Download Quantum Algorithms

Quantum Computing
Lecture 1
Michele Mosca

Course Outline
http://cacr.math.uwaterloo.ca/~mmosca/outlinef00.htm


Web page
http://cacr.math.uwaterloo.ca/~mmosca/quantumcoursef00.htm
Course Notes:
“Quantum Computation and Quantum
Information” by Nielsen and Chuang
(available at the UW Bookstore)
Overview
Chapter 1: General Introduction
 Chapter 2: Intro to Quantum Mechanics
 Chapter 3: Intro to Computer Science
 Chapter 4: More on Quantum Circuits
 Chapters 5,6: Quantum Algorithms
 Chapter 7: Physical Realizations
 Chapter 10: Quantum Error Correction
 Chapter 8: General Quantum Operations

Reading for General
Introduction
Chapter 1 of Text. Sections 1.1-1.5
 Introduction posted on web page

I will cover parts of the introduction in Lecture 1.
I will continue in Lecture 5 and subsequent lectures.
Introduction to Quantum
Mechanics
Lectures 2,3 and 4
 Parts of Chapter 2, as instructed by
Prof. Mann

General Introduction
Strong Church-Turing thesis states that a
probabilistic Turing machine (ie a classical
computer that can make fair coin flips) can
efficiently simulate any realistic model of
computing
 Therefore if we are interested in which
problems can be solved efficiently on a
realistic model of computation, we can
restrict attention to a probabilistic Turing
machine (or an equivalent model)

Physics and Computation
Information is stored in a physical medium
and manipulated by physical processes
 Therefore the laws of physics dictate the
capabilities and limitations of any
information processor
 The “classical” laws of physics are (in
macroscopic systems moving relatively
slowly) a good approximation to the laws of
physics

Physics and Computation
Realisations are getting smaller (and
faster) and reaching a point where
“classical” physics is not longer a sufficient
model for the laws of physics
Physics and Computation
However the theory of quantum physics is a
much better approximation to the laws of
physics
 The probabilistic Turing machine is implicitly
a “classical” device and it is not known in
general how to use it simulate quantum
mechanical systems [Fey82]
 A computer designed to exploit the quantum
features of Nature (a quantum computer)
seems to violate the Strong Church-Turing
thesis

Physics and Computation
Is a quantum computer realistic? Answer
seems to be YES (chapter 10)
 If the quantum computers are a reasonable
model of computation, and classical devices
cannot efficiently simulate them, then the
strong Church-Turing thesis needs to be
modified to state that a quantum Turing
machine can efficiently simulate any realistic
model of computation

Quantum Communication and
Cryptography

By exploiting the quantum mechanical
behaviour of the communication medium, we
can detect eavesdroppers (leading to
quantum cryptography, section 12.6) and
solve distributed computation tasks more
efficiently. Unfortunately, we won’t be
covering this in this course, but we will lay
the foundation for further reading in
quantum information theory.
A beam-splitter
0
0
1
1
50%
50%
The simplest explanation is that the beamsplitter acts as a classical coin-flip,
randomly sending each photon one way or
the other.
Quantum Interference
0
0
1
1
100%
The simplest explanation must be wrong,
since it would predict a 50-50 distribution.
More experimental data
0
1


0 sin  2 
2  
1 cos  
2
2

A new theory
The particle can exist in a linear combination
or superposition of the two paths
0
1
i
1
0 
1
2
2


0 sin  2 
2  
1 cos  
2
2

i
ei
0 
1
2
2
ei  1
i(ei  1)
0 
1
2
2
Probability Amplitude and
Measurement
If the photon is measured when it is in the
state 0 0  1 1
then we get 0 with
2
probability  0
0

1

0
1
0
2
1
2
2
 0  1
2
1
Quantum Operations
The operations are induced by the apparatus
linearly, that is, if 0  i 0  1 1
2
then
and
2
1
i
1 
0 
1
2
2
 i
 0 0  1 1   0 
0 
 2
i

  0
 1
2

1


1   1 
2 

1 

 0   0
2

1
i

0 
1
2
2 
1
i 
 1
1
2
2
Quantum Operations
Any linear operation that takes states
2
2
satisfying
0 0  1 1
 0  1  1
and maps them to states
satisfying
'0 0  '1 1
must be UNITARY
'
0

2
'
1

2
1
Linear Algebra
0
corresponds to
1
corresponds to
0 0  1 1 corresponds to
1
 
0
0
 
1
1
 0   0 
 0    1     
0
 1   1 
Linear Algebra
corresponds to

corresponds to






i
2
1
2
1 

2
i 

2
1 0 


i 
0 e 
Linear Algebra
0

corresponds to






i
2
1
2
1 

2
i 

2

1
0

 


i  
0 e  


i
2
1
2
1 

2
i 

2
1
 
0
Linear Algebra
u00 u01 
U

 u10 u11 
is unitary if and only if

u
u
u


00
01
t
00
UU  


u10 u11  u01*
*
u10   1 0
I
*   

u11  0 1 

*
Abstraction
The two position states of a photon in
a Mach-Zehnder apparatus is just one
example of a quantum bit or qubit
Except when addressing a particular
physical implementation, we will simply
talk about “basis” states 0 and 1
unitary operations like
H
and

where
and
H

1
2
1
2
1 

2
1 

2
corresponds to






corresponds to
1 0 


i 
0 e 
An arrangement like
0

is represented with a network like
0
H

H
More than one qubit
If we concatenate two qubits

0
0  1 1
 
0
0  1 1

we have a 2-qubit system with 4 basis states
0 0  00
0 1  01
1 0  10
1 1  11
and we can also describe the state as
00 00  01 01  10 10  11 11
or by the vector
  0 0 


  0    0    0 
           
 1 0  1  1
 
 1 1
More than one qubit
In general we can have arbitrary
superpositions
00 0 0  01 0 1  10 1 0  11 1 1
2
2
2
2
 00   01  10  11  1
where there is no factorization into the
tensor product of two independent qubits.
These states are called entangled.
Measuring multi-qubit
systems
If we measure both bits of
00 0 0  01 0 1  10 1 0  11 1 1
we get x y
with probability  xy
2