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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
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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
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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
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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
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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
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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
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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
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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
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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:
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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
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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⟩
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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⟩
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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⟩
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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−=−
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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
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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
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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
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Structure among subsystemsStructure among subsystems
V
UW
qubits:
#2
#1
#4
#3
time
unitary operations measurements
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Quantum circuitsQuantum circuits
|0⟩
|1⟩
|1⟩
|0⟩
|1⟩
|0⟩
1
0
1
0
1
1
Computation is “feasible” if circuit-size scales polynomially
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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
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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≡
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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
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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)
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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)
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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
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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
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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?
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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
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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 ...
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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⟩
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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⟩)
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Quantum algorithm (Quantum algorithm (33) ) 3. Distinguishes between ±(|0⟩ + |1⟩) and ±(|0⟩ – |1⟩)
H
±(|0⟩ + |1⟩) ±|0⟩
±(|0⟩ – |1⟩) ±|1⟩
H
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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
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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
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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? ____
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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⟩)
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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.)
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