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CS 497 Frontiers of Computer Science
Introduction to Quantum Computing
Lecture 1 of 2
Richard Cleve
David R. Cheriton School of Computer Science
Institute for Quantum Computing
University of Waterloo
January 16 & 18, 2007
1
Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
2
Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
3
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
4
Quantum mechanical effects
Additional nuisances to overcome?
or
New types of behavior to make use of?
[Shor ’94]: polynomial-time algorithm for factoring integers
on a quantum computer
This could be used to break most of the existing public-key
cryptosystems, including RSA, and elliptic curve crypto
[Bennett, Brassard ’84]: provably secure codes with short keys
5
Also with quantum information:
•
•
•
•
Faster algorithms for combinatorial search problems
Fast algorithms for simulating quantum mechanics
Communication savings in distributed systems
More efficient notions of “proof systems”
Quantum information theory
is a generalization of the
classical information theory
that we all know—which is
based on probability theory
quantum
information
theory
classical
information
theory
6
Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
7
Classical and quantum 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
8
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
9
Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
10
Basic operations on qubits (I)
(0) Initialize qubit to |0 or to |1
Recall 0   
0
1
 
0
1  
1
†
(1) Apply a unitary operation U (formally U U = I )
conjugate transpose
Examples:
Rotation by :
cos   sin 
 sin  cos  


0 1 
NOT (bit flip):  x  X  

1
0


Maps |0  |1
|1  |0
1 0
Phase flip:  z  Z  

0

1


Maps |0  |0
|1  |1
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Basic operations on qubits (II)
1
More examples of unitary operations:
(unitary  rotation)
1
Hadamard: H 
2
1 1
1  1



0  1 
2
1  1

0  1 
2
1  1
H0 
1
H1 
1
Reflection
about this line
H0
0
H1
 
2  1
 
2   1
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Basic operations on qubits (III)
(3) Apply a “standard” measurement:
0 + 1

0 with prob 
2
1 with prob 
2
ψ1 1
ψ0
||2
0
||2
… and the quantum state collapses
to 0 or 1
() There exist other quantum operations, but they
can all be “simulated” by the aforementioned types
Example: measurement with respect to a different
orthonormal basis {ψ0, ψ1}
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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 … but statistical difference is detectable
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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
15
Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
16
Entanglement
Suppose that two qubits are in states:  0   1
' 0  ' 1
The state of the combined system their tensor product:
α 0
 β 1 α' 0  β' 1   αα' 00  αβ ' 01  βα' 10  ββ' 11
Question: what are the states of the individual qubits for
1. 12 00  12 01  12 10  12 11 ?
1
2
2.
Answers:
1.

1
2
00 
0 
1
2
1
2
11 ?
1

1
2
0 
1
2
1

2. ?? ... this is an entangled state
17
Structure among subsystems
qubits:
#1
time
U
W
#2
V
#3
#4
unitary operations
measurements
18
Quantum circuits
0
1
1
0
1
1
0
0
1
1
0
1
Computation is “feasible” if circuit-size scales polynomially
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Example of a one-qubit gate
applied to a two-qubit system
(do nothing)
U
u00 u01 
U 

u
u
 10 11 
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
Question: what happens if U is applied to the first qubit?
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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
H
H
0 − 1
0 − 1
H
H

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Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
23
Multiplication problem
Input: two n-bit numbers (e.g. 101 and 111)
Output: their product (e.g. 100011)
• “Grade school” algorithm takes O(n2) steps
• Best currently-known classical algorithm costs
O(n log n loglog n)
• Best currently-known quantum method: same
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Factoring problem
Input: an n-bit number (e.g. 100011)
Output: their product (e.g. 101, 111)
• Trial division costs  2n/2
⅓ log⅔ n
n
• Best currently-known classical algorithm costs O(2
)
• Hardness of factoring is the basis of the security of many
cryptosystems (e.g. RSA)
• Shor’s quantum algorithm costs 
n2
[ O(n2 log n loglog n)
]
• Implementation would break RSA and other cryptosystems25
How do quantum algorithms work?
Given a polynomial-time classical algorithm for f :{0,1}n → T,
it is straightforward to construct a quantum algorithm that
1
creates the state:
x, f ( x )
2
n

x
This is not performing “exponentially many computations at
polynomial cost”
The most straightforward way of extracting information from
the state yields just (x, f (x)) for a random x{0,1}n
But we can make some interesting tradeoffs:
instead of learning about any (x, f (x)) point, one can learn
something about a global property of f
26
Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
27
Deutsch’s problem
Let f : {0,1} → {0,1}
f
There are four possibilities:
x f1(x)
x f2(x)
x f3(x)
x f4(x)
0
1
0
1
0
1
0
1
0
0
1
1
0
1
1
0
Goal: determine f(0)  f(1)
Any classical method requires two queries
What about a quantum method?
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Reversible black box for f
a
b
a
Uf
alternate
notation:
f
b  f(a)
A classical algorithm: 0
(still requires 2 queries)
f
0
f
1
f(0)  f(1)
2 queries + 1 auxiliary operation
29
Quantum algorithm for Deutsch
2
0
H
1
H
3
f
H
1
1 query + 4 auxiliary operations
f(0)  f(1)
H
1

2
1 1
1  1


How does this algorithm work?
Each of the three H operations can be seen as playing
a different role ...
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Quantum algorithm (1)
2
0
H
1
H
3
H
f
1
1. Creates the state 0 – 1, which is an eigenvector of
NOT with eigenvalue –1
I with eigenvalue +1
This causes f to induce a phase shift of (–1) f(x) to x
x
0 – 1
f
(–1) f(x)x
0 – 1
31
Quantum algorithm (2)
2. Causes f to be queried in superposition (at 0 + 1)
0
H
(–1) f(0)0 + (–1) f(1)1
f
0 – 1
0 – 1
x f1(x)
x f2(x)
x f3(x)
x f4(x)
0
1
0
1
0
1
0
1
0
0
(0 + 1)
1
1
0
1
1
0
(0 – 1)
32
Quantum algorithm (3)
3. Distinguishes between (0 + 1) and (0 – 1)
(0 + 1)
H
0
(0 – 1)
H
1
33
Summary of Deutsch’s algorithm
Makes only one query, whereas two are needed classically
extracts phase differences from
produces superpositions
of inputs to f : 0 + 1
(–1) f(0)0 + (–1) f(1)1
2
0
H
1
H
3
f
H
f(0)  f(1)
1
constructs eigenvector so f-queries
induce phases: x  (–1) f(x)x
34
Contents of lecture 1
1.
2.
3.
4.
5.
6.
7.
Preliminary remarks
Quantum states
Unitary operations & measurements
Subsystem structure & quantum circuit diagrams
Introductory remarks about quantum algorithms
Deutsch’s parity algorithm
One-out-of-four search algorithm
35
One-out-of-four search
Let f : {0,1}2 → {0,1} have the property that there is exactly
one x  {0,1}2 for which f (x) = 1
Four possibilities: x f00(x)
00
01
10
11
1
0
0
0
x f01(x)
x f10(x)
x f11(x)
00
01
10
11
00
01
10
11
00
01
10
11
0
1
0
0
0
0
1
0
0
0
0
1
Goal: find x  {0,1}2 for which f (x) = 1
What is the minimum number of queries classically? ____
Quantumly? ____
36
Quantum algorithm (I)
Black box for 1-4 search:
x1
x1
f
x2
y
x2
y  f(x1,x2)
Start by creating phases in superposition of all inputs to f:
0
0
1
H
H
H
f
Input state to query?
(00 + 01 + 10 + 11)(0 – 1)
Output state of query?
((–1) f(00)00 + (–1) f(01)01 + (–1) f(10)10 + (–1) f(11)11)(0 – 1)
37
Quantum algorithm (II)
 Apply the U that maps
H
0
f
U
 ψ00, ψ01, ψ10, ψ11 to
H
0
 00, 01, 10, 11 (resp.)
H
1
Output state of the first two qubits in the four cases:
Case of f00?
Case of f01?
Case of f10?
Case of f11?
ψ00 = – 00 + 01 + 10 + 11
ψ01 = + 00 – 01 + 10 + 11
ψ10 = + 00 + 01 – 10 + 11
ψ11 = + 00 + 01 + 10 – 11
What noteworthy property do these states have?
Orthogonal!
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