Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”)

1. Quantum ComputingQuantum Computing
Lecture on Linear Algebra
Sources:Sources: Angela
Antoniu, Bulitko, Rezania,
Chuang, Nielsen

2. Goals:Goals:
• Review circuit fundamentals
• Learn more formalisms and different notations.
• Cover necessary math more systematically
• Show all formal rules and equations

3. Introduction to Quantum MechanicsIntroduction to Quantum Mechanics
• This can be found inThis can be found in MarinescuMarinescu and inand in Chuang andChuang and
NielsenNielsen
• Objective
– To introduce all of the fundamental principles of Quantum
mechanics
• Quantum mechanics
– The most realistic known description of the world
– The basis for quantum computing and quantum information
• Why Linear Algebra?
– LA is the prerequisite for understanding Quantum Mechanics
• What is Linear Algebra?
– … is the study of vector spaces… and of
– linear operations on those vector spaces

4. Linear algebra -Linear algebra -Lecture objectivesLecture objectives
• Review basic concepts from Linear Algebra:
– Complex numbers
– Vector Spaces and Vector Subspaces
– Linear Independence and Bases Vectors
– Linear Operators
– Pauli matrices
– Inner (dot) product, outer product, tensor product
– Eigenvalues, eigenvectors, Singular Value Decomposition (SVD)
• Describe the standard notations (the Dirac notations)
adopted for these concepts in the study of Quantum
mechanics
• … which, in the next lecture, will allow us to study the
main topic of the Chapter: the postulates of quantum
mechanics

5. Review: Complex numbersReview: Complex numbers
• A complex number is of the form
where and i2
=-1
• Polar representation
• With the modulus or magnitude
• And the phase
• Complex conjugateconjugate
nnn
ibaz +=
R,ba nn
∈Czn
∈
where, R,θueuz nn
i
nn
n
∈= θ
22
nn
n bau +=




=
n
n
n
a
barctanθ
( )nnnn
iuz θθ sincos += ( ) nnnnn
ibaibaz −=+=
∗∗

6. Review: The Complex Number SystemReview: The Complex Number System
• Another definitions and Notations:
• It is the extension of the real number system via closure
under exponentiation.
• (Complex) conjugate:
c* = (a + bi)* ≡ (a − bi)
• Magnitude or absolute value:
|c|2
= c*c = a2
+b2
1-≡i )( RC ∈∈ a,b,c
bc
ac
bac
≡
≡
+=
][
][
Im
Re
i
“Real” axis
+
+i
−
−i
“Imaginary”
axis
The “imaginary”
unit
a
b c
22*
))(( bababaccc +=+−=≡ ii

7. Review: ComplexReview: Complex
ExponentiationExponentiation
• Powers of i are complex
units:
• Note:
eπi/2
= i
eπi
= −1
e3π i/2
= − i
e2π i
= e0
= 1
θθ sincos iiθ
+≡e
+1
+i
−1
−i
eθi
θ
Z1=2 eZ1=2 e πi
Z1Z122
= (2 e= (2 e πi
)2
= 2 2
(eeπi
)2
= 4= 4 (ee πi
)2
==
4 e4 e 22πi
44
22

8. Recall:Recall: What is a qubit?What is a qubit?
• A bit has two possible states
• Unlike bits, a qubit can be in a state other than
• We can form linear combinations of states
• A qubit state is a unit vector in a two-dimensional
complex vector space
0 or 1
0 or 1
0 1ψ α β= +

9. Properties of QubitsProperties of Qubits
• Qubits are computational basis states
- orthonormal basis
- we cannot examine a qubit to determine its quantum state
- A measurement yields
0 for
1 for
ij ij
i j
i j
i j
δ δ
≠
= = 
=
2
0 with probability α
2
1 with probability β
2 2
where 1α β+ =

10. (Abstract) Vector Spaces(Abstract) Vector Spaces
• A concept from linear algebra.
• A vector space, in the abstract, is any set of objects that can be
combined like vectors, i.e.:
– you can add them
• addition is associative & commutative
• identity law holds for addition to zero vector 0
– you can multiply them by scalars (incl. −1)
• associative, commutative, and distributive laws hold
• Note: There is no inherent basis (set of axes)
– the vectors themselves are the fundamental objects
– rather than being just lists of coordinates

11. VectorsVectors
• Characteristics:
– Modulus (or magnitude)
– Orientation
• Matrix representation of a vector
[ ] vector)(row,,
dualitsandcolumn),(a
1
1
∗∗
==








=
n
n
zz
z
z


vv
v
τ
This is adjoint, transpose andThis is adjoint, transpose and
next conjugatenext conjugate
Operations
on vectors

12. Vector Space, definition:Vector Space, definition:
• A vector space (of dimension n) is a set of n vectors
satisfying the following axioms (rules):
– Addition: add any two vectors and pertaining to a
vector space, say Cn
, obtain a vector,
the sum, with the properties :
• Commutative:
• Associative:
• Any has a zero vector (called the origin):
• To every in Cn
corresponds a unique vector - v such as
– Scalar multiplication:  next slide
v '
v








+
+
=+
'
'
11
'
nn zz
zz
vv
vvvv +=+ ''
( ) ( )'''''' vvvvvv ++=++
v0v =+v
v
( ) 0vv =−+
Operations
on vectors

13. Vector Space (cont)Vector Space (cont)
ScalarScalar multiplication:multiplication: foranyscalarforanyscalar
 Multiplication by scalars is Associative:Multiplication by scalars is Associative:

distributive with respect to vectoraddition:distributive with respect to vectoraddition:
 Multiplication by vectors isMultiplication by vectors is
distributive with respect to scalaraddition:distributive with respect to scalaraddition:
A VectorsubspaceA Vectorsubspace in anin an n-dimensional vectorn-dimensional vector spacespace isis
a non-empty subset of vectors satisfying the same axiomsa non-empty subset of vectors satisfying the same axioms
hatsuch way tinproduct,scalarthe,
vectoraistherevvectorand
1








=
∈∈
n
n
zz
zz
z
CCz
v
( ) ( ) vv '' zzzz =
vv =1
( ) '' vvvv zzz +=+
( ) vvv '' zzzz +=+
in such way thatin such way that
Operations
on vectors

14. Linear AlgebraLinear Algebra

15. Vector SpacesVector Spaces
Complex numberComplex number
fieldfield

16. CCnn

17. Spanning Set and Basis vectorsSpanning Set and Basis vectors
OrOr SPANNING SET forCSPANNING SET forCnn
: any set of: any set of nn vectors such thatvectors such that any
vector in the vectorspacein the vectorspace CCnn
can be written using thecan be written using the nn basebase
vectorsvectors
Example forC2
(n=2):
which is a linearcombination of the 2-dimensional basis
vectors and 0 1
Spanning setSpanning set
is a set of allis a set of all
such vectorssuch vectors
for any alphafor any alpha
and betaand beta

18. Bases and LinearBases and Linear
IndependenceIndependence
Always exists!Always exists!
in the spacein the space
Red and blueRed and blue
vectors add to 0,vectors add to 0,
are not linearlyare not linearly
independentindependent
LinearlyLinearly
independentindependent
vectorsvectors

19. BasisBasis

20. Bases for CBases for Cnn

21. So far we talked only about vectors and operations onSo far we talked only about vectors and operations on
them. Now we introduce matricesthem. Now we introduce matrices
Linear OperatorsLinear Operators
A is linearA is linear
operatoroperator

22. Hilbert spacesHilbert spaces
• A Hilbert space is a vector space in which the
scalars are complex numbers, with an inner
product (dot product) operation • : H×H → C
– Definition of inner product:
x•y = (y•x)* (* = complex conjugate)
x•x ≥ 0
x•x = 0 if and only if x = 0
x•y is linear, under scalar multiplication
and vector addition within both x and y
x
y
x•y/|x|
yxyx ≡•
Another notation often used:
“Component”
picture:
“bracket”
Black dot is anBlack dot is an
inner productinner product

23. Vector Representation of StatesVector Representation of States
• Let S={s0, s1, …} be a maximal set of
distinguishable states, indexed by i.
• The basis vector vi identified with the ith
such state
can be represented as a list of numbers:
s0 s1 s2 si-1 si si+1
vi = (0, 0, 0, …, 0, 1, 0, … )
• Arbitrary vectors v in the Hilbert space can then be
defined by linear combinations of the vi:
),,( 10 ccc
i
ii == ∑ vv
∑=
i
iyxi
*
yx

24. Dirac’s Ket NotationDirac’s Ket Notation
• Note: The inner product
definition is the same as the
matrix product of x, as a
conjugated row vector, times
y, as a normal column vector.
• This leads to the definition, for state s, of:
– The “bra” 〈s| means the row matrix [c0* c1* …]
– The “ket” |s〉 means the column matrix →
• The adjoint operator †
takes any matrix M
to its conjugate transpose M†
≡ MT
*, so
† †
∑=
i
iyxi
*
yx
[ ]










=

 2
1
*
2
*
1 y
y
xx
“Bracket”











2
1
c
c
You have to be familiarYou have to be familiar
with these three notationswith these three notations

25. LinearLinear OperatorsOperators
New spaceNew space

26. Properties: Unitary
and Hermitian
( ) kkk ∀= ,Iσσ
τ
( ) kk σσ =
τ
Pauli Matrices =Pauli Matrices = examplesexamples
X is like inverterX is like inverter
This is adjointThis is adjoint

27. Pay attention
to this
notation
Matrices to transform betweenMatrices to transform between
basesbases

28. Examples of operatorsExamples of operators
Similar to HadamardSimilar to Hadamard

29. This is new, we
did not use inner products yet
Inner Products of vectorsInner Products of vectors
We already talked
about this when we
defined Hilbert spacespace
Be able to prove these properties from definitions
Complex numbersComplex numbers

30. Slightly other formalism for InnerSlightly other formalism for Inner
ProductsProducts
Be familiar withBe familiar with
various formalismsvarious formalisms

31. Example: InnerExample: Inner
Product on CProduct on Cnn

32. NormsNorms

33. Outer Products ofOuter Products of
vectorsvectors
This is KroneckerThis is Kronecker
operationoperation

34. We will illustrate how this
can be used formally to
create unitary and other
matrices
Outer Products of vectorsOuter Products of vectors
|u> <v||u> <v| is an outeris an outer
productproduct of |u> andof |u> and
|v>|v>
|u> is from U, ||u> is from U, |
v> is from V.v> is from V.
|u><v||u><v| is a mapis a map
VV UU

35. Eigenvalues of
matrices are used in
analysis and synthesis
Eigenvectors of linear operatorsEigenvectors of linear operators andand
their Eigenvaluestheir Eigenvalues

36. Eigenvalues andEigenvalues and EigenvectorsEigenvectors
versus diagonalizable matricesversus diagonalizable matrices
Eigenvector ofEigenvector of
Operator AOperator A

37. Diagonal Representations of matricesDiagonal Representations of matrices
Diagonal matrixDiagonal matrix
From last slideFrom last slide

38. Adjoint OperatorsAdjoint Operators
This is veryThis is very
important, weimportant, we
have used it manyhave used it many

39. Normal andNormal and
Hermitian OperatorsHermitian Operators
But not necessarily
equal identity

40. Unitary OperatorsUnitary Operators

41. Unitary and Positive Operators:Unitary and Positive Operators:
some propertiessome properties and a new notationand a new notation
Other notation for adjointOther notation for adjoint
(Dagger is also used(Dagger is also used
Positive operatorPositive operator
Positive definitePositive definite
operatoroperator

42. Hermitian Operators: someHermitian Operators: some
propertiesproperties in different notationin different notation
These are important and
useful properties of our
matrices of circuits

43. Tensor ProductsTensor Products of Vectorof Vector
SpacesSpaces
Note variousNote various
notationsnotations
Notation for vectors inNotation for vectors in
space Vspace V

44. Tensor Product of twoTensor Product of two
MatricesMatrices

45. Tensor Products of vectors andTensor Products of vectors and
Tensor Products of OperatorsTensor Products of Operators
Properties of tensor
products for vectors
Tensor productTensor product
for operatorsfor operators

46. Properties of Tensor ProductsProperties of Tensor Products
of vectorsof vectors and operatorsand operators
We repeat them inWe repeat them in
different notationdifferent notation
herehere
These can be vectors of any sizeThese can be vectors of any size

47. Functions ofFunctions of
OperatorsOperators
I is the identity
matrix
X is the Pauli X
matrix
Matrix of
Pauli
rotation X
θθ sincos iiθ
+≡e
Remember also this:

48. For Normal OperatorsFor Normal Operators
there is also Spectralthere is also Spectral
DecompositionDecomposition
If A is represented like this Then f(A) can be represented like
this

49. Trace of a matrix andTrace of a matrix and aa
Commutator of matricesCommutator of matrices

50. Quantum NotationQuantum Notation
(Sometimes denoted by bold fonts)(Sometimes denoted by bold fonts)
(Sometimes called Kronecker(Sometimes called Kronecker
multiplication)multiplication)
Review to rememberReview to remember

51. Exam ProblemsExam Problems
Review systematically from basicReview systematically from basic
Dirac elementsDirac elements
.|a|a〉〉|a|a〉〉
|a|a〉〉|a|a〉〉 xx
|a|a〉〉〈〈a|a| xx
vectorvector
numbernumber
matrixmatrix
numbernumber
|a|a〉〉 〈〈a|a|xx
The most important new idea that we introduced inThe most important new idea that we introduced in
this lecture is inner products, outer products,this lecture is inner products, outer products,
eigenvectors and eigenvalues.eigenvectors and eigenvalues.

52. Exam ProblemsExam Problems
• Diagonalization of unitary matrices
• Inner and outer products
• Use of complex numbers in quantum theory
• Visualization of complex numbers and Bloch Sphere.
• Definition and Properties of Hilbert Space.
• Tensor Products of vectors and operators – properties and proofs.
• Dirac Notation – all operations and formalisms
• Functions of operators
• Trace of a matrix
• Commutator of a matrix
• Postulates of Quantum Mechanics.
• Diagonalization
• Adjoint, hermitian and normal operators
• Eigenvalues and Eigenvectors

53. Bibliography & acknowledgementsBibliography & acknowledgements
• Michael Nielsen and Isaac Chuang, Quantum
Computation and Quantum Information, Cambridge
University Press, Cambridge, UK, 2002
• R. Mann,M.Mosca, Introduction to Quantum
Computation, Lecture series, Univ. Waterloo, 2000
http://cacr.math.uwaterloo.ca/~mmosca/quantumcourse
• Paul Halmos, Finite-Dimensional Vector Spaces,
Springer Verlag, New York, 1974

54. • Covered in 2003, 2004, 2005, 2007
• All this material is illustrated with examples in
next lectures.