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October 1 & 3, 2007 1
Introduction to Quantum ComputingIntroduction to Quantum ComputingLecture 1 of 2 Lecture 1 of 2
http://www.cs.uwaterloo.ca/~cleve/CS497-F07
CS 497 Frontiers of Computer ScienceCS 497 Frontiers of Computer Science
Richard CleveDavid R. Cheriton School of Computer Science
Institute for Quantum ComputingUniversity of Waterloo
2
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
3
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
4
Moore’s LawMoore’s Law
• 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
Following trend … atomic scale in 15-20 years
Quantum mechanical effects occur at this scale:
1975 1980 1985 1990 1995 2000 2005104
105
106
107
108
109number of transistors
year
5
Quantum mechanical effectsQuantum mechanical effectsAdditional nuisances to overcome?
orNew 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
6
Also with quantum information: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 theoryis a generalization of the classical information theorythat we all know—which is based on probability theory classical
informationtheory
quantum information
theory
7
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
8
Classical and quantum systemsClassical and quantum systems
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
111
110
101
100
011
010
001
000
ppppppppProbabilistic states:
1=∑x
xp
0≥∀ xpx,
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
111
110
101
100
011
010
001
000
ααααααααQuantum states:
12=∑
xxα
Cαx x ∈∀ ,
∑=x
x xαψDirac notation: |000⟩, |001⟩, |010⟩, …, |111⟩ are basis vectors,
so
9
DiracDirac bra/bra/ketket notationnotation
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
dα
αα
M
2
1Ket: |ψ⟩ always denotes a column vector, e.g.
Bracket: ⟨φ|ψ⟩ denotes ⟨φ|⋅|ψ⟩, the inner product of
|φ⟩ and |ψ⟩
Bra: ⟨ψ| always denotes a row vector that is the conjugate transpose of |ψ⟩, e.g. [ α*
1 α*2 … α*
d ]
⎥⎦
⎤⎢⎣
⎡=
01
0 ⎥⎦
⎤⎢⎣
⎡=
10
1Convention:
10
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
11
Basic operations on Basic operations on qubitsqubits (I)(I)
⎥⎦
⎤⎢⎣
⎡θθθ−θ
cossinsincos
Rotation by θ:
(1) Apply a unitary operation U (formally U†U = I )
Examples:
(0) Initialize qubit to |0⟩ or to |1⟩ ⎥⎦
⎤⎢⎣
⎡=
01
0 ⎥⎦
⎤⎢⎣
⎡=
10
1Recall
conjugate transpose
⎥⎦
⎤⎢⎣
⎡==σ
0110
XxNOT (bit flip):Maps |0⟩ → |1⟩
|1⟩ → |0⟩
⎥⎦
⎤⎢⎣
⎡−
==σ1001
ZzPhase flip: Maps |0⟩ → |0⟩|1⟩ → −|1⟩
12
Basic operations on Basic operations on qubitsqubits (II)(II)
⎥⎦
⎤⎢⎣
⎡−
=1111
21H
More examples of unitary operations:
Hadamard:
(unitary ≈ rotation)
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
++
=+=11
2
1
2
1 100H
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+
=−=11
2
1
2
1 101H
|0⟩
|1⟩
Reflection about this line
H|0⟩
H|1⟩
13
Basic operations on Basic operations on qubitsqubits (III)(III)(3) Apply a “standard” measurement:
α|0⟩ + β|1⟩a 2
2
β
α
probwith1 probwith0
(∗) 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⟩}
|α|2
|β|2
|0⟩
|1⟩
|ψ0⟩
|ψ1⟩
… and the quantum state collapses to |0⟩ or |1⟩
14
Distinguishing between two statesDistinguishing between two states
Question 1: can we distinguish between the two cases?
Let be in state or
Distinguishing procedure:1. apply H2. 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 perfectlydistinguished … but statistical difference is detectable
( )102
1+=+ ( )10
2
1−=−
15
Operations on Operations on nn--qubitqubit statesstates
Unitary operations: ⎟⎠⎞⎜
⎝⎛∑∑
xx
xx xαxα Ua
… and the quantum state collapses
∑x
x xα
Measurements:
2111
2001
2000
probwith111
probwith001 probwith000
α
αα
MMM
⎧
⎩⎨a
(U†U = I )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
111
001
000
α
αα
M
16
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
17
EntanglementEntanglement
( )( ) 11'10'01'00'1'0'10 βββααβααβαβα +++=++The state of the combined system their tensor product:
??
Suppose that two qubits are in states: 10 β+α 1'0' β+α
11100100 21
21
21
21 −−+
Question: what are the states of the individual qubits for1. ?
2. ?11002
12
1 +
( )( )10102
12
12
12
1 +−Answers: 1.2. ... this is an entangled state
18
Structure among subsystemsStructure among subsystems
V
UW
qubits:
#2
#1
#4
#3
time
unitary operations measurements
19
Quantum circuitsQuantum circuits
|0⟩
|1⟩
|1⟩
|0⟩
|1⟩
|0⟩
1
0
1
0
1
1
Computation is “feasible” if circuit-size scales polynomially
20
Example of a oneExample of a one--qubitqubit gate gate applied to a twoapplied to a two--qubitqubit systemsystem
⎥⎦
⎤⎢⎣
⎡=
1110
0100
uuuu
U
U
(do nothing)
The resulting 4x4 matrix is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=⊗
1110
0100
1110
0100
0000
0000
uuuu
uuuu
UI|0⟩|0⟩ → |0⟩U|0⟩|0⟩|1⟩ → |0⟩U|1⟩|1⟩|0⟩ → |1⟩U|0⟩|1⟩|1⟩ → |1⟩U|1⟩
Maps basis states as:
Question: what happens if U is applied to the first qubit?
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ControlledControlled--UU gatesgates
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
1110
0100
0000
00100001
uuuu
U
|0⟩|0⟩ → |0⟩|0⟩|0⟩|1⟩ → |0⟩|1⟩|1⟩|0⟩ → |1⟩U|0⟩|1⟩|1⟩ → |1⟩U|1⟩
Maps basis states as:
Resulting 4x4 matrix is controlled-U =
⎥⎦
⎤⎢⎣
⎡=
1110
0100
uuuu
U
22
ControlledControlled--NOTNOT (CNOT)(CNOT)
Note: “control” qubit may change on some input states!
X
|a⟩
|b⟩ |a⊕b⟩
|a⟩≡
|0⟩ + |1⟩
|0⟩ − |1⟩|0⟩ − |1⟩
|0⟩ − |1⟩ H
H
H
H≡
23
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
24
Multiplication problemMultiplication problem
• “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
Input: two n-bit numbers (e.g. 101 and 111)
Output: their product (e.g. 100011)
25
Factoring problemFactoring problem
• Trial division costs ≈ 2n/2
• Best currently-known classical algorithm costs O(2n⅓ log⅔n)• Hardness of factoring is the basis of the security of many
cryptosystems (e.g. RSA)
• Shor’s quantum algorithm costs ≈ n2 [O(n2 logn loglogn) ]• Implementation would break RSA and other cryptosystems
Input: an n-bit number (e.g. 100011)
Output: their product (e.g. 101, 111)
26
How do quantum algorithms work?How do quantum algorithms work?
This is not performing “exponentially many computations at polynomial cost”
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
Given a polynomial-time classical algorithm for f :{0,1}n → T, it is straightforward to construct a quantum algorithm that creates the state: ∑
xxfxn )(,
2
1
The most straightforward way of extracting information from the state yields just (x, f (x)) for a random x∈{0,1}n
27
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
28
Deutsch’s problemDeutsch’s problemLet f : {0,1} → {0,1} fThere are four possibilities:
x f1(x)01
00
x f2(x)01
11
x f3(x)01
01
x f4(x)01
10
Goal: determine f(0) ⊕ f(1)
Any classical method requires two queries
What about a quantum method?
29
ReversibleReversible black box for black box for ff
Uf
a
b
a
b⊕ f(a)
falternate notation:
A classical algorithm: (still requires 2 queries)
f f0
0
1
f(0) ⊕ f(1)
2 queries + 1 auxiliary operation
30
Quantum algorithm for Deutsch Quantum algorithm for Deutsch
H f
H
H
|1⟩
|0⟩ f(0) ⊕ f(1)
1 query + 4 auxiliary operations ⎥⎦
⎤⎢⎣
⎡−
=1111
21H1
2 3
How does this algorithm work?
Each of the three H operations can be seen as playing a different role ...
31
Quantum algorithm (Quantum algorithm (11) ) H f
H
H
|1⟩
|0⟩
1
2 3
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⟩
f
|0⟩ – |1⟩
|x⟩ (–1) f(x)|x⟩
|0⟩ – |1⟩
32
Quantum algorithm (Quantum algorithm (22) ) 2. Causes f to be queried in superposition (at |0⟩ + |1⟩)
f
|0⟩ – |1⟩
|0⟩ (–1) f(0)|0⟩ + (–1) f(1)|1⟩
|0⟩ – |1⟩
H
x f1(x)01
00
x f2(x)01
11
x f3(x)01
01
x f4(x)01
10
±(|0⟩ + |1⟩) ±(|0⟩ – |1⟩)
33
Quantum algorithm (Quantum algorithm (33) ) 3. Distinguishes between ±(|0⟩ + |1⟩) and ±(|0⟩ – |1⟩)
H
±(|0⟩ + |1⟩) ±|0⟩
±(|0⟩ – |1⟩) ±|1⟩
H
34
Summary of Deutsch’s algorithm Summary of Deutsch’s algorithm
H f
H
H
|1⟩
|0⟩ f(0) ⊕ f(1)
1
2 3
constructs eigenvector so f-queries induce phases: |x⟩ (–1) f(x)|x⟩
produces superpositionsof inputs to f : |0⟩ + |1⟩
extracts phase differences from
(–1) f(0)|0⟩ + (–1) f(1)|1⟩
Makes only one query, whereas two are needed classically
35
Contents of lecture 1Contents of lecture 11. Preliminary remarks2. Quantum states3. Unitary operations & measurements4. Subsystem structure & quantum circuit diagrams5. Introductory remarks about quantum algorithms6. Deutsch’s parity algorithm7. One-out-of-four search algorithm
36
OneOne--outout--ofof--four searchfour searchLet 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)00011011
1000
Goal: find x ∈ {0,1}2 for which f (x) = 1
x f01(x)00011011
0100
x f10(x)00011011
0010
x f11(x)00011011
0001
What is the minimum number of queries classically? ____
Quantumly? ____
37
Quantum algorithm (I)Quantum algorithm (I)
f|x1⟩|x2⟩|y⟩
|x2⟩|x1⟩
|y ⊕ f(x1,x2)⟩
Black box for 1-4 search:
((–1) f(00)|00⟩ + (–1) f(01)|01⟩ + (–1) f(10)|10⟩ + (–1) f(11)|11⟩)(|0⟩ – |1⟩)Output state of query?
Start by creating phases in superposition of all inputs to f:
Input state to query?fHH
H|1⟩
|0⟩|0⟩ (|00⟩ + |01⟩ + |10⟩ + |11⟩)(|0⟩ – |1⟩)
38
Quantum algorithm (II)Quantum algorithm (II)
Output state of the first two qubits in the four cases:
fHH
H|1⟩
|0⟩|0⟩
Case of f00?|ψ01⟩ = + |00⟩ – |01⟩ + |10⟩ + |11⟩|ψ10⟩ = + |00⟩ + |01⟩ – |10⟩ + |11⟩|ψ11⟩ = + |00⟩ + |01⟩ + |10⟩ – |11⟩
What noteworthy property do these states have?
U
|ψ00⟩ = – |00⟩ + |01⟩ + |10⟩ + |11⟩Case of f01?Case of f10?Case of f11?
Orthogonal!
Apply the U that maps |ψ00⟩, |ψ01⟩, |ψ10⟩, |ψ11⟩ to |00⟩, |01⟩, |10⟩, |11⟩ (resp.)
39
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