quantum parallelism and the exact simulation of physical systems

Quantum Parallelism and the Quantum Parallelism and the Exact Exact

Simulation of Physical SystemsSimulation of Physical Systems

Dan Cristian Marinescu Dan Cristian Marinescu School of Computer Science School of Computer Science

University of Central FloridaUniversity of Central Florida

Orlando, Florida 32816Orlando, Florida 32816, , USAUSA

Computing Frontiers, Ischia, April 14, Computing Frontiers, Ischia, April 14, 20042004

22

Frontier(s)…Frontier(s)…from Webster’s unabridged from Webster’s unabridged dictionary.dictionary.

The part of a settled or civilized country The part of a settled or civilized country nearest to an unsettled or uncivilized nearest to an unsettled or uncivilized region.region.

Any new or incompletely investigated field Any new or incompletely investigated field of learning or thought.of learning or thought.


33

What is a Quantum computer?What is a Quantum computer? A device that harnesses quantum physical A device that harnesses quantum physical

phenomena such as phenomena such as entanglemententanglement and and superpositionsuperposition. .

The laws of quantum mechanics differ The laws of quantum mechanics differ radically from the laws of classical physics. radically from the laws of classical physics.

The unit of information, the The unit of information, the qubit qubit can exist can exist as a 0, or 1, or, simultaneously, as both 0 as a 0, or 1, or, simultaneously, as both 0 and 1. and 1.


44

Does quantum computing Does quantum computing represent the represent the

frontiers of computing? frontiers of computing? Is it for real? Can we actually build quantum Is it for real? Can we actually build quantum

computers? - computers? - Very likely, but it will take some Very likely, but it will take some time….time….

If so, what would a quantum computer allow us to If so, what would a quantum computer allow us to do that is either unfeasible or impractical with do that is either unfeasible or impractical with today’s most advanced systems? – today’s most advanced systems? – Exact Exact simulation of physical systems, among other things.simulation of physical systems, among other things.

Once we have quantum computers do we need new Once we have quantum computers do we need new algorithms? – algorithms? – Yes, we need quantum algorithms. Yes, we need quantum algorithms.

Is it so different from our current thinking that it Is it so different from our current thinking that it requires a substantial change in the way we requires a substantial change in the way we educate our students? – educate our students? – Yes, it does. Yes, it does.


55

Quantum computers: now and Quantum computers: now and thenthen

All we have at this time is a All we have at this time is a 7 (seven) 7 (seven) qubit quantum computerqubit quantum computer able to compute able to compute the prime factors of a small integer, 15. the prime factors of a small integer, 15.

To break a code with a key size of 1024 To break a code with a key size of 1024 bits requires bits requires more than 3,000 qubits and more than 3,000 qubits and 108 quantum gates.108 quantum gates.


66

Approximate computer simulation Approximate computer simulation of physical systemsof physical systems

Eniac and the Manhattan project. The first Eniac and the Manhattan project. The first programs to run, simulation of physical processes. programs to run, simulation of physical processes.

Computer simulation – new approach to scientific Computer simulation – new approach to scientific discovery, complementing the two well established discovery, complementing the two well established methods of science: experiment and theory. methods of science: experiment and theory.

Approximate simulation – based upon a Approximate simulation – based upon a modelmodel that that abstracts some properties of interest of a physical abstracts some properties of interest of a physical system. system.


77

Exact simulation of physical Exact simulation of physical systemssystems

How far do we want to go at the microscopic How far do we want to go at the microscopic level? Molecular, atomic, quantum? - level? Molecular, atomic, quantum? - All of the All of the above. above.

What about cosmic level? What about cosmic level? - Yes, of course. - Yes, of course. Is it important? - Is it important? - - Yes (Feynman, 1981) . - Yes (Feynman, 1981) . Who will benefit? – Who will benefit? –

Natural sciences Natural sciences physics, chemistry, biology, physics, chemistry, biology, astrophysics, cosmology,….astrophysics, cosmology,….

Application Application nanotechnology, smart materials, drug nanotechnology, smart materials, drug design,…design,…


88

Large problem state spaceLarge problem state space From black hole thermodynamics – a system enclosed From black hole thermodynamics – a system enclosed

by a surface with area A has a number of observable by a surface with area A has a number of observable states:states:

c = 3 x10c = 3 x101010 cm/sec cm/sech = 1.054 x 10h = 1.054 x 10-34-34 Joules/second Joules/secondG = 6.672 x 10G = 6.672 x 10-8 -8 cmcm33 g g-1-1 sec sec-2-2

For an object with a radius of 1 Km For an object with a radius of 1 Km N(A) = e N(A) = e8080


99

AcknowledgmentsAcknowledgments

Some of the material presented is from the bookSome of the material presented is from the book

Approaching Quantum ComputingApproaching Quantum Computing by Dan C. Marinescu and Gabriela M. Marinescuby Dan C. Marinescu and Gabriela M. Marinescu

to be published by Prentice Hall in June 2004to be published by Prentice Hall in June 2004

Work supported by National Science Foundation Work supported by National Science Foundation grants MCB9527131, DBI0296107,ACI0296035, and grants MCB9527131, DBI0296107,ACI0296035, and EIA0296179.EIA0296179.


1010

ContentsContents

Computing and the Laws of PhysicsComputing and the Laws of Physics Quantum Mechanics & ComputersQuantum Mechanics & Computers Qubits and Quantum GatesQubits and Quantum Gates Quantum ParallelismQuantum Parallelism Deutsch’s AlgorithmDeutsch’s Algorithm Virus Structure Determination and Drug Virus Structure Determination and Drug

Design Design SummarySummary


1111

The limits of solid-state technologiesThe limits of solid-state technologies

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

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

Reliability will also be affectedReliability will also be affected to increase the speed we need increasingly smaller to increase the speed we need increasingly smaller

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

particles are subject to particles are subject to Heissenberg uncertaintyHeissenberg uncertainty


1212

Power dissipation and circuit densityPower dissipation and circuit density

The computer technology vintage year 2000 The computer technology vintage year 2000 requires some requires some 3 x 103 x 10-18-18 Joules per elementary Joules per elementary operation. operation.

In 1992 Ralph Merkle from Xerox PARC calculated In 1992 Ralph Merkle from Xerox PARC calculated that a that a 1 GHz computer operating at room 1 GHz computer operating at room temperature, with 10temperature, with 101818 gates packed in a volume gates packed in a volume of about 1 cmof about 1 cm3 3 would dissipate 3 MW of power.would dissipate 3 MW of power. A small city with 1,000 homes each using 3 KW would A small city with 1,000 homes each using 3 KW would

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

such circuits.such circuits.


1313

Energy consumption of a logic circuit

Speed of individual logic gates

S

E

(a) (b)

Heat removal for a circuit with densely packedlogic gates poses tremendous challenges.


1414

ContentsContents

Computing and the Laws of PhysicsComputing and the Laws of Physics Quantum Mechanics & ComputersQuantum Mechanics & Computers Qubits and Quantum GatesQubits and Quantum Gates Quantum ParallelismQuantum Parallelism Deutsch’s AlgorithmDeutsch’s Algorithm Virus Structure Determination and Drug Virus Structure Determination and Drug

Design Design SummarySummary


1515

A happy marriage…A happy marriage…

The two greatest discoveries of the 20-th The two greatest discoveries of the 20-th centurycentury quantum mechanicsquantum mechanics stored program computersstored program computers

led to the idea of led to the idea of

quantum computingquantum computing andand

quantum information theoryquantum information theory


1616

Quantum; Quantum mechanicsQuantum; Quantum mechanics

QuantumQuantum Latin word meaning some Latin word meaning some quantity. In physics it is used with the same quantity. In physics it is used with the same meaning as the word meaning as the word discrete discrete in in mathematics. mathematics.

Quantum mechanics Quantum mechanics a mathematical a mathematical model of the physical world.model of the physical world.


1717

Heissenberg’s uncertainty principleHeissenberg’s uncertainty principle

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


1818

Milestones in quantum computingMilestones in quantum computing 1961 - 1961 - Rolf LandauerRolf Landauer decrees that computation is physical decrees that computation is physical

and studies heat generation.and studies heat generation. 1973 - 1973 - Charles BennetCharles Bennet studies the logical reversibility of studies the logical reversibility of

computations. computations. 1981 - 1981 - Richard FeynmanRichard Feynman suggests that physical systems suggests that physical systems

including quantum systems can be simulated exactly with including quantum systems can be simulated exactly with quantum computers. quantum computers.

1982 - 1982 - Peter BeniofPeter Beniof develops quantum mechanical models develops quantum mechanical models of Turing machines.of Turing machines.

1984 - 1984 - Charles BennetCharles Bennet and and Gilles BrassardGilles Brassard introduce introduce quantum cryptography.quantum cryptography.

1985 - 1985 - David DeutschDavid Deutsch reinterprets the Church-Turing reinterprets the Church-Turing conjecture.conjecture.

1993 - 1993 - Bennet, Brassard, Crepeau, Josza, Peres, WootersBennet, Brassard, Crepeau, Josza, Peres, Wooters discover quantum teleportation.discover quantum teleportation.

1994 - 1994 - Peter ShorPeter Shor develops a clever algorithm for factoring develops a clever algorithm for factoring large integers. large integers.


1919

Deterministic versus probabilistic photon Deterministic versus probabilistic photon behaviorbehavior

(b)(a)

D1

D2

D3

D5

D7

Detector D1

Detector D2

Beam splitter

Incident beam of light

Reflected beam

Transmitted beam


2020

(a)

(b)

S

E

(c)

E

S

A

intensity = I intensity = I/2

(d)

E

S

A B

intensity = 0

(e)

E

S

A B

intensity = I/8

C

0|

1

|


2121

ContentsContents

Computing and the Laws of PhysicsComputing and the Laws of Physics Quantum Mechanics & ComputersQuantum Mechanics & Computers Qubits and Quantum GatesQubits and Quantum Gates Quantum ParallelismQuantum Parallelism Virus Structure Determination and Drug Virus Structure Determination and Drug

DesignDesign SummarySummary


2222

One qubitOne qubit Mathematical abstractionMathematical abstraction Vector in a two dimensional complex vector space Vector in a two dimensional complex vector space

(Hilbert space)(Hilbert space) Dirac’s notationDirac’s notation

ket ket column vector column vector bra bra row vector row vector

bra bra dual vector (transpose and complex conjugate) dual vector (transpose and complex conjugate)

|


2323

State descriptionState description

O

1V

(a)

= q

O

1V

(b)

0 0

1 1q1

q0

q1 = q

q045o

30o


2424

b

| 0 >

| 1 >

z

x

y

ψ|

r


2525

A bit versus a qubitA bit versus a qubit

A bitA bit Can be in two distinct states, 0 and 1Can be in two distinct states, 0 and 1 A measurement does not affect the stateA measurement does not affect the state

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

state that is a linear combination of the basis statestate that is a linear combination of the basis state When we measure the qubit we find itWhen we measure the qubit we find it

in state with probability in state with probability in state with probability in state with probability

0| 1|10 10

0|1|

20 ||2

1 ||


2626

0

1

0

1

(a) One bit (b) One qubit

Superposition states

Basis (logical) state 1

Basis (logical) state 0


2727

Qubit measurementQubit measurement0

1

Possible states of one qubit beforethe measurement

The state of the qubit afterthe measurement

p1

p0


2828

Two qubitsTwo qubits

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

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

with probabilities with probabilities

11,10,01,00

11,10,01,00

211

210

201

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


2929

Two qubitsTwo qubits

11100100 11100100

1|||||||| 211

210

201

200


3030

Measuring two qubitsMeasuring two qubits

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

After the measurement the state is certainAfter the measurement the state is certain, it is , it is

00, 01, 10, or 11 like in the case of a classical 00, 01, 10, or 11 like in the case of a classical two bit system.two bit system.


3131

Measuring two qubits (cont’d)Measuring two qubits (cont’d)

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

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

1 with probability1 with probability

201

200 ||||

211

210 ||||


3232


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

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

201

200

01000

||||

0100

I

211

210

11011

||||

1101

I

I0

I1


3333


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

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

210

200

10000

||||

1000

II

II0

II1

211

201

11011

||||

1110

II


3434

Bell states - a special state of a pair of Bell states - a special state of a pair of qubitsqubits

If andIf and

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

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

2

11100 01001

11|1I 00|0

I

00|0II 11|1

II


3535

This is an amazing result!This is an amazing result!

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

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


3636

Entanglement (Entanglement (Verschrankung) Verschrankung)

Discovered by Schrodinger.Discovered by Schrodinger. An An entangled pairentangled pair is a single quantum is a single quantum

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

Einstein called entanglement “Einstein called entanglement “Spooky Spooky action at a distanceaction at a distance".".


3737

Classical gatesClassical gates

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

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

information.information.


3838

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


3939

One-qubit gate

10 10

G

2221

1211

gg

ggG

1

0

2221

1211

1

0

gg

gg

10 '1

'0


4040

One qubit gatesOne qubit gates

I I identity gate; leaves a qubit identity gate; leaves a qubit unchanged.unchanged.

X or NOT gateX or NOT gate transposes the transposes the components of an input qubit.components of an input qubit.

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


4141

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

Identity transformation, Pauli matrices, Identity transformation, Pauli matrices, HadamardHadamard


4242

CNOT a two qubit gateCNOT a two qubit gate

Two inputsTwo inputs Control Control TargetTarget

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

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

addition modulo 2+O

Target input

Control input

O+


4343

CNOT

CNOTV

0100

1000

0010

0001

CNOTG

CNOTCNOTCNOT VGW


4444

The two input qubits of a two qubit The two input qubits of a two qubit gatesgates

10 10

10 10

11

01

10

00

1

0

1

0

CNOTV


4545

State space dimension of classical and State space dimension of classical and quantum systemsquantum systems

Individual state spaces of n particles combine Individual state spaces of n particles combine quantum mechanically through the tensor product. If quantum mechanically through the tensor product. If X and Y are vectors, then X and Y are vectors, then their tensor product X Y is also a vector, but its their tensor product X Y is also a vector, but its

dimension is:dimension is: dim(X) x dim(Y) dim(X) x dim(Y) while the vector product X x Y has dimension while the vector product X x Y has dimension dim(X)+dim(Y). dim(X)+dim(Y).

For example, if dim(X)= dim(Y)=10, then the tensor For example, if dim(X)= dim(Y)=10, then the tensor product of the two vectors has dimension 100 while product of the two vectors has dimension 100 while the vector product has dimension 20.the vector product has dimension 20.


4646

Parallelism and Quantum computersParallelism and Quantum computers

In quantum systems In quantum systems the amount of the amount of parallelism increases exponentially with the parallelism increases exponentially with the size of the systemsize of the system, thus with the number of , thus with the number of qubits (e.g. a 21 qubit quantum computer qubits (e.g. a 21 qubit quantum computer is twice as powerful as a 20 qubit quantum is twice as powerful as a 20 qubit quantum computer). computer).

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


4747

ContentsContents

Computing and the Laws of PhysicsComputing and the Laws of Physics Quantum Mechanics & Quantum ComputersQuantum Mechanics & Quantum Computers Qubits and Quantum GatesQubits and Quantum Gates Quantum ParallelismQuantum Parallelism Deutsch’s AlgorithmDeutsch’s Algorithm Virus Structure Determination and Drug Virus Structure Determination and Drug



4848

A quantum circuitA quantum circuit

Given a function f(x) we can construct a reversible Given a function f(x) we can construct a reversible quantum circuit consisting of Fredking gates only, quantum circuit consisting of Fredking gates only, capable of transforming two qubits as followscapable of transforming two qubits as follows

The function f(x) is hardwired in the circuitThe function f(x) is hardwired in the circuit

| x > | x >

| y >

Uf

| y o f(x )>+


4949

A quantum circuit (cont’d)A quantum circuit (cont’d)

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

| x > | x >

| 0 >

Uf

| f(x )>


5050

A quantum circuit (cont’d)A quantum circuit (cont’d)

Now apply the first qubit through a Hadamad gate.Now apply the first qubit through a Hadamad gate.

The resulting sate of the circuit isThe resulting sate of the circuit is

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

| 0 >

Uf

H| 0 > | 0 >

)2

10(0

f

2

)1(1)0(0 ff

2

10


5151

Quantum parallelismQuantum parallelism

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

Quantum parallelism allows us to Quantum parallelism allows us to construct the construct the entire truth table of a quantum gate array having entire truth table of a quantum gate array having 2n entries at once2n entries at once. In a classical system we can . In a classical system we can compute the truth table in one time step with 2compute the truth table in one time step with 2nn gate arrays running in parallel, or we need 2gate arrays running in parallel, or we need 2nn time time steps with a single gate array.steps with a single gate array.

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


5252

Uf

| x > | x >

| y > | y O f(x) >+

(m-dimensional)

(k-dimensional) (n=m+k)-dimensional)


5353

2

100

H

)]10()10()10[(2

1

n

12

02

1n

xn

x

000)( HHH

12

0

12

0

12

0

))(,2

1)0,(

2

1)0,

2

1(

nnn

xn

xfn

xnf xfxxUxU


5454

ContentsContents

Computing and the Laws of PhysicsComputing and the Laws of Physics Quantum Mechanics and ComputersQuantum Mechanics and Computers Qubits and Quantum GatesQubits and Quantum Gates Quantum ParallelismQuantum Parallelism Deutsch’s AlgorithmDeutsch’s Algorithm Virus Structure Determination and Drug Virus Structure Determination and Drug



5555

Deutsch’s problemDeutsch’s problem Consider a black box characterized by a Consider a black box characterized by a

transfer function that maps a single input bit x transfer function that maps a single input bit x into an output, f(x). It takes the same amount of into an output, f(x). It takes the same amount of time, T, to carry out each of the four possible time, T, to carry out each of the four possible mappings performed by the transfer function mappings performed by the transfer function f(x) of the black box:f(x) of the black box:

f(0) = 0 f(0) = 0 f(0) = 1f(0) = 1 f(1) = 0 f(1) = 0 f(1) = 1f(1) = 1 The problem posed is to distinguish if The problem posed is to distinguish if )1()0(

)1()0(

ff

ff


5656

0 f(0) 1 f(1)

2T

0 f(0)

1 f(1)

T

(a) (b)

O+

| x >

| y > f(x) >

| x >

| y >

T(c)

Uf


5757

| y > + f(x)

Uf

| x >

| y >

| x >

O

| 0 > H

H| 1 >

H

0 1 2 3

A quantum circuit to solve Deutsch’s A quantum circuit to solve Deutsch’s problemproblem


5858

1111

1111

1111

1111

2

1

11

11

2

1

11

11

2

11 HHG

2

10

2

10)11100100(

2

11

1

1

1

1

2

1

0

0

1

0

1111

1111

1111

1111

2

1011 G

0

0

1

0

1

0

0

1100

2

10 x

2

10 y


5959

2

)(1)(

2

)(1)(0 xfxfxfxf

)(2

10)( xfxfy

2

10)1()( )(

xfxfy

1)(

2

10

0)(2

10

)(

xfif

xfifxfy


6060

1)1()0(

1

1

1

1

2

1

2

10

2

10

0)1()0(

1

1

1

1

2

1

2

10

2

10

))((2

ffif

ffif

xfyx

0)1(,1)0(

1

1

1

1

2

1

2

10

2

10

1)1(,0)0(

1

1

1

1

2

1

2

10

2

10

))((2

ffif

ffif

xfyx

2

10 x

2

10 y


6161

)1()0(

1

1

1

1

2

1

)1()0(

1

1

1

1

2

1

))((2

ffif

ffif

xfyx


6262

1010

0101

1010

0101

2

1

10

01

11

11

2

13 IHG

)1()0(2

101

1

1

0

0

2

1

1

1

1

1

2

1

1010

0101

1010

0101

2

1

)1()0(2

100

0

0

1

1

2

1

1

1

1

1

2

1

1010

0101

1010

0101

2

1

233

ffif

ffif

G


6363

Evrika!!Evrika!!

By measuring the first output qubit qubit we are By measuring the first output qubit qubit we are able to determine performing a able to determine performing a single evaluation.single evaluation.

)1()0(1

)1()0(0)1()0(

ffif

ffifff

2

10)1()0(3

ff

)1()0( ff


6464

ContentsContents

Computing and the Laws of PhysicsComputing and the Laws of Physics Quantum Mechanics and ComputersQuantum Mechanics and Computers Qubits and Quantum GatesQubits and Quantum Gates Quantum ParallelismQuantum Parallelism Deutsch’s AlgorithmDeutsch’s Algorithm Virus Structure Determination and Drug Virus Structure Determination and Drug



6565

Sindbis virus reconstruction and pseudo-atomic modeling. The reconstruction computed at ~11Å (top half of first two panels) and ~22Å (bottom half of first two panels), viewed along a two-fold axis and represented as a surface-shaded solid (left panel) and as a thin, non-equatorial section (middle panel). The 11Å reconstruction, which shows significantly greater detail compared to that in the 22Å map available approximately one year ago, provides more accurate data for fitting atomic models as illustrated in the right panel, which is an enlarged view of the boxed area in the middle panel.


6666

Final remarksFinal remarks

A tremendous progress has been made in A tremendous progress has been made in quantum computing and quantum quantum computing and quantum information theory during the past decade. information theory during the past decade.

Motivation Motivation the incredible impact this the incredible impact this discipline could have on how we store, discipline could have on how we store, process, and transmit data and knowledge in process, and transmit data and knowledge in this information age.this information age.


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Final remarks (cont’d)Final remarks (cont’d)

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

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

integersintegers with applications to cryptography. with applications to cryptography. Quantum search algorithmsQuantum search algorithms speedup considerably the speedup considerably the

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

communication systems communication systems because eavesdropping on a because eavesdropping on a quantum communication channel can be detected with quantum communication channel can be detected with high probability.high probability.


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SummarySummary

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

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

quantum parallelism and the exact simulation of physical systems

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