Quantum Computing - PowerPoint PPT Presentation

About This Presentation
Title:

Quantum Computing

Description:

Linear algebra -Lecture objectives. Review basic concepts from Linear Algebra: Complex numbers ... A concept from linear algebra. ... – PowerPoint PPT presentation

Number of Views:60
Avg rating:3.0/5.0
Slides: 55
Provided by: marekpe
Learn more at: http://web.cecs.pdx.edu
Category:

less

Transcript and Presenter's Notes

Title: Quantum Computing


1
Quantum Computing
  • Lecture on Linear Algebra

Sources Angela Antoniu, Bulitko, Rezania,
Chuang, Nielsen
2
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 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

4
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

5
Review Complex numbers
  • A complex number is of the form
    where and
    i2-1
  • Polar representation
  • With the modulus or
    magnitude
  • And the phase
  • Complex conjugate

6
Review 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
  • c2 cc a2b2

The imaginaryunit
i
c
b

?
a
Real axis
Imaginaryaxis
?i
7
Review Complex Exponentiation
e?i
i
?
  • Powers of i are complex units
  • Note
  • e?i/2 i
  • e?i ?1
  • e3? i /2 ? i
  • e2? i e0 1

?1
1
?i
Z12 e ?i
Z12 (2 e ?i)2 2 2 (e ?i)2 4 (e ?i )2 4 e
2?i
2
4
8
Recall 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

9
Properties of Qubits
  • Qubits are computational basis states
  • - orthonormal basis
  • - we cannot examine a qubit to determine its
    quantum state
  • - A measurement yields

10
(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
Vectors
  • Characteristics
  • Modulus (or magnitude)
  • Orientation
  • Matrix representation of a vector

Operations on vectors
This is adjoint, transpose and next conjugate
12
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

Operations on vectors
13
Vector Space (cont)
  • Scalar multiplication for any scalar
  • Multiplication by scalars is Associative
  • distributive with respect to vector addition
  • Multiplication by vectors is
  • distributive with respect to scalar addition
  • A Vector subspace in an n-dimensional vector
    space is a non-empty subset of vectors satisfying
    the same axioms

in such way that
Operations on vectors
14
Linear Algebra
15
Vector Spaces
Complex number field
16
Cn
17
Spanning Set and Basis vectors
  • 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 (n2)

Spanning set is a set of all such vectors for any
alpha and beta
which is a linear combination of the
2-dimensional basis vectors and
18
Bases and Linear Independence
Linearly independent vectors
in the space
Red and blue vectors add to 0, are not linearly
independent
Always exists!
19
Basis
20
Bases for Cn
21
So far we talked only about vectors and
operations on them. Now we introduce matrices
Linear Operators
A is linear operator
22
Hilbert spaces
  • A Hilbert space is a vector space in which the
    scalars are complex numbers, with an inner
    product (dot product) operation ? HH ? 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

Black dot is an inner product
Componentpicture
y
Another notation often used
x
x?y/x
bracket
23
Vector Representation of States
  • Let Ss0, 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 si1
  • 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

24
Diracs Ket Notation
You have to be familiar with these three
notations
  • Note The inner productdefinition 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, so?s can be
    defined as s?, and x?y xy.

Bracket
25
Linear Operators
New space
26
Pauli Matrices examples
X is like inverter
  • Properties Unitary
  • and Hermitian

This is adjoint
27
Matrices to transform between bases
Pay attention to this notation
28
Examples of operators
Similar to Hadamard
29
This is new, we did not use inner products yet
Inner Products of vectors
We already talked about this when we defined
Hilbert space
Complex numbers
Be able to prove these properties from
definitions
30
Slightly other formalism for Inner Products
Be familiar with various formalisms
31
Example Inner Product on Cn
32
Norms
33
Outer Products of vectors
This is Kronecker operation
34
Outer Products of vectors
ugt ltv is an outer product of ugt and vgt
ugt is from U, vgt is from V. ugtltv is a map V? U
We will illustrate how this can be used formally
to create unitary and other matrices
35
Eigenvectors of linear operators and their
Eigenvalues
Eigenvalues of matrices are used in analysis and
synthesis
36
Eigenvalues and Eigenvectors versus
diagonalizable matrices
Eigenvector of Operator A
37
Diagonal Representations of matrices
From last slide
Diagonal matrix
38
Adjoint Operators
This is very important, we have used it many
times already
39
Normal and Hermitian Operators
But not necessarily equal identity
40
Unitary Operators
41
Unitary and Positive Operators some properties
and a new notation
Other notation for adjoint (Dagger is also used
Positive operator
Positive definite operator
42
Hermitian Operators some properties in different
notation
These are important and useful properties of our
matrices of circuits
43
Tensor Products of Vector Spaces
Notation for vectors in space V
Note various notations
44
Tensor Product of two Matrices
45
Tensor Products of vectors and Tensor Products of
Operators
Properties of tensor products for vectors
Tensor product for operators
46
Properties of Tensor Products of vectors and
operators
These can be vectors of any size
We repeat them in different notation here
47
Functions of Operators
I is the identity matrix
X is the Pauli X matrix
Remember also this
Matrix of Pauli rotation X
48
For Normal Operators there is also Spectral
Decomposition
If A is represented like this
Then f(A) can be represented like this
49
Trace of a matrix and a Commutator of matrices
50
Review to remember
Quantum Notation
(Sometimes denoted by bold fonts)
(Sometimes called Kronecker multiplication)
51
Exam Problems
Review systematically from basic Dirac elements
.a?
number
a?
a?
a? x
vector
?a x
number
a?
matrix
a?
?a
x
The most important new idea that we introduced in
this lecture is inner products, outer products,
eigenvectors and eigenvalues.
52
Exam 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 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/quantumcours
    ef00.htm
  • 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.
Write a Comment
User Comments (0)
About PowerShow.com