu|Avi = hu|vi
All eigenvalues of a unitary operator have modulus 1.
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Tensor Products
If U is a vector space of dimension m and V one of dimension n
then U ⊗ V is a space of dimension mn.
Writing |uvi for the vectors in U ⊗ V:
0
0
• |(u + u )vi = |uvi + |u vi
• |u(v + v 0 )i = |uvi + |uv 0 i
• z|uvi = |(zu)vi = |u(zv)i
Given linear operators A : U → U and B : V → V, we can define
an operator A ⊗ B on U ⊗ V by
(A ⊗ B)|uvi = |(Au), (Bv)i
Tensor Products
In matrix terms,


A11 B
A12 B
···


 A21 B
A⊗B =
..


.

Am1 B
A22 B
..
.


A2m B 




· · · Amm B
Am2 B
···
..
.
A1m B