Unformatted text preview: Quantum Computing Quantum
Lecture on Linear Algebra
Sources: Angela Sources:
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 Mechanics Introduction
• This can be found in Marinescu and in Chuang and This Marinescu Nielsen Nielsen • 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 Lecture objectives Linear
• 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 • A complex number z ∈ C z = a + ib
n n n n iθ n n n n n Review: Complex numbers Review:
is o2 the forma ,b ∈ R f
n n where u e , where u ,θ and i =1 z= ∈R • Polar representation u = a +b
n 2 n 2 n • With • And the phase magnitude θthe modulus or bn a n =arctan n z = u ( cosθ + i sin θ • Complex conjugate conjugate
n n n n ) zn = ( an + ibn ) = an − ibn
∗ ∗ Review: The Complex Number System Review:
• Another definitions and Notations: • 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 I m [c ] ≡ b b c a • (Complex) conjugate:
c* = (a + bi)* ≡ (a − bi) − + “Real” axis • Magnitude or absolute value:
c2 = c*c = a2+b2 “Imaginary” − i axis c ≡ c*c = (a − bi )(a + bi ) = a 2 + b 2 Review: Complex Exponentiation Exponentiation
+i eθ i • Powers of i are complex units: − 1 − i θ +1 e θi ≡ cosθ + i sin θ
Z1=2 e π i Z1=2 Z12 = (2 e π i)2 = 2 2 (e πi)2 = 4 (e πi )2 = 4 e 2π i
2 4 • Note: eπi/2 = i eπ i = − 1 e3π i /2 = − i e2π i = e0 = 1 Recall: What is a qubit? Recall:
• A bit has two possible states
0 or 1 • Unlike bits, a qubit can be in a state other than 0 or 1
• We can form linear combinations of states ψ =α 0 + β 1
• A qubit state is a unit vector in a twodimensional complex vector space Properties of Qubits Properties
• Qubits are computational basis states
 orthonormal basis i j = δij 0 for i ≠ j δij = 1 for i = j  we cannot examine a qubit to determine its quantum state  A measurement yields 0 with probability α 2 1 with probability β
2 2 2 where α + β = 1 (Abstract) Vector Spaces (Abstract)
• 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 Vectors Vectors
• 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)
This is adjoint, transpose and This next conjugate next Operations on vectors Vector Space, definition: Vector
• 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' = ' zn + zn • • • • the sum, with the properties : v + v' = v' + v Commutative: 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 v + (− v ) = 0 such as v + 0 = v – Scalar multiplication: next slide Operations on vectors Scalar multiplication: for any scalar Scalar multiplication: Vector Space (cont) Vector
1v = v z ∈ C and vector v ∈ C n there is a vector zz1 z v = , the scalar product, in ssuch way at in uch way th that zz n Multiplication by scalars is Associative: Multiplication z ( z ' v ) = ( zz ') v distributive with respect to vector addition: z ( v + v' ) = z v + z v' Multiplication by vectors is Multiplication distributive with respect to scalar addition: distributive Operations on vectors ( z + z ') v = z v + z ' v A Vector subspace in an ndimensional vector ndimensional Vector
space is a nonempty subset of vectors satisfying the same
axioms Linear Algebra Linear Vector Spaces Vector
Complex number field n C Spanning Set and Basis vectors
n Or SPANNING SET for C : any set of n vectors such that Or SPANNING any Spanning set Spanning is a set of all any vector in the vector space C can be written using the n such vectors base vectors for any alpha and beta and Example for C2 (n=2): n which is a linear combination of the 2dimensional basis vectors 0 and 1 Bases and Linear Independence Independence Linearly Linearly independent vectors vectors in the space Red and blue Red vectors add to 0, are not linearly independent independent Always exists! Basis Basis n Bases for C Bases So far we talked only about vectors and operations on them. Now we introduce matrices them. Linear Operators
A is linear is operator operator • 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:
Black dot is an Black inner product inner Hilbert spaces Hilbert
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 Another within both x and vector additionnotation often used:and y x x• y ≡ x y
“bracket” “Component” picture: y x•y/x Vector Representation of States Vector
• 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 si1 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 = ∑ ci vi = (c0 , c1 , )
i x y = ∑ x * yi i
i Dirac’s Ket Notation Dirac’s You have to be familiar You with these three notations with • Note: The inner product x y = ∑ x * yi definition is the same as the “Bracket” i y1 matrix product of x, as a * * = [ x1 x2 ] y2 conjugated row vector, times y, as a normal column vector. • This leads to the definition, for state s, of:
i – 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
† † c1 c 2 Linear Operators Operators New space Pauli Matrices = examples examples Pauli
X is like inverter Properties: Unitary and Hermitian ( σ ) σ = I, ∀k
τ (σ ) = σ
τ
k k k k This is adjoint Matrices to transform between Matrices bases bases
Pay attention to this notation Examples of operators Examples Similar to Hadamard Inner Products of vectors did no This is new t use , inner we produ cts y et Complex numbers Complex We already talked about this when we defined Hilbert space Be able to prove these properties from definitions Slightly other formalism for Inner Products Products Be familiar with various formalisms various Example: Inner
n Product on C Product Norms Norms Outer Products of vectors Outer Products of vectors This is This Kronecker operation operation Outer Products of vectors Outer
u> <v is an outer u> product of u> and product v> v>
u> is from U,  v> is from V. u><v is a map u><v is V U We will illustrate how this can be used formally to create unitary and other matrices Eigenvectors of linear operators and Eigenvectors their Eigenvalues their Eigenvalues of matrices are used in analysis and synthesis Eigenvalues and Eigenvectors
versus diagonalizable matrices versus
Eigenvector of Eigenvector Operator A Operator Diagonal Representations of matrices Diagonal
From last slide Diagonal matrix Adjoint Operators Adjoint This is very This important, we have used it many Normal and Hermitian Operators
But not necessarily equal identity Unitary Operators Unitary and Positive Operators: some properties and a new notation and
Other notation for adjoint Other (Dagger is also used (Dagger Positive operator Positive definite Positive operator operator Hermitian Operators: some properties in different notation in These are important and useful properties of our matrices of circuits Tensor Products of Vector of
Spaces Spaces Notation for vectors in Notation space V space Note various notations Tensor Product of two Matrices Tensor Products of vectors and Tensor Products of Operators Properties of tensor products for vectors Tensor product Tensor for operators for Properties of Tensor Products of vectors and operators and
These can be vectors of any size We repeat them in We different notation here here Functions of Operators Operators
I is the identity matrix X is the Pauli X matrix
Matrix of Pauli rotation X Remember also this: e θi ≡ cosθ + i sin θ For Normal Operators there is also Spectral Decomposition Decomposition If A is represented like this Then f(A) can be represented like this Trace of a matrix and a Commutator of matrices Commutator Review to remember Quantum Notation Quantum
(Sometimes denoted by bold fonts) (Sometimes called Kronecker (Sometimes multiplication) multiplication) Exam Problems Exam
Review systematically from basic Review Dirac elements Dirac a〉 .a〉 a〉 x a〉 〈a x a〉 a a〉
x number vector number matrix 〈a The most important new idea that we introduced in The this lecture is inner products, outer products, eigenvectors and eigenvalues. eigenvectors Exam Problems Exam
• • • • • • • • • • • • • • 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 & acknowledgements Bibliography • 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, FiniteDimensional Vector Spaces, Springer Verlag, New York, 1974 • Covered in 2003, 2004, 2005, 2007 • All this material is illustrated with examples in next lectures. ...
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 Spring '09
 WHITE
 Linear Algebra, Vectors, Vector Space, Hilbert space

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