Title: Quantum Computing
1Quantum Computing
- Lecture on Linear Algebra
-
Sources Angela Antoniu, Bulitko, Rezania,
Chuang, Nielsen
2Goals
- Review circuit fundamentals
- Learn more formalisms and different notations.
- Cover necessary math more systematically
- Show all formal rules and equations
3Introduction 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
4Linear 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
5Review 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
6Review 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
7Review 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
8Recall 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
9Properties 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
11Vectors
- Characteristics
- Modulus (or magnitude)
- Orientation
- Matrix representation of a vector
-
Operations on vectors
This is adjoint, transpose and next conjugate
12Vector 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
13Vector 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
14Linear Algebra
15Vector Spaces
Complex number field
16Cn
17Spanning 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
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
18Bases and Linear Independence
Linearly independent vectors
in the space
Red and blue vectors add to 0, are not linearly
independent
Always exists!
19Basis
20Bases for Cn
21So far we talked only about vectors and
operations on them. Now we introduce matrices
Linear Operators
A is linear operator
22Hilbert 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
23Vector 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
24Diracs 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
25Linear Operators
New space
26Pauli Matrices examples
X is like inverter
- Properties Unitary
- and Hermitian
This is adjoint
27Matrices to transform between bases
Pay attention to this notation
28Examples of operators
Similar to Hadamard
29This 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
30Slightly other formalism for Inner Products
Be familiar with various formalisms
31Example Inner Product on Cn
32Norms
33Outer Products of vectors
This is Kronecker operation
34Outer 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
35Eigenvectors of linear operators and their
Eigenvalues
Eigenvalues of matrices are used in analysis and
synthesis
36Eigenvalues and Eigenvectors versus
diagonalizable matrices
Eigenvector of Operator A
37Diagonal Representations of matrices
From last slide
Diagonal matrix
38Adjoint Operators
This is very important, we have used it many
times already
39Normal and Hermitian Operators
But not necessarily equal identity
40Unitary Operators
41Unitary and Positive Operators some properties
and a new notation
Other notation for adjoint (Dagger is also used
Positive operator
Positive definite operator
42Hermitian Operators some properties in different
notation
These are important and useful properties of our
matrices of circuits
43Tensor Products of Vector Spaces
Notation for vectors in space V
Note various notations
44Tensor Product of two Matrices
45Tensor Products of vectors and Tensor Products of
Operators
Properties of tensor products for vectors
Tensor product for operators
46Properties of Tensor Products of vectors and
operators
These can be vectors of any size
We repeat them in different notation here
47Functions of Operators
I is the identity matrix
X is the Pauli X matrix
Remember also this
Matrix of Pauli rotation X
48For Normal Operators there is also Spectral
Decomposition
If A is represented like this
Then f(A) can be represented like this
49Trace of a matrix and a Commutator of matrices
50Review to remember
Quantum Notation
(Sometimes denoted by bold fonts)
(Sometimes called Kronecker multiplication)
51Exam 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.
52Exam 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
53Bibliography 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.