Introduction to Quantum Computing Lecture 1 of 2 http://www.cs.uwaterloo.ca/~cleve/CS497-F07

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Introduction to Quantum ComputingLecture 1 of 2
http//www.cs.uwaterloo.ca/cleve/CS497-F07
CS 497 Frontiers of Computer Science

• Richard Cleve
• David R. Cheriton School of Computer Science
• Institute for Quantum Computing
• University of Waterloo

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Contents of lecture 1

1. Preliminary remarks
2. Quantum states
3. Unitary operations measurements
4. Subsystem structure quantum circuit diagrams
5. Introductory remarks about quantum algorithms
6. Deutschs parity algorithm
7. One-out-of-four search algorithm

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Contents of lecture 1

1. Preliminary remarks
2. Quantum states
3. Unitary operations measurements
4. Subsystem structure quantum circuit diagrams
5. Introductory remarks about quantum algorithms
6. Deutschs parity algorithm
7. One-out-of-four search algorithm

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Moores Law
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

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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
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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
knowwhich is based on probability theory
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Contents of lecture 1

• Preliminary remarks
• Quantum states
• Unitary operations measurements
• Subsystem structure quantum circuit diagrams
• Introductory remarks about quantum algorithms
• Deutschs parity algorithm
• One-out-of-four search algorithm

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Classical and quantum systems
Probabilistic states
Quantum states
Dirac notation 000?, 001?, 010?, , 111? are
basis vectors, so
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Dirac bra/ket notation
Ket ??? always denotes a column vector, e.g.
Convention
Bra ??? always denotes a row vector that is the
conjugate transpose of ???, e.g. ?1 ?2 ?
?d
Bracket ?f??? denotes ?f?????, the inner product
of ?f? and ???
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Contents of lecture 1

1. Preliminary remarks
2. Quantum states
3. Unitary operations measurements
4. Subsystem structure quantum circuit diagrams
5. Introductory remarks about quantum algorithms
6. Deutschs parity algorithm
7. One-out-of-four search algorithm

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Basic operations on qubits (I)
(0) Initialize qubit to 0? or to 1?
(1) Apply a unitary operation U (formally UU
I )
Examples
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Basic operations on qubits (II)
?1?
More examples of unitary operations (unitary ?
rotation)
Hadamard
?0?
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Basic operations on qubits (III)
?1?
??1?
(3) Apply a standard measurement
??0?
?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 ??0?, ??1?
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Distinguishing between two states
Let be in state
or
Question 1 can we distinguish between the two
cases?

• Distinguishing procedure
• apply H
• measure

This works because H ?? ?0? and H ?-? ?1?
Question 2 can we distinguish between ?0? and
???
Since theyre not orthogonal, they cannot be
perfectly distinguished but statistical
difference is detectable
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Operations on n-qubit states
(UU I )
and the quantum state collapses
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Contents of lecture 1

1. Preliminary remarks
2. Quantum states
3. Unitary operations measurements
4. Subsystem structure quantum circuit diagrams
5. Introductory remarks about quantum algorithms
6. Deutschs parity algorithm
7. One-out-of-four search algorithm

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Entanglement
Suppose that two qubits are in states
The state of the combined system their tensor
product
Answers 1. 2.
??
... this is an entangled state
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Structure among subsystems
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Quantum circuits
Computation is feasible if circuit-size scales
polynomially
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Example of a one-qubit gate applied to a
two-qubit system
Question what happens if U is applied to the
first qubit?
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Controlled-U gates
U
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?
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Controlled-NOT (CNOT)
Note control qubit may change on some input
states!
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Contents of lecture 1

1. Preliminary remarks
2. Quantum states
3. Unitary operations measurements
4. Subsystem structure quantum circuit diagrams
5. Introductory remarks about quantum algorithms
6. Deutschs parity algorithm
7. One-out-of-four search algorithm

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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
• Best currently-known classical algorithm costs
  O(2n? log? n )
• Hardness of factoring is the basis of the
  security of many cryptosystems (e.g. RSA)
• Shors quantum algorithm costs ? n2 O(n2
  log n loglog n)
• Implementation would break RSA and other
  cryptosystems

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How do quantum algorithms work?
Given a polynomial-time classical algorithm for
f 0,1n ? T, it is straightforward to construct
a quantum algorithm that creates the state
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?0,1n
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
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Contents of lecture 1

1. Preliminary remarks
2. Quantum states
3. Unitary operations measurements
4. Subsystem structure quantum circuit diagrams
5. Introductory remarks about quantum algorithms
6. Deutschs parity algorithm
7. One-out-of-four search algorithm

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Deutschs problem
Let f 0,1 ? 0,1
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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Reversible black box for f
a
a
Uf
b
b ? f(a)
2 queries 1 auxiliary operation
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Quantum algorithm for Deutsch
H
H
f(0) ? f(1)
?0?
H
?1?
1 query 4 auxiliary operations
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
3
1
1. Creates the state ?0? ?1?, which is an
eigenvector of
This causes f to induce a phase shift of (1)
f(x) to ?x?
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Quantum algorithm (2)
2. Causes f to be queried in superposition (at
?0? ?1?)
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 (3)
3. Distinguishes between ?(?0? ?1?) and
?(?0? ?1?)
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Summary of Deutschs algorithm
Makes only one query, whereas two are needed
classically
produces superpositions of inputs to f ?0?
?1?
extracts phase differences from (1) f(0)?0?
(1) f(1)?1?
constructs eigenvector so f-queries induce
phases ?x? ? (1) f(x)?x?
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Contents of lecture 1

1. Preliminary remarks
2. Quantum states
3. Unitary operations measurements
4. Subsystem structure quantum circuit diagrams
5. Introductory remarks about quantum algorithms
6. Deutschs parity algorithm
7. One-out-of-four search algorithm

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One-out-of-four search
Let f 0,12 ? 0,1 have the property that
there is exactly one x ? 0,12 for which f (x)
1
Four possibilities
x f00(x)
00 01 10 11 1 0 0 0
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
Goal find x ? 0,12 for which f (x) 1
What is the minimum number of queries
classically? ____
Quantumly? ____
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Quantum algorithm (I)
Black box for 1-4 search
Start by creating phases in superposition of all
inputs to 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?)
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Quantum algorithm (II)
? Apply the U that maps ? ??00?, ??01?, ??10?,
??11? to ? ?00?, ?01?, ?10?, ?11? (resp.)
Output state of the first two qubits in the four
cases
Case of f00?
??00? ?00? ?01? ?10? ?11?
??01? ?00? ?01? ?10? ?11?
Case of f01?
??10? ?00? ?01? ?10? ?11?
Case of f10?
??11? ?00? ?01? ?10? ?11?
Case of f11?
What noteworthy property do these states have?
Orthogonal!
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THE END