QuantumQuantum ComputingComputing

Lecture on Linear Algebra

Sources:Sources: Angela Antoniu, Bulitko, Rezania, Chuang, Nielsen IntroductionIntroduction toto QuantumQuantum MechanicsMechanics

•• ReviewReview Chapters 1 and 2 fromfrom ChuangChuang andand 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 LinearLinear algebraalgebra --LectureLecture objectivesobjectives

• 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:Review: TheThe ComplexComplex NumberNumber SystemSystem

• It is the extension of the real number system via closure under exponentiation. i ≡ -1 c = a + bi (c ∈C, a,b ∈R) The “imaginary” Re [c] ≡ a +i unit Im [c] ≡ b b c • (Complex) conjugate: − a + c* = (a + bi)* ≡ (a − bi) “Real” axis “Imaginary” • Magnitude or absolute value: −i axis |c|2 = c*c = a2+b2 c ≡ c*c = (a − bi )(a + bi ) = a2 + b2 Review:Review: ComplexComplex ExponentiationExponentiation

+i eθi θ • Powers of i areθ complex units: −1 +1 θi e ≡ cos + i sinθ −i

•Note: eπi/2 = i eπi = −1 e3π i /2 = − i e2π i = e0 = 1 Recall:Recall: WhatWhat isis aa qubitqubit??

• A qubit has two possible states 0or1 • Unlike bits, a quibit can be in a state other than 0or1 • We can form linear combinations of states ψ =+α 01β • A quibit state is a unit vector in a two dimensional complex vector space PropertiesProperties ofof QubitsQubits • Qubits are computational basis states - orthonormal basis

0 for ij≠ ij==δδ  ij ij 1 for ij=

- we cannot examine a qubit to determine its quantum state - A measurement yields 0 with probability α 2 1 with probability β 2

where αβ22+=1 ComplexComplex numbersnumbers ∈ • A complex number z n C is of the form a ,b ∈ R n n = + 2 where z n a n ib n and i =-1 • Polar representation iθ = n ∈ zn une , where un ,θn R = 2 + 2 • With u n a b the modulus or magnitude n n

• And the phase θ =  b n  n arctan    a n 

• Complex conjugateconjugate

= ()θ + θ ∗ = ()+ ∗ = − zn un cos n isin n zn an ibn an ibn (Abstract)(Abstract) VectorVector SpacesSpaces

• 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 •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 “Component” x•x = 0 if and only if x = 0 picture: x•y is linear, under scalar multiplication y and vector addition within both x and y Another notation often used: x x • y ≡ x y x•y/|x| “bracket” VectorVector RepresentationRepresentation ofof StatesStates

•Let S={s0, s1, …} be a maximal set of distinguishable states, indexed by i. th • The basis vector vi identified with the i 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: = =  v ∑civi (c0 ,c1, ) i x y = ∑ x* y • And the inner product is given by: i i i Dirac’sDirac’s KetKet NotationNotation

•Note: The inner product x y = ∑ x* y i i definition is the same as the “Bracket” i  y  matrix product of x, as a 1 = []* * o   conjugated row vector, times x1 x2 y2  y, as a normal column vector.  p  • This leads to the definition, for state s, of: –The “bra” 〈s| means the row matrix [c * c * …] 0 1 c  –The “ket” |s〉 means the column matrix →  1  c • The adjoint operator † takes any matrix M  2     to its conjugate transpose M† ≡ MT*, so   〈s| can be defined as |s〉†, and x•y = x†y. VectorsVectors • Characteristics: – Modulus (or magnitude) – Orientation • Matrix representation of a vector

z1  v =    (a column), and its dual   zn  τ = = []∗  ∗ v v z1 , , zn (row vector) VectorVector Space,Space, definition:definition: • A vector space (of dimension n) is a set of n vectors satisfying the following axioms (rules): – Addition: add any two vectors v and v ' pertaining to a vector space, say Cn, obtain a vector,  + '  z1 z1 v + v' =    the sum, with the properties :  + '  zn zn  • Commutative: v + v' = v' + v • Associative: ()v + v' + v'' = v + ()v' + v'' • Any v has a zero vector (called the origin): + = • To every v in Cn corresponds a unique vector - v such as v 0 v v + ()− v = 0 –Scalar multiplication:
next slide VectorVector SpaceSpace ((contcont))
Scalar multiplication: for any scalar z ∈ C and vector v ∈ C n there is a vector

 zz1  z v =   , the scalar product, inin such such way way t hat that 1 v = v   zz n 

Multiplication by scalars is Associative:
z()z' v = ()zz' v distributive with respect to vector addition: z()v + v' = z v + z v'  Multiplication by vectors is distributive with respect to scalar addition: ()z + z' v = z v + z' v

AA VectorVector subspacesubspace in an n-dimensional vector spacespace is a non-empty subset of vectors satisfying the same axioms BasisBasis vectorsvectors
Or SPANNING SET for Cn: any set of n vectors such that any vector in the vector space Cn can be written using the n base vectors
Example for C2 (n=2):

which is a linear combination of the 2 dimensional basis vectors 0 and 1 BasesBases andand LinearLinear IndependenceIndependence

AlwaysAlways exists!exists! (Sometimes denoted by bold fonts)

(Sometimes called Kronecker multiplication)

X is like inverter

τ
Properties: Unitary ()= ∀ σ k σ k I, k ()τ = and Hermitian σ k σ k

EigenvaluesEigenvalues andand EigenvectorsEigenvectors

MoreMore onon InnerInner ProductsProducts Hilbert Space: Orthogonality: Norm:

Orthonormal basis:

EigenvaluesEigenvalues andand EigenvectorsEigenvectors HermitianHermitian OperatorsOperators UnitaryUnitary andand PositivePositive OperatorsOperators

MoreMore onon TensorTensor ProductsProducts FunctionsFunctions ofof OperatorsOperators TraceTrace andand CommutatorCommutator

BibliographyBibliography && acknowledgementsacknowledgements

• 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/quantumcou rsef00.htm

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