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![Page 1: October 1 & 3, 20071 Introduction to Quantum Computing Lecture 1 of 2 Introduction to Quantum Computing Lecture 1 of 2 cleve/CS497-F07.](https://reader035.fdocuments.net/reader035/viewer/2022062718/56649eba5503460f94bc2709/html5/thumbnails/1.jpg)
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 Cleve David 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
109
number 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 theory is a generalization of the classical information theory that we all know—which is based on probability theory
classical information
theory
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
p
p
p
p
p
p
p
pProbabilistic states:
1x
xp
0 xpx,
111
110
101
100
011
010
001
000
α
α
α
α
α
α
α
αQuantum states:
12
xxα
Cαx x ,
x
x xαψ
Dirac notation: |000, |001, |010, …, |111 are basis vectors,
so
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Dirac bra/ket notationDirac bra/ket notation
d
2
1Ket: ψ always denotes a column vector, e.g.
Bracket: φψ denotes φψ, the inner
product of φ and ψ
Bra: ψ always denotes a row vector that is the
conjugate transpose of ψ, e.g. [ *1 *
2 *d ]
0
10
1
01Convention:
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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 qubits (I)Basic operations on qubits (I)
cossin
sincosRotation by :
(0) Initialize qubit to |0 or to |1
(1) Apply a unitary operation U (formally U†U = I )
Examples:
0
10
1
01Recall
conjugate transpose
01
10XxNOT (bit flip):
Maps |0 |1|1 |0
10
01ZzPhase flip: Maps |0 |0
|1 |1
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Basic operations on qubits (II)Basic operations on qubits (II)
11
11
2
1HHadamard:
More examples of unitary operations: (unitary rotation)
1
1
2
1
2
1 100H
1
1
2
1
2
1 101H
0
1
Reflection about this line
H0
H1
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Basic operations on qubits (III)Basic operations on qubits (III)(3) Apply a “standard” measurement:
0 + 1
2
2
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}
||2
||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 perfectly distinguished … but statistical difference is detectable
102
1 10
2
1
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Operations on Operations on nn-qubit states-qubit states
Unitary operations:
xx
xx xαxα U
… and the quantum state collapses
x
x xα
Measurements:
2
111
2
001
2
000
probwith111
probwith001
probwith000
α
α
α
(U†U = I )
111
001
000
α
α
α
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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 1'0'
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 one-qubit gate Example of a one-qubit gate applied to a two-qubit systemapplied to a two-qubit system
1110
0100
uu
uuU
U
(do nothing)
The resulting 4x4 matrix is
1110
0100
1110
0100
00
00
00
00
uu
uu
uu
uu
UI00 0U0 01 0U1 10 1U0 11 1U1
Maps basis states as:
Question: what happens if U is applied to the first qubit?
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Controlled-Controlled-UU gates gates
1110
0100
00
00
0010
0001
uu
uu
U
00 00 01 01 10 1U0 11 1U1
Maps basis states as:
Resulting 4x4 matrix is controlled-U =
1110
0100
uu
uuU
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Controlled-Controlled-NOTNOT (CNOT)(CNOT)
Note: “control” qubit may change on some input states!
X
a
b ab
a
≡
0 + 1
0 − 10 − 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 log⅔
n
)• 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 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: x
xfxn
)(,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 problem
Let f : {0,1} → {0,1} f
There are four possibilities:
x f1(x)
0
1
0
0
x f2(x)
0
1
1
1
x f3(x)
0
1
0
1
x f4(x)
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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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
11
11
2
1H
How does this algorithm work?
Each of the three H operations can be seen as playing a different role ...
1
2 3
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Quantum algorithm (Quantum algorithm (11) ) H f
H
H
1
0
1. Creates the state 0 – 1, which is an eigenvector of
1
2 3
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)
0
1
0
0
x f2(x)
0
1
1
1
x f3(x)
0
1
0
1
x f4(x)
0
1
1
0
(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 superpositions
of 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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One-out-of-four searchOne-out-of-four 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)
00
01
10
11
1
0
0
0
Goal: find x {0,1}2 for which f (x) = 1
x f01(x)
00
01
10
11
0
1
0
0
x f10(x)
00
01
10
11
0
0
1
0
x f11(x)
00
01
10
11
0
0
0
1
What is the minimum number of queries classically? ____
Quantumly? ____
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Quantum algorithm (I)Quantum algorithm (I)
fx1x2y
x2x1
y f(x1,x2)
((–1) f(00)00 + (–1) f(01)01 + (–1) f(10)10 + (–1) f(11)11)(0 – 1)
Output state of query?
Black box for 1-4 search:
Start by creating phases in superposition of all inputs to f:
Input state to query?fH
H
H1
00 (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:
fH
H
H1
00
Case of f00?ψ01 = + 00 – 01 + 10 + 11ψ10 = + 00 + 01 – 10 + 11ψ11 = + 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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