introduction to quantum computingcleve/courses/cs497-f07/cs497l1.pdf · 26 how do quantum...

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

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

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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

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:

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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⟩

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

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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⟩)

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