Linear Algebra - Quantum Computing - Lecture Slides, Slides of Computer Science

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Quantum Computing

Lecture on Linear Algebra

Goals:

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

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 Docsity.com

Complex

numbers

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 :

| c |^2 = c*c = a^2 + b^2

i ≡ - (^1) ( c ∈ C , a,b ∈ R )

c b

c a

c a b

[ ]

[ ]

Im

Re

i

“Real” axis

+ i

− i

“Imaginary”

axis

The “imaginary”

unit

a

b c

c ≡ c *^ c = ( a − bi )( a + bi ) = a^2 + b^2

Review: Complex

Exponentiation

• Powers of i are complex

units:

• Note:

e π i /2^ = i

e π i^ = − 1

e^3 π^ i^ /2^ = − i

e^2 π^ i^ = e^0 = 1

cos θ i sin θ

θi e ≡ +

• i

− 1 − i

e^ θ i

Z1=2 e πi

Z1^2 = (2 e πi)^2 = 2 2 (e πi)^2 = 4 (e πi^ )^2

= 4 e 2 πi

(^24)

Properties of Qubits

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

0 for

1 for

ij ij

i j

i j

i j

δ δ

2 0 with probability α

2 1 with probability β

2 2 where α + β = 1

Vector

Spaces

Vectors

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

[ (^) , , ] (^) (row vector)

(a column), and its dual

1

1

= = ∗^ ∗

n

n

z z

z

z

v v

v

τ

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 pertaining to a

vector space, say C n , 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

v (^) v '

' 1 1

zn z n

z z

v v 

v + v ' = v ' + v ( v + v ' ) + v '' = v +( v ' + v '' )

v v^ +^0 =^ v

v

v + (− v ) = 0

Operations

on vectors

Linear Algebra

Vector Spaces

over

Complex

Number Field

C

n