Quantum Computing Lecture on Linear Algebra Sources Angela

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

Introduction to Quantum Mechanics • This can be found in Marinescu and in Chuang and 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 • 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 numbers • A complex number where • Polar representation • With is of the form and i 2=-1 the modulus or magnitude • And the phase • Complex conjugate

Review: The Complex Number System • Another definitions: It is the extension of the real number system via closure under exponentiation. The “imaginary” unit • (Complex) conjugate: c* = (a + bi)* (a bi) • Magnitude or absolute value: |c|2 = c*c = a 2+b 2 +i b c a + “Real” axis “Imaginary” i axis

Review: Complex Exponentiation e i +i • Powers of i are complex units: • Note: e i/2 = i e i = 1 e 3 i /2 = i e 2 i = e 0 = 1 1 +1 i Z 1=2 e i Z 12 = (2 e i)2 = 2 2 (e i)2 = 4 (e i )2 2 = 4 e 2 i 2 4

Recall: What is a qubit? • A qubit 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

Properties of Qubits • Qubits are computational basis states - orthonormal basis - we cannot examine a qubit to determine its quantum state - A measurement yields

(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

Vectors • Characteristics: – Modulus (or magnitude) – Orientation • Matrix representation of a vector This is adjoint, transpose and next conjugate Operations on vectors

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 vector space, say Cn, obtain a vector, pertaining to a the sum, with the properties : • • Commutative: Associative: Any has a zero vector (called the origin): To every in Cn corresponds a unique vector - v – Scalar multiplication: next slide such as Operations on vectors

Vector Space (cont) n. Scalar multiplication: for any scalar in such way that n Multiplication by scalars is Associative: n distributive with respect to vector addition: Operations on vectors l Multiplication by vectors is distributive with respect to scalar addition: l. A Vector subspace in an n-dimensional vector space is a non-empty subset of vectors satisfying the same axioms

Hilbert 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: Black dot is an inner x y = (y x)* (* = complex conjugate) product 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 x x y/|x| Another notation often used: “bracket”

Vector Representation of States • Let S={s 0, s 1, …} 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: s 0 s 1 s 2 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:

Dirac’s Ket Notation You havr to be familiar with these three notations • Note: The inner product definition is the same as the “Bracket” 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 [c 0* c 1* …] – The “ket” |s means the column matrix • The adjoint operator † takes any matrix M to its conjugate transpose M† MT*, so s| can be defined as |s †, and x y = x†y.

Linear Algebra

Vector Spaces Complex number field

n C

Spanning Set and Basis vectors n. 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 Spanning set n. Example for C 2 (n=2): is a set of all such vectors for any alpha and beta which is a linear combination of the 2 -dimensional basis vectors and

Bases and Linear Independence Linearly independent vectors in the space Red and blue vectors add to 0, are not linearly independent Always exists!

Basis

Bases for n C

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

Linear Operators New space

Pauli Matrices X is like inverter n. Properties: Unitary and Hermitian This is adjoint

Matrices Pay attention to this notation

Examples of operators

did no This is new t use , inner we produ cts Inner Products of vectors Complex numbers yet We already talked about this when we defined Hilbert space Be able to prove these properties from definitions

Slightly other formalism for Inner Products Be familiar with various formalisms

Example: Inner n Product on C

Norms

Outer Products of vectors This is Kronecker operation

Eigenvectors of linear operators and their Eigenvalues of matrices are used in analysis and synthesis

Eigenvalues and Eigenvectors versus diagonalizable matrices Eigenvector of Operator A

Diagonal Representations of matrices Diagonal matrix

Adjoint Operators This is very important, we have used it many times already

Normal and Hermitian Operators But not necessarily equal identity

Unitary Operators

Exam Problems • Diagonalization of unitary matrices

Unitary and Positive Operators: some properties Other notation for adjoint (Dagger is also used Positive operator Positive definite operator

Hermitian Operators: some properties in different notation These are important and useful properties of our matrices of circuits

Tensor Products of Vector Spaces Notation for vectors in space V Note various notations

Tensor Products of vectors and Tensor Products of Operators Properties of tensor products for vectors Tensor product for operators

Tensor Product of two Matrices

Properties of Tensor Products of vectors and operators These can be vectors of any size We repeat them in different notation here

Functions of Operators I is the identity matrix Matrix of Pauli rotation X Spectral decomposition eigenvalues

Trace and Commutator

Polar Decomposition Left polar decomposition Right polar decomposition

Eigenvalues and Eigenvectors More on Inner Products Hilbert Space: Orthogonality: Norm: Orthonormal basis:

Review to remember Quantum Notation (Sometimes denoted by bold fonts) (Sometimes called Kronecker multiplication)

Bibliography & 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/quantumcou rsef 00. htm • Paul Halmos, Finite-Dimensional Vector Spaces, Springer Verlag, New York, 1974