January 16 & 18, 2007 1

Introduction to Quantum ComputingIntroduction to Quantum ComputingLecture 2 of 2Lecture 2 of 2

Richard Cleve David R. Cheriton School of Computer Science

Institute for Quantum ComputingUniversity of Waterloo

CS 497 Frontiers of Computer ScienceCS 497 Frontiers of Computer Science

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Recap of previous lectureRecap of previous lecture

Quantum states on n qubits are

2n-dimensional unit vectors

00 01

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The basic operations on them are:• unitary operations (rotations)• measurements, that project on

to the basis states

quantum states

quantum circuits

Quantum algorithms so far:

• f(0) f(1) [Deutsch]

• one-out-of-four search

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one-out-of-one-out-of-NN search?search?

Natural question: what about search problems in spaces larger than four (and without uniqueness conditions)?

For spaces of size eight (say), the previous method breaks down—the state vectors will not be orthogonal

Later on, we’ll see how to search a space of size N with

O(N ) queries ...

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Contents of lecture 2Contents of lecture 21. 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 algorithm8. Shor’s period finding algorithm9. Grover’s search algorithm10. Concluding remarks

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Contents of lecture 2Contents of lecture 21. 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 algorithm8. Shor’s period finding algorithm9. Grover’s search algorithm10. Concluding remarks

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Period-findingPeriod-finding

r

Classically this is very hard in the general case—essentially it is as hard as finding a collision, which costs 2O(n) queries

Yet Quantum algorithms can determine r very efficiently: with only O(1) queries to f

This is the basis of Shor’s factoring algorithm ...

x y

Given: f :{0,1}n → T such that f is (strictly) r-periodic, with

unknown period r

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ApplicationApplication of period-finding algorithm of period-finding algorithm

Example: let a = 4 and m = 35 (note that gcd(4,35) = 1)

•41 mod 35 = 4•42 mod 35 = 16•43 mod 35 = 29•44 mod 35 = 11•45 mod 35 = 9•46 mod 35 = 1•47 mod 35 = 4•48 mod 35 = 16• :

Question: what is r in this case?

The sequence is cyclic because the set Zm* ={a {1,2,…, m 1} : gcd(x, m ) = 1} is a group under multiplication mod m

Answer: r = 6

Order-finding problem ( factoring)

Input: m (an n-bit integer) and a < m such that gcd(x, m ) = 1

Output: the minimum r > 0 such that ar = 1 (mod m )

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Application of period-finding algorithmApplication of period-finding algorithm

No classical polynomial-time algorithm is known for this problem — in fact, the factoring problem reduces to it

Order-finding reduces to finding the period of the function f (x) = a

x mod m, which can be computed in polynomial time

A circuit computing the function f is substituted into the black-box:

Order-finding problem ( factoring)

Input: m (an n-bit integer) and a < m such that gcd(x, m ) = 1

Output: the minimum r > 0 such that ar = 1 (mod m )

Ua,m

HV

x1x2xn

f(x)000

x1x2xn

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SketchSketch of period-finding algorithm of period-finding algorithmConstruct x |x, f (x) and then measure second register to get

the following state:

k r r r r r r r(k random)

|k + |k + r + |k + 2r + |k + 3r + … + |k + (s 1)r

Measuring this state yields just a uniformly random value

More is needed to extract r, a global property of f ...

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Quantum Fourier transformQuantum Fourier transform

2113121

13963

12642

132

1

1

1

1

11111

1

)()()(

)(

)(

NNNN

N

N

N

ωωωω

ωωωω

ωωωω

ωωωω

NNF

It’s a unitary operation on n qubits

(an NN matrix, where N = 2n)

where = e2i/N

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Period inversion propertyPeriod inversion propertyApplying the quantum Fourier transform to the state |k + |k + r + |k + 2r + |k + 3r + … + |k + (s 1)r

yields the state|0 + k |s + 2k |2s + 3k |3s + … + (r

1)k |(r 1)s

Note: there is no longer an offset k (it’s now part of the “phase”)

k r r r r r r r

= e2i / r

s s s s s s

Measure to get multiple of s, from which r can be deduced

s = r 1

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Computing the QFTComputing the QFT

H

4

8 4

816 4

32 16 8 4

H

H

H

H

11

11

2

1

mπie /2000

0100

0010

0001

Hm

Quantum circuit for F32:

Gates:

For F2n costs O(n2) gates

rev

erse

ord

er

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QuantumQuantum algorithmalgorithm forfor order-findingorder-finding

HHH

000001

HH 4

4 8H

after measuring these qubits, can calculate r using classical methods

Ua,M x,y = x,axy mod m (poly-size circuit)

Quantum Fourier transform

Number of gates for a constant success probability is: O(n2 log n loglog n)

Ua,m

HV

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Two-dimensional periodicityTwo-dimensional periodicity

Given: f :{0,1}n {0,1}n → T with a two-dimensional repeating pattern

Goal: find a simple description of this periodic strucuture

Quantum algorithms can also solve this very efficiently, and this is the basis of Shor’s discrete logarithm algorithm [1994]

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Hidden subgroup problemHidden subgroup problemLet G be a known group and H be an unknown subgroup of G

Let f : G → T have the property f (x) = f ( y) iff x y1 H

(i.e., x and y are in the same right coset of H )

Problem: given a method for computing f, determine H

Example: G = Sn , the symmetric group (permutations of {1,2,…, n})

… alas no efficient quantum has been found for this version of HSP, despite significant effort by many people

Interesting fact: a fast algorithm for this leads to a fast algorithm for the graph isomorphism problem

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Contents of lecture 2Contents of lecture 21. 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 algorithm8. Shor’s period finding algorithm9. Grover’s search algorithm10. Concluding remarks

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Prelude: factoring vs. NPPrelude: factoring vs. NP

3-CNF-SAT

factoring

P

NP

PSPACE

co-NP

Is factoring an NP-hard problem?

But factoring hasn’t been shown to be NP-hard

Moreover, there is “evidence” that it is not NP-hard:factoring NPco-NP

If so, then every problem in NP is solvable by a poly-time quantum algorithm!

If factoring is NP-hard then NP = co-NP

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Quantum search problemQuantum search problem

Given: a black box computing f : {0,1}n {0,1}

Goal: find x {0,1}n such that f (x) = 1

Classically, using probabilistic procedures, order 2n queries are necessary to succeed — even with probability ¾ (say)

x1

xny

xn

x1

y f(x1,...,xn)

UfQuery:

[Grover ’96]

Grover’s quantum algorithm makes only O(2n) queries

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Applications of quantum searchApplications of quantum search

nn xxxxxxxxxx,...,xf 515324311

The function f could be realized as a 3-CNF formula:

In fact, the search could be for a certificate for any problem in NP

The resulting quantum algorithms appear to be quadratically more efficient than the best classical algorithms known

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Prelude to Grover’s algorithm:Prelude to Grover’s algorithm:

two reflections = a rotationtwo reflections = a rotation

1

2

1

2

Consider two lines with intersection angle :

reflection 1

reflection 2

Net effect: rotation by angle 2, regardless of starting vector

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Grover’s algorithm IGrover’s algorithm I

D

x1

xny

xn

x1

y f(x1,...,xn)

UfUf x(0 1) = (1) f(x)

x(0

1)

HHHH

Basic operations used:

nnn

nnn

nnn

D

2212222

2222122

2222221

Query:

Diffusion:

Hadamard:

Costs only O(n) gates

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Grover’s algorithm IIGrover’s algorithm II

1. construct state H 0...02. repeat k times:

apply D Uf to state

3. measure state, to get x{0,1}n, and check if f (x) =1

0

0

H Uf D

iteration 1

Uf D

iteration 2

Uf D

iteration 3

Uf D

iteration 4

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Grover’s algorithm IIIGrover’s algorithm III

Algorithm: (D Uf )k H 0...0

target

target (assume unique)

H0...0 = x x 2

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2

Since D Uf is a composition of two reflections, it is a rotation

by 2, where sin() 1/N We want (2k+1)(1/N) /2, so set k (/4)N

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Contents of lecture 2Contents of lecture 21. 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 algorithm8. Shor’s period finding algorithm and factoring9. Grover’s search algorithm10. Concluding remarks

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ConclusionConclusion

Quantum computing challenges this thesis

Extended Chruch-Turing Thesis: any polynomial-time algorithm can be simulated by a probabilistic polynomial-time Turing machine

Either:

• The Extended Chruch-Turing Thesis is false

• Quantum mechanics as we understand it is false

• There is a classical poly-time factoring algorithm

There are many curious properties of quantum information in the context of computation, communication, and cryptography

Waterloo has many people in the faculties of Mathematics, Science, and Engineering working in quantum computing (please see www.iqc.ca more information)

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Some possible project topicsSome possible project topics1. Efficient “quantum proofs”: an example is the group non-

membership problem

2. Nonlocal effects: apparently paradoxical tricks that can be performed with entangled states

3. Quantum error-correcting codes: important for physical implementations of quantum information processing

4. Alternate models of quantum computation: an example is the measurement-based model

5. Quantum walks: the quantum analogues of random walks, and their algorithmic applications

6. Quantum cryptography: information-theoretical security in a public-key setting

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