Quantum ComputingQuantum Computing

Lecture on Linear Algebra

Sources:Sources: Angela Antoniu, Bulitko, Rezania, Chuang, Nielsen

Goals:Goals:

• Review circuit fundamentals

• Learn more formalisms and different notations.

• Cover necessary math more systematically

• Show all formal rules and equations

Introduction to Quantum MechanicsIntroduction to Quantum Mechanics• This can be found in This can be found in MarinescuMarinescu and in and in Chuang and Chuang 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

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

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

nnCzn

where, R,θueuz nn

i

nn

n

22

nnn bau

n

nn a

barctan

nnnn iuz sincos nnnnn ibaibaz

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

Review: Complex Review: Complex ExponentiationExponentiation

• Powers of i are complex units:

• Note:ei/2 = iei = 1e3 i /2 = ie2 i = e0 = 1

sincos iθi e

+1

+i

1

i

ei

Z1=2 e Z1=2 e i

Z1Z12 2 = (2 e = (2 e i)2 = 2 2 (ee i)2 = 4 = 4 (e e i )2 = = 4 e4 e 2 2i

4422

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 1

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 ji j

i j

20 with probability 2

1 with probability

2 2where 1

(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

VectorsVectors• Characteristics:

– Modulus (or magnitude)– Orientation

• Matrix representation of a vector

vector)(row ,,

dual its and column), (a

1

1

n

n

zzz

z

vv

v

This is adjoint, transpose and This is adjoint, transpose and next conjugatenext conjugate

Operations on vectors

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

zzvv

vvvv ''

'''''' vvvvvv

v0v vv

0vv Operations on vectors

Vector Space (cont)Vector Space (cont)Scalar Scalar multiplication:multiplication: for any scalar for any scalar

Multiplication by scalars is Associative:Multiplication by scalars is Associative:

distributive with respect to vector addition: distributive with respect to vector addition:

Multiplication by vectors isMultiplication by vectors is

distributive with respect to scalar addition:distributive with respect to scalar addition:

A Vector subspace A Vector subspace in an in an n-dimensional n-dimensional vectorvector space space is a non-empty subset of vectors is a non-empty subset of vectors satisfying the same axioms satisfying the same axioms

hat such way tin product,scalar the,

vectora is therev vector and 1

n

n

zz

zzz

CCz

v

vv '' zzzz

vv 1

'' vvvv zzz

vvv '' zzzz

in such way thatin such way that

Operations on vectors

Linear AlgebraLinear Algebra

Vector SpacesVector SpacesComplex numberComplex number

fieldfield

CCnn

Spanning Set and Basis vectors Spanning Set and Basis vectors Or Or SPANNING SET for CSPANNING SET for Cnn: any set of : any set of nn vectors vectors such that such that any vector in the vector space in the vector space CCnn can be can be written using thewritten using the n n base vectors base vectors

Example for C2 (n=2):

which is a linear combination of the 2-dimensional basis vectors and 0 1

Spanning set Spanning set is a set of all is a set of all such vectors such vectors for any alpha for any alpha and betaand beta

Bases and Linear Bases and Linear IndependenceIndependence

Always exists!Always exists!

in the spacein the space

Red and blue Red and blue vectors add to 0, vectors add to 0, are not linearly are not linearly independentindependent

Linearly Linearly independent independent vectorsvectors

BasisBasis

Bases for CBases for Cnn

So far we talked only about vectors and operations on So far we talked only about vectors and operations on them. Now we introduce matricesthem. Now we introduce matrices

Linear OperatorsLinear Operators

A is linear A is linear

operatoroperator

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:

xy = (yx)* (* = complex conjugate)

xx 0

xx = 0 if and only if x = 0

xy is linear, under scalar multiplication and vector addition within both x and y

x

y

xy/|x|yxyx

Another notation often used:

“Component”picture:

“bracket”

Black dot is an Black dot is an

inner productinner product

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:

• And the inner product is given by: ),,( 10 ccc

iii vv

i

iyxi

*yx

Dirac’s Ket NotationDirac’s Ket Notation• Note: The inner product

definition is the same as thematrix product of x, as aconjugated row vector, timesy, 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 Mto its conjugate transpose M† MT*, sos| can be defined as |s†, and xy = x†y.

i

iyxi

*yx

2

1*

2*

1 y

y

xx

“Bracket”

2

1

c

c

You have to be familiar You have to be familiar

with these three notationswith these three notations

Linear Linear OperatorsOperators

New spaceNew space

Properties: Unitary

and Hermitian

kkk ,Iσσ

kk σσ

Pauli Matrices = Pauli Matrices = examplesexamples

X is like inverterX is like inverter

This is adjointThis is adjoint

Pay attention to this notation

Matrices to transform between Matrices to transform between basesbases

Examples of operatorsExamples of operators

Similar to HadamardSimilar to Hadamard

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 space space

Be able to prove these properties from definitions

Complex numbersComplex numbers

Slightly other formalism for Inner Slightly other formalism for Inner ProductsProducts

Be familiar withBe familiar with

various formalismsvarious formalisms

Example: Inner Example: Inner Product on CProduct on Cnn

NormsNorms

Outer Products of Outer Products of vectorsvectors

This is Kronecker This is Kronecker

operationoperation

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 outer is an outer productproduct of |u> and of |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 U U

Eigenvalues of matrices are used in analysis and synthesis

Eigenvectors of linear operators Eigenvectors of linear operators and and their Eigenvaluestheir Eigenvalues

Eigenvalues and Eigenvalues and Eigenvectors Eigenvectors versus diagonalizable matricesversus diagonalizable matrices

Eigenvector of Eigenvector of Operator AOperator A

Diagonal Representations of matricesDiagonal Representations of matrices

Diagonal matrixDiagonal matrix

From last slideFrom last slide

Adjoint OperatorsAdjoint Operators

This is very This is very

important, we important, we

have used it many have used it many

times alreadytimes already

Normal and Normal and Hermitian Operators Hermitian Operators

But not necessarily equal identity

Unitary Operators Unitary Operators

Unitary and Positive Operators: Unitary and Positive Operators: some properties some properties and a new notationand a new notation

Other notation for adjoint Other notation for adjoint (Dagger is also used(Dagger is also used

Positive operatorPositive operator

Positive definite Positive definite operatoroperator

Hermitian Operators: some Hermitian Operators: some properties properties in different notationin different notation

These are important and useful properties of our matrices of circuits

Tensor Products Tensor Products of Vectorof Vector SpacesSpaces

Note variousNote various

notationsnotations

Notation for vectors in Notation for vectors in

space Vspace V

Tensor Product of two Tensor Product of two Matrices Matrices

Tensor Products of vectors and Tensor Products of vectors and Tensor Products of Operators Tensor Products of Operators

Properties of tensor products for vectors

Tensor product Tensor product for operatorsfor operators

Properties of Tensor Products Properties of Tensor Products of vectors of vectors and operatorsand operators

We repeat them in We repeat them in different notation different notation herehere

These can be vectors of any sizeThese can be vectors of any size

Functions of Functions of OperatorsOperators

I is the identity matrix

X is the Pauli X matrix

Matrix of Pauli rotation X sincos iθi e

Remember also this:

For Normal Operators For Normal Operators there is also Spectral there is also Spectral

DecompositionDecomposition

If A is represented like this Then f(A) can be represented like this

Trace of a matrix and Trace of a matrix and a a Commutator of matricesCommutator of matrices

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

Exam ProblemsExam ProblemsReview systematically from basic Review systematically from basic

Dirac elementsDirac elements

.|a|a|a|a|a|a|a|a xx

|a|aa| a| xx

vectorvector

numbernumber

matrixmatrix

numbernumber

|a|a a|a|xx

The most important new idea that we introduced in The 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.

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

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/quantumcoursef00.htm

• Paul Halmos, Finite-Dimensional Vector Spaces, Springer Verlag, New York, 1974

• Covered in 2003, 2004, 2005, 2007

• All this material is illustrated with examples in next lectures.