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

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

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

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

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

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

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

Classical and quantum systems

Probabilistic states

Quantum states

Dirac notation 000?, 001?, 010?, , 111? are

basis vectors, so

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 ???

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

Basic operations on qubits (I)

(0) Initialize qubit to 0? or to 1?

(1) Apply a unitary operation U (formally UU

I )

Examples

Basic operations on qubits (II)

?1?

More examples of unitary operations (unitary ?

rotation)

Hadamard

?0?

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?

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

Operations on n-qubit states

(UU I )

and the quantum state collapses

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

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

Structure among subsystems

Quantum circuits

Computation is feasible if circuit-size scales

polynomially

Example of a one-qubit gate applied to a

two-qubit system

Question what happens if U is applied to the

first qubit?

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?

Controlled-NOT (CNOT)

Note control qubit may change on some input

states!

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

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

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

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

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

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?

Reversible black box for f

a

a

Uf

b

b ? f(a)

2 queries 1 auxiliary operation

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 ...

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?

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?)

Quantum algorithm (3)

3. Distinguishes between ?(?0? ?1?) and

?(?0? ?1?)

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?

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

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? ____

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?)

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!

THE END