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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 (2008)
Richard Cleve
DC 2117
[email protected]
1
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
2
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
3
Quantum algorithms
Classical deterministic:
START
FINISH
Classical probabilistic:
p1
p2
START
p3
FINISH
Quantum:
α1
α2
START
α3
FINISH
POSITIVE
NEGATIVE
4
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
5
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
6
General course information
Background:
• classical algorithms and complexity
• linear algebra
• probability theory
Evaluation:
• 5 assignments (12% each)
• project presentation (40%)
Recommended texts:
An Introduction to Quantum Computation, P. Kaye, R. Laflamme, M.
Mosca (Oxford University Press, 2007). Primary reference.
Quantum Computation and Quantum Information, Michael A. Nielsen
and Isaac L. Chuang (Cambridge University Press, 2000). Secondary
reference.
7
Basic framework
of quantum
information
8
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
9
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)
10
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
11
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


12
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′}
13
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 …
14
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
15
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
16
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:
17
Structure among subsystems
qubits:
#1
time
U
W
#2
V
#3
#4
unitary operations
measurements
18
Quantum computations
Quantum circuits:
0
1
1
0
1
1
0
0
1
1
0
1
“Feasible” if circuit-size scales polynomially
19
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
20
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 
21
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
22
Introduction to
Quantum Information Processing
CS 467 / CS 667
Phys 667 / Phys 767
C&O 481 / C&O 681
Lecture 2 (2008)
Richard Cleve
DC 2117
[email protected]
23
Superdense
coding
24
How much classical information in n qubits?
2n1 complex numbers apparently needed to describe
an arbitrary n-qubit pure quantum state:
000000 + 001001 + 010010 +  + 111111
Does this mean that an exponential amount of
classical information is somehow stored in n qubits?
Not in an operational sense ...
For example, Holevo’s Theorem (from 1973) implies:
one cannot convey more than n classical bits of
information in n qubits
25
Holevo’s Theorem
Easy case:
ψ
n qubits
U
Hard case (the general case):
b1
b2
b3
bn
b1b2 ... bn certainly
cannot convey more
than n bits!
ψ
n qubits
m qubits
0
0
0
0
0
U
b1
b2
b3
bn
bn+1
bn+2
bn+3
bn+4
bn+m
The difficult proof is beyond
the scope of this course
26
Superdense coding (prelude)
Suppose that Alice wants to convey two classical bits to Bob
sending just one qubit
ab
Alice
Bob
ab
By Holevo’s Theorem, this is impossible
27
Superdense coding
In superdense coding, Bob is allowed to send a qubit
to Alice first
ab
Alice
Bob
ab
How can this help?
28
How superdense coding works
1. Bob creates the state 00 + 11 and sends the first qubit
to Alice
2. Alice: if a = 1 then apply X to qubit
if b = 1 then apply Z to qubit
send the qubit back to Bob
ab
state
00
00 + 11
01
00 − 11
10
01 + 10
11
01 − 10
0 1 
X 

1
0


1 0
Z 

0

1


Bell basis
3. Bob measures the two qubits in the Bell basis
29
Measurement in the Bell basis
Specifically, Bob applies
input
H
output
00 + 11
00
01 + 10
01
00 − 11
10
01 − 10
11
to his two qubits ...
and then measures them, yielding ab
This concludes superdense coding
30
Introduction to
Quantum Information Processing
CS 467 / CS 667
Phys 667 / Phys 767
C&O 481 / C&O 681
Lecture 3 (2008)
Richard Cleve
DC 2117
[email protected]
31
Teleportation
32
Recap
• n-qubit quantum state: 2n-dimensional unit vector
†
2n2n linear
• Unitary op:
operation U such that U U = I
†
(where U denotes the conjugate transpose of U )
U0000 = the 1st column of U
U0001 = the 2nd column of U
the columns of U
:
are orthonormal
:
:
:
:
:
U1111 = the (2n)th column of U
33
Incomplete measurements (I)
Measurements up until now
are with respect to orthogonal
one-dimensional subspaces:
2
0
The orthogonal subspaces
can have other dimensions:
2
1
span of 0 and 1
(qutrit)
34
Incomplete measurements (II)
Such a measurement on 0 0 + 1 1 + 2 2
(renormalized)
results in
00 + 11 with prob 02 + 12
2
with prob 22
35
Measuring the first qubit of a
two-qubit system
0000 + 0101 + 1010 + 1111
Defined as the incomplete measurement with respect
to the two dimensional subspaces:
• span of 00 & 01 (all states with first qubit 0), and
• span of 10 & 11 (all states with first qubit 1)
Result is the mixture 0000 + 0101 with prob 002 + 012
1010 + 1111 with prob 102 + 112
36
Easy exercise: show that measuring the first qubit and
then measuring the second qubit gives the same result
as measuring both qubits at once
37
Teleportation (prelude)
Suppose Alice wishes to convey a qubit to Bob by sending
just classical bits
0 + 1
0 + 1
If Alice knows  and , she can send approximations of them
―but this still requires infinitely many bits for perfect precision
Moreover, if Alice does not know  or , she can at best
acquire one bit about them by a measurement
38
Teleportation scenario
In teleportation, Alice and Bob also start with a Bell state
0 + 1
(1/2)(00 + 11)
and Alice can send two classical bits to Bob
Note that the initial state of the three qubit system is:
(1/2)(0 + 1)(00 + 11)
= (1/2)(000 + 011 + 100 + 111)
39
How teleportation works
Initial state: (0 + 1)(00 + 11)
(omitting the 1/2 factor)
= 000 + 011 + 100 + 111
= ½(00 + 11)(0 + 1)
+ ½(01 + 10)(1 + 0)
+ ½(00 − 11)(0 − 1)
+ ½(01 − 10)(1 − 0)
Protocol: Alice measures her two qubits in the Bell basis
and sends the result to Bob (who then “corrects” his state) 40
What Alice does specifically
Alice applies
to her two qubits, yielding:
H
½00(0 + 1)
+ ½01(1 + 0)
+ ½10(0 − 1)
+ ½11(1 − 0)
(00, 0 + 1)
(01, 1 + 0)
(10, 0 − 1)
(11, 1 − 0)
with prob. ¼
with prob. ¼
with prob. ¼
with prob. ¼
Then Alice sends her two classical bits to Bob, who then
adjusts his qubit to be 0 + 1 whatever case occurs
41
Bob’s adjustment procedure
Bob receives two classical bits a, b from Alice, and:
if b = 1 he applies X to qubit
if a = 1 he applies Z to qubit
yielding:
00,
01,
0 1 
X 

1
0


1 0
Z 

0  1
0 + 1
X(1 + 0) = 0 + 1
Z(0 − 1) = 0 + 1
11, ZX(1 − 0) = 0 + 1
10,
Note that Bob acquires the correct state in each case
42
Summary of teleportation
0 + 1
Alice
H
a
b
00 + 11
Bob
X
Z
0 + 1
Quantum circuit exercise: try to work through the
details of the analysis of this teleportation protocol
43
No-cloning
theorem
44
Classical information can be copied
a
a
a
a
0
a
0
a
What about quantum information?
ψ
ψ
0
ψ
?
45
Candidate:
works fine for ψ = 0 and ψ = 1
... but it fails for ψ = (1/2)(0 + 1) ...
... where it yields output (1/2)(00 + 11)
instead of ψψ = (1/4)(00 + 01 + 10 + 11)
46
No-cloning theorem
Theorem: there is no valid quantum operation that maps
an arbitrary state ψ to ψψ
Proof:
ψ
0
0
ψ
Let ψ and ψ′ be two input states,
yielding outputs ψψg and
ψ′ψ′g′ respectively
g
Since U preserves inner products:
ψ
U
ψψ′ = ψψ′ψψ′gg′ so
ψψ′(1− ψψ′gg′) = 0 so
ψψ′ = 0 or 1
47