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Introduction to
Quantum Information Processing
CS 467 / CS 667
Phys 667 / Phys 767
C&O 481 / C&O 681
Lecture 1 (2005)
Richard Cleve
DC 3524
cleve@cs.uwaterloo.ca
1
Overview
2
Moore’s Law
109
number of
transistors
108
107
106
105
year
104
1975
1980
1985
1990
1995
2000
2005
Following trend … atomic scale in 15-20 years
Quantum mechanical effects occur at this scale:
• Measuring a state (e.g. position) disturbs it
• Quantum systems sometimes seem to behave
as if they are in several states at once
• Different evolutions can interfere with each other
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Quantum mechanical effects
Additional nuisances to overcome?
or
New types of behavior to make use of?
[Shor, 1994]: polynomial-time algorithm for
factoring integers on a quantum computer
This could be used to break most of the existing
public-key cryptosystems on the internet, such
as RSA
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Quantum algorithms
Classical deterministic:
START
FINISH
Classical probabilistic:
p1
p2
START
p3
FINISH
Quantum:
α1
α2
START
α3
FINISH
POSITIVE
NEGATIVE
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Also with quantum information:
• Faster algorithms for combinatorial search [Grover ’96]
• Unbreakable codes with short keys [Bennett, Brassard ’84]
• Communication savings in distributed systems
[C, Buhrman ’97]
• More efficient “proof systems” [Watrous ’99]
… and an extensive quantum
information theory arises, which
generalizes classical information
theory
For example: a theory of quantum
error-correcting codes
quantum
information
theory
classical
information
theory
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This course covers the basics of
quantum information processing
Topics include:
• Quantum algorithms and complexity theory
• Quantum information theory
• Quantum error-correcting codes*
• Physical implementations*
• Quantum cryptography
• Quantum nonlocality and communication complexity
* Jonathan Baugh
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General course information
Background:
• classical algorithms and complexity
• linear algebra
• probability theory
Evaluation:
• 3 assignments (15% each)
• midterm exam (20%)
• written project (35%)
Recommended text:
“Quantum Computation and Quantum Information”
by Nielsen and Chuang (available at the UW Bookstore)
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Basic framework
of quantum
information
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Types of information
is quantum information digital or analog?
probabilistic
digital:
1
1
analog:
1
r
0
0
1
0
p
q
• Probabilities p, q  0, p + q = 1
• Cannot explicitly extract p and q
(only statistical inference)
• In any concrete setting, explicit
state is 0 or 1
• Issue of precision (imperfect ok)
0
r  [0,1]
• Can explicitly extract r
• Issue of precision for
setting & reading state
• Precision need not be
perfect to be useful
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Quantum (digital) information
0
1
1
0
0
1


• Amplitudes ,   C, ||2 + ||2 = 1
• Explicit state is α 
β 
 
• Cannot explicitly extract  and 
(only statistical inference)
• Issue of precision (imperfect ok)
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Dirac bra/ket notation
Ket: ψ always denotes a column vector, e.g.
1
Convention: 0   
0
0
1  
1
 1 
 
 2
  
 
 d 
Bra: ψ always denotes a row vector that is the conjugate
transpose of ψ, e.g. [ *1 *2  *d ]
Bracket: φψ denotes φψ, the inner product of
φ and ψ
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Basic operations on qubits (I)
(0) Initialize qubit to |0 or to |1
†
(1) Apply a unitary operation U (U U = I )
Examples:
cos   sin 
Rotation: 

sin

cos



1
Hadamard: H 
2
1 1
1  1


0 1 
NOT (bit flip):  x  X  

1
0


1 0
Phase flip:  z  Z  

0

1


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Basic operations on qubits (II)
(3) Apply a “standard” measurement:
0 + 1

0 with prob 
2
1 with prob 
2
ψ′ 1
ψ
||2
0
||2
… and the quantum state collapses
() There exist other quantum operations, but they
can all be “simulated” by the aforementioned types
Example: measurement with respect to a different
orthonormal basis {ψ, ψ′}
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Distinguishing between two states
Let
be in state  
1
2
0
 1  or  
1
2
0
1
Question 1: can we distinguish between the two cases?
Distinguishing procedure:
1. apply H
2. measure
This works because H + = 0 and H − = 1
Question 2: can we distinguish between 0 and +?
Since they’re not orthogonal, they cannot be perfectly
distinguished …
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n-qubit systems
Probabilistic states:  p000 
p 
 001 
x, px  0
 p010 


p
 011 
px  1
 p100 
x


 p101 
p 
 110 
 p111 

Quantum states:
x, αx  C
α
x
2
x
1
α000 
α 
 001 
α010 


α
 011 
α100 


α101 
α 
 110 
α111 
Dirac notation: |000, |001, |010, …, |111 are basis vectors,
so
ψ   αx x
x
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Operations on n-qubit states
Unitary operations:
†
(U U = I )
Measurements:
α
x
x
x
 αx x

x
α 000 
α 
 001 
 
 
α111 
U   α

x
x
x 

?

000 with prob α000
2
with prob α001
2
 001

 111


2
with prob α111
… and the quantum state collapses
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Entanglement
Product state (tensor/Kronecker product):
α 0
 β 1 α' 0  β' 1   αα' 00  αβ ' 01  βα' 10  ββ' 11
Example of an entangled state:
1
2
00 
1
2
11
… can exhibit interesting “nonlocal” correlations:
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Structure among subsystems
qubits:
#1
time
U
W
#2
V
#3
#4
unitary operations
measurements
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Quantum computations
Quantum circuits:
0
1
1
0
1
1
0
0
1
1
0
1
“Feasible” if circuit-size scales polynomially
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Example of a one-qubit gate
applied to a two-qubit system
(do nothing)
u00 u01 
U 

u
u
 10 11 
U
The resulting 4x4 matrix is
Maps basis states as:
00  0U0
01  0U1
10  1U0
11  1U1
0
u00 u01 0
u

u
0
0

I  U   10 11
0
0 u00 u01 


0 u10 u11 
0
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Controlled-U gates
U
u00 u01 
U 

u
u
 10 11 
Resulting 4x4 matrix is
controlled-U =
Maps basis states as:
00  00
01  01
10  1U0
11  1U1
1
0

0

0
0 0
0
1 0
0 
0 u00 u01 

0 u10 u11 
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Controlled-NOT (CNOT)
a
a
b
ab
≡
X
Note: “control” qubit may change on some input states
0 + 1
0 − 1
0 − 1
0 − 1
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