Introduction to Quantum Computing & Quantum Information Theory

Introduction to Quantum Computing and

Quantum Information Theory

Contents

I. Computing and the Laws of PhysicsII. A Happy Marriage; Quantum Mechanics & ComputersIII. Qubits and Quantum GatesIV. Quantum ParallelismV. Summary

Technological limits

For the past two decades we have enjoyed Gordon Moore’s law. But all good things may come to an end…

We are limited in our ability to increase the density and the speed of a computing engine.

Reliability will also be affectedto increase the speed we need increasingly smaller

circuits (light needs 1 ns to travel 30 cm in vacuum)smaller circuits systems consisting only of a few

particles subject to Heissenberg uncertainty

Power dissipation, circuit density, and speed

In 1992 Ralph Merkle from Xerox PARC calculated that a 1 GHz computer operating at room temperature, with 1018 gates packed in a volume of about 1 cm3 would dissipate 3 MW of power. A small city with 1,000 homes each using 3 KW would

require the same amount of power; A 500 MW nuclear reactor could only power some 166

such circuits.

Talking about the heat…

The heat produced by a super dense computing engine is proportional with the number of elementary computing circuits, thus, with the volume of the engine. The heat dissipated grows as the cube of the radius of the device.

To prevent the destruction of the engine we have to remove the heat through a surface surrounding the device. Henceforth, our ability to remove heat increases as the square of the radius while the amount of heat increases with the cube of the size of the computing engine.

Contents


A happy marriage…The two greatest discoveries of the 20-th

centuryquantum mechanicsstored program computers

produced quantum computing and quantum information theory

Quantum; Quantum mechanics

Quantum is a Latin word meaning some quantity. In physics it is used with the same meaning as the word discrete in mathematics, i.e., some quantity or variable that can take only sharply defined values as opposed to a continuously varying quantity. The concepts continuum and continuous are known from geometry and calculus

smallest possible discrete unit of any physical property

Quantum mechanics is a mathematical model of the physical world

Heissenberg uncertainty principle

Heisenberg uncertainty principle says we cannot determine both the position and the momentum of a quantum particle with arbitrary precision.

In his Nobel prize lecture on December 11, 1954 Max Born says about this fundamental principle of Quantum Mechanics : ``... It shows that not only the determinism of classical physics must be abandoned, but also the naive concept of reality which looked upon atomic particles as if they were very small grains of sand. At every instant a grain of sand has a definite position and velocity. This is not the case with an electron. If the position is determined with increasing accuracy, the possibility of ascertaining its velocity becomes less and vice versa.''

A revolutionary approach to computing and communication

We need to consider a revolutionary rather than an evolutionary approach to computing.

Quantum theory does not play only a supporting role by prescribing the limitations of physical systems used for computing and communication.

Quantum properties such as uncertainty, interference, and entanglement

form the foundation of a new brand of theory, the quantum information theory where computational and communication processes rest upon fundamental physics.

Deterministic versus probabilistic photon behavior

(b)(a)

D1

D2

D3

D5

D7

Detector D1

Detector D2

Beam splitter

Incident beam of light

Reflected beam

Transmitted beam

A mathematical model to describe the state of a quantum system

10 10

|,| 10

1|||| 21

20

are complex numbersare complex numbers

Superposition and uncertainty

In this model a state is a superposition of two basis states, “0” and “1”This state is unknown before we make a measurement.After we perform a measurement the system is no longer

in an uncertain state but it is in one of the two basis states. is the probability of observing the outcome “1” is the probability of observing the outcome “0”

10 10

1|||| 21

20

20 ||

21 ||

The superposition probability rule

If an event may occur in two or more indistinguishable ways then the probability amplitude of the event is the sum of the probability amplitudes of each case considered separately (sometimes known as Feynmann rule).

Quantum computers

In quantum systems the amount of parallelism increases exponentially with the size of the system, thus with the number of qubits. This means that the price to pay for an exponential increase in the power of a quantum computer is a linear increase in the amount of matter and space needed to build the larger quantum computing engine.

A quantum computer will enable us to solve problems with a very large state space.

Contents


A bit versus a qubit

A bitCan be in two distinct states, 0 and 1A measurement does not affect the state

A qubit can be in state or in state or in any other

state that is a linear combination of the basis stateWhen we measure the qubit we find it

in state with probability in state with probability

0| 1|10 10

0|1|

20 ||

21 ||

0

1

0

1

(a) One bit (b) One qubit

Superposition states

Basis (logical) state 1

Basis (logical) state 0

Other states of a qubit

12

10

2

1

12

30

2

1

The Boch sphere representation of one qubit

A qubit in a superposition state is represented as a vector connecting the center of the Bloch sphere with a point on its periphery.

The two probability amplitudes can be expressed using Euler angles.

b

| 0 >

| 1 >

z

x

y

ψ|

r

b

z

xy

Two qubits

Represented as vectors in a 2-dimensional Hilbert space with four basis vectors

When we measure a pair of qubits we decide that the system it is in one of four states

with probabilities

11,10,01,00

11,10,01,00

211

210

201

200 ||,||,||,||

Two qubits

11100100 11100100

1|||||||| 211

210

201

200

Measuring two qubits

Before a measurement the state of the system consisting of two qubits is uncertain (it is given by the previous equation and the corresponding probabilities).

After the measurement the state is certain, it is 00, 01, 10, or 11 like in the case of a classical

two bit system.

Measuring two qubits (cont’d)

What if we observe only the first qubit, what conclusions can we draw?

We expect that the system to be left in an uncertain sate, because we did not measure the second qubit that can still be in a continuum of states. The first qubit can be 0 with probability 1 with probability

201

200 ||||

211

210 ||||


211

210

11011

||||

1101

I

Call the post-measurement state when we measure the first qubit and find it to be 0.

Call the post-measurement state when we measure the first qubit and find it to be 1.

201

200

01000

||||

0100

I

I0

I1


Call the post-measurement state when we measure the second qubit and find it to be 0.

Call the post-measurement state when we measure the second qubit and find it to be 1.

210

200

10000

||||

1000

II

211

201

11011

||||

1110

II

II0

II1

Bell states - a special state of a pair of qubits

2

11100 If and

When we measure the first qubit we get the post measurement state

When we measure the second qubit we get the post mesutrement state

01001

11|1I00|0

I

00|0II 11|1

II

This is an amazing result!

The two measurements are correlated, once we measure the first qubit we get exactly the same result as when we measure the second one.

The two qubits need not be physically constrained to be at the same location and yet, because of the strong coupling between them, measurements performed on the second one allow us to determine the state of the first.

Entanglement

Entanglement is an elegant, almost exact translation of the German term Verschrankung used by Schrodinger who was the first to recognize this quantum effect.

An entangled pair is a single quantum system in a superposition of equally possible states. The entangled state contains no information about the individual particles, only that they are in opposite states.

The important property of an entangled pair is that the measurement of one particle influences the state of the other particle. Einstein called that “Spooky action at a distance".

The spin

In quantum mechanics the intrinsic angular moment, the spin, is quantized and the values it may take are multiples of the rationalized Planck constant.

The spin of an atom or of a particle is characterized by the spin quantum number s , which may assume integer and half-integer values. For a given value of s the projection of the spin on any axis may assume 2s + 1 values ranging from - s to $ + s by unit steps, in other words the spin is quantized.

More about the spin

There are two classes of quantum particlesfermions - spin one-half particles such as the

electrons. The spin quantum numbers of fermions can be s=+1/2 and s=-1/2

bosons - spin one particles. The spin quantum numbers of bosons can be s=+1, s=0, and s=-1

The spin of the electron

The electron has spin s = 1 /2 and the spin projection can assume the values + ½ referred to as spin up, and -1/2 referred to as spin down.

Light and photons

Light is a form of electromagnetic radiation; the wavelength of the radiation in the visible spectrum varies from red to violet.

Light can be filtered by selectively absorbing some color ranges and passing through others.

A polarization filter is a partially transparent material that transmits light of a particular polarization.

Photons

Photons differ from the spin 1/2 electrons in two ways: (1) they are massless and (2) have spin $1$.

A photon is characterized by its vector momentum (the vector momentum determines

the frequency) and polarization.

In the classical theory light is described as having an electric field which oscillates either vertically, the light is x-polarized, or horizontally, the light is y-polarized in a plane perpendicular to the direction of propagation, the z-axis.

The two basis vectors are |h> and |v>

Classical gates

Implement Boolean functions.Are not reversible (invertible). We cannot

recover the input knowing the output. This means that there is an iretriviable loss of

information

NOT gate

AND gate

x y = NOT(x)

x y z0

111 1 1

0 00 0

0 0

x y z0

111 1

10

0

00

NAND gate

x

yz = (x) AND (y)

x

yz = (x) NAND (y) 1

1

x y0

011

0

11

x y z0

111 1

10

00

x

yz = (x) OR (y)OR gate

x y z0

111 1

100

0

x

yz = (x) NOR (y)NOR gate 0

00

x y z0

111 1

10

00

x

yz = (x) XOR (y)XOR gate

1

0

0

One qubit gates

Transform an input qubit into an output qubitCharacterized by a 2 x 2 matrix with complex

coefficients

One-qubit gate

10 10

G

2221

1211

gg

ggG

1

0

2221

1211

1

0

gg

gg

10 '1

'0

One qubit gates

I identity gate; leaves a qubit unchanged. X or NOT gate transposes the components of an

input qubit. Y gate. Z gate flips the sign of a qubit. H the Hadamard gate.

One qubit gates

2221

1211

gg

ggG

2212

2111

gg

ggGT

*22

*12

*21

*11

gg

ggG

Igggggggg

ggggggggGG

22*2212

*1221

*2211

*12

22*2112

*1121

*2111

*11

Identity transformation, Pauli matrices, Hadamard

10

010 I

10

013 Z

0

02 i

iY

01

101 X

11

11

2

1H

10 10

10 01

10 01 ii

10 10

2

10

2

1010

CNOT a two qubit gate

Two inputsControl Target

The control qubit is transferred to the output as is.The target qubit

Unaltered if the control qubit is 0Flipped if the control qubit is 1.

addition modulo 2+O

Target input

Control input

O+

CNOT

CNOTV

0100

1000

0010

0001

CNOTG

CNOTCNOTCNOT VGW

The two input qubits of a two qubit gates

10 10

10 10

11

01

10

00

1

0

1

0

CNOTV

Two qubit gates

0100

1000

0010

0001

CNOTG

1011111001010000 CNOTG

Two qubit gates

01

11

10

00

11

01

10

00

0100

1000

0010

0001

CNOTW

CNOTCNOTCNOT VGW

11100100 01111000 CNOTW

)10(1)10(0 011100 CNOTW

Final comments on the CNOT gate

CNOT preserves the control qubit (the first and the second component of the input vector are replicated in the output vector) and flips the target qubit (the third and fourth component of the input vector become the fourth and respectively the third component of the output vector).

The CNOT gate is reversible. The control qubit is replicated at the output and knowing it we can reconstruct the target input qubit.

)10(1)010(0 011100 CNOTW

Fredkin gate

Three input and three output qubitsOne controlTwo target

When the control qubit is0 the target qubits are replicated to the output1 the target qubits are swapped

a

b

c

a'

b'

c'

a

b

0

a

b

0

a

b

1

b

a

1

0

Input Outputa b c a' b' c'

0 0

0

0

0

0

0

0 0

0

0 0

0 0

01

1

1

1

1 1 1 1


1 1

1

1

1

1

1

0 0

0

0 1

0 0

11

1

0

0

1 1 1 1


0 0

0

0

0

0

0

0 0

0

0 0

0 0

01

1

1

1

1 1 1 1

0 1 0

1

1

1

1

1

1

1

1

0 0

0

0 0

0

1

1

1 1

1

1

11

0

0

(a) (b)

(c)

(d) (e)

0

b

c

bc

bc

c'

1

0

c

c

c

c'

a=0 a=1 & b=0 a’= bc & b’ = bc a’= c & b’ = c

Threequbitgate

Toffoli gate

Three input and three output qubitsTwo controlOne target

When both control qubit are 1 the target qubit is flippedotherwise the target qubit is not changed.

Toffoli gate is universal. It may emulate an AND and a NOT gate

a

b

c

a

b

c O ab+

(a)

a a

(b) (c)

1 1

b b

1 NAND(ab) 0

b b

b

Controlled H gate

HH

(a) (b)

Generic one qubit controlled gate

(b)

A CB

| c >

| t >

(a)

U

| c >

| t >

U

|c0>

|c1>

|c2>

|c3>

|c4>

|c5>

| w0>=| 0 >

| 0>

| t >

Control

Target

| 0>

| 0>

| 0>

| 0>

| w1>=| 0 >

| w2>=| 0 >

| w3>=| 0 >

| w4>=| 0 >

|c0>

|c1>

|c2>

|c3>

|c4>

|c5>

Contents


A quantum circuitGiven a function f(x) we can construct a

reversible quantum circuit consisting of Fredking gates only, capable of transforming two qubits as follows

The function f(x) is hardwired in the circuit

| x > | x >

| y >

Uf

| y o f(x )>+

A quantum circuit (cont’d)

If the second input is zero then the transformation done by the circuit is

| x > | x >

| 0 >

Uf

| f(x )>

A quantum circuit (cont’d)

Now apply the first qubit through a Hadamad gate.

The resulting sate of the circuit is

The output state contains information about f(0) and f(1).

| 0 >

Uf

H| 0 > | 0 >

)2

10(0

f

2

)1(1)0(0 ff

2

10

O

(c)

Uf

H

| 0 >

+

| x >

f(x) >

| x >

| y| y >

(a)

| x >

| f(x) >

| x >

| 0 >

(b)

| 0 >

Uf Uf

Quantum parallelism

The output of the quantum circuit contains information about both f(0) and f(1). This property of quantum circuits is called quantum parallelism.

Quantum parallelism allows us to construct the entire truth table of a quantum gate array having 2n entries at once. In a classical system we can compute the truth table in one time step with 2n gate arrays running in parallel, or we need 2n time steps with a single gate array.

We start with n qubits, each in state |0> and we apply a Welsh-Hadamard transformation.

Advantages

• Much more powerful

• Faster

• Smaller

• Improvements to science

• Can improve on practical personal electronics

• what we are waiting for….

Christmas!

Difficulties

• Hard to control quantum particles

• Lots of heat

• Expensive

• Difficult to build

• Not enough is known about quantum mechanics

Future Work

• Silicon Quantum Computer

• It may become technology sooner than we expect

• New algorithms and communication

• Maximum exploitation

• Simulate other quantum systems.

Contents


Final remarks

A tremendous progress has been made in the area of quantum computing and quantum information theory during the past decade. Thousands of research papers, a few solid reference books, and many popular-science books have been published in recent years in this area.

The growing interest in quantum computing and quantum information theory is motivated by the incredible impact this discipline could have on how we store, process, and transmit data and knowledge in this information age.

Final remarks (cont’d)

Computer and communication systems using quantum effects have remarkable properties. Quantum computers enable efficient simulation of the most

complex physical systems we can envision.Quantum algorithms allow efficient factoring of large

integers with applications to cryptography. Quantum search algorithms speedup considerably the

process of identifying patterns in apparently random data. We can guarantee the security of our quantum

communication systems because eavesdropping on a quantum communication channel can always be detected.

Final remarks (cont’d)

Building a quantum computer faces tremendous technological and theoretical challenges.

At the same time, we witness a faster rate of progress in quantum information theory where applications of quantum cryptography seem ready for commercialization. Recently, a successful quantum key distribution experiment over a distance of some 100 km has been announced.

Gallary

D-Wave's Quantum Computer

Summary

Quantum computing and quantum information theory is truly an exciting field.

It is too important to be left to the physicists alone….

Thank You

Technology

Introduction to Quantum Computing & Quantum Information Theory