Quantum Computing

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Quantum Computing

• Lecture on Linear Algebra

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

• Review circuit fundamentals
• Learn more formalisms and different notations.
• Cover necessary math more systematically
• Show all formal rules and equations

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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

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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

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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

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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
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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
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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

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Properties of Qubits

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

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(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

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Vectors

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

Operations on vectors
This is adjoint, transpose and next conjugate
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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
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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
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Linear Algebra
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Vector Spaces
Complex number field
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Cn
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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
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Bases and Linear Independence
Linearly independent vectors
in the space
Red and blue vectors add to 0, are not linearly
independent
Always exists!
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Basis
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Bases for Cn
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So far we talked only about vectors and
operations on them. Now we introduce matrices
Linear Operators
A is linear operator
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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
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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

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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
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Linear Operators
New space
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Pauli Matrices examples
X is like inverter

• Properties Unitary
• and Hermitian

This is adjoint
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Matrices to transform between bases
Pay attention to this notation
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Examples of operators
Similar to Hadamard
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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
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Slightly other formalism for Inner Products
Be familiar with various formalisms
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Example Inner Product on Cn
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Norms
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Outer Products of vectors
This is Kronecker operation
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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
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Eigenvectors of linear operators and their
Eigenvalues
Eigenvalues of matrices are used in analysis and
synthesis
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Eigenvalues and Eigenvectors versus
diagonalizable matrices
Eigenvector of Operator A
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Diagonal Representations of matrices
From last slide
Diagonal matrix
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Adjoint Operators
This is very important, we have used it many
times already
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Normal and Hermitian Operators
But not necessarily equal identity
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Unitary Operators
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Unitary and Positive Operators some properties
and a new notation
Other notation for adjoint (Dagger is also used
Positive operator
Positive definite operator
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Hermitian Operators some properties in different
notation
These are important and useful properties of our
matrices of circuits
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Tensor Products of Vector Spaces
Notation for vectors in space V
Note various notations
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Tensor Product of two Matrices
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Tensor Products of vectors and Tensor Products of
Operators
Properties of tensor products for vectors
Tensor product for operators
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Properties of Tensor Products of vectors and
operators
These can be vectors of any size
We repeat them in different notation here
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Functions of Operators
I is the identity matrix
X is the Pauli X matrix
Remember also this
Matrix of Pauli rotation X
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For Normal Operators there is also Spectral
Decomposition
If A is represented like this
Then f(A) can be represented like this
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Trace of a matrix and a Commutator of matrices
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Review to remember
Quantum Notation
(Sometimes denoted by bold fonts)
(Sometimes called Kronecker multiplication)
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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.
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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

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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

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• Covered in 2003, 2004, 2005, 2007
• All this material is illustrated with examples in
  next lectures.