the strange world abstract of quantum computing · pdf file2 lomonaco, samuel j., jr., a...

1

QuantumQuantumComputing ?Computing ?

Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr.Dept. of Comp. Sci. & Electrical EngineeringDept. of Comp. Sci. & Electrical Engineering

University of Maryland Baltimore CountyUniversity of Maryland Baltimore CountyBaltimore, MD 21250Baltimore, MD 21250

Email: Email: [email protected]@UMBC.EDUWebPageWebPage: : http://www.csee.umbc.edu/~lomonacohttp://www.csee.umbc.edu/~lomonaco

The Strange World The Strange World ofof

Quantum ComputingQuantum Computing

LL--OO--OO--P FundP FundTeleportation

AbstractAbstractThis talk will give an introductory

overview of quantum computing in an intuitive and conceptual fashion. No prior knowledge of quantum mechanics will be assumed.

A copy of the PowerPoint slides for this talk, as well others, can be found at:

www.csee.umbc.edu/~lomonaco/Lectures.html

Other PowerPoint talks on QC can be Other PowerPoint talks on QC can be found at:found at:

http://www.csee.umbc.edu/~lomonaco/Lectures.htmlhttp://www.csee.umbc.edu/~lomonaco/Lectures.html

LomonacoLomonaco LibraryLibrary

2

Lomonaco, Samuel J., Jr., Lomonaco, Samuel J., Jr., A Rosetta stone A Rosetta stone for quantum mechanics with an introduction to for quantum mechanics with an introduction to quantum computationquantum computation, in AMS PSAPM/58, , in AMS PSAPM/58, (2002), pages 3 (2002), pages 3 –– 65.65.

Talk ObjectiveTalk Objective

Communicate the fundamental Intuitionsinvolved in Quantum Computation (QC) and in Quantum Information Science (QIS).

This is an elementary, This is an elementary, not an advanced talknot an advanced talk

The Big PictureThe Big PictureThe Theory of Computation and Information Theory are being used to probe the boundaries of Quantum Mechanics (QM).

QMQM

Theory ofTheory ofComputationComputation

InformationInformationTheoryTheory

So far no “cracks” have appeared !!!

The Big PictureThe Big PictureThe Theory of Computation and Information Theory are being used to probe the boundaries of Quantum Mechanics (QM).An Example:Question: When is a physical system behaving Quantum Mechanically, i.e., non-classically ???

One Answer: When Simon’s Quantum Algorithm runs on the system in polytime.

Ergo, the system is behaving non-classically !!!

? ? ? Why ? ? ?? ? ? Why ? ? ?QuantumQuantum

ComputationComputation• Limits of small scale integrationLimits of small scale integrationtechnology to be reached 2010technology to be reached 2010--20202020

• No Longer !No Longer ! Moore’s LawMoore’s Law, i.e., every, i.e., everyyear & a half, double the computing poweryear & a half, double the computing powerat half the price. at half the price. No Longer !No Longer !

• A whole new industry will be built aroundA whole new industry will be built aroundthe new & emerging quantum technologythe new & emerging quantum technology

CollisionCollision CourseCourse

QuantumQuantumComputationComputation

MultiMulti--DisciplinaryDisciplinary

Math

CompSci EE

Physics

3

TheTheClassicalClassical

WorldWorld

ClassicalClassicalShannonShannon

BitBit

0 or 1

DecisiveIndividual

CopyingMachine

OutIn

ClassicalClassical BitsBits CanCan BeBe CopiedCopied

TheTheQuantumQuantum

WorldWorld

IntroducingIntroducingthethe QubitQubit

? ? ?? ? ?

Quantum BitQuantum BitQubitQubit

IndecisiveIndividual

Can be both 0 & 1at the same time !!!

Quantum Representations Quantum Representations of Qubitsof Qubits

ExampleExample 11.. A spinA spin-- particleparticle12

Spin Up Spin Down11

00

4

Quantum Representations Quantum Representations of Qubits (Cont.)of Qubits (Cont.)

ExampleExample 22.. Polarization States of a PhotonPolarization States of a Photon

1 0

1 0

or,

,

H =Where does a Qubit live ?Where does a Qubit live ?

HomeDef. A Hilbert Space is a vector space over together with an inner product such that

H , : H H

The elements of will be called The elements of will be called ketskets, and , and will be denoted bywill be denoted by label

H

1) & 1 2 1 2, , ,u u v u v u v 1 2 1 2, , ,vu u vu vu 2) , ,u v u v 3) , ,u v v u4) Cauchy seq in , 1 2, ,u u H lim nn

u

H

A A QubitQubit is a is a quantum quantum systemsystem whose whose statestate is is represented by a represented by a KetKetlying in a 2lying in a 2--D Hilbert D Hilbert SpaceSpace H

0 10 1

Superposition of StatesSuperposition of States

A typical Qubit is ???

where 2 20 1 1

The above Qubit is in a SuperpositionSuperposition of statesand

It is simultaneously both and !!!0 1

10

Amplitudes

“Collapse” of the Wave Function“Collapse” of the Wave Function

0 10 1

Observer

Qubit

i

Kets as Column Vectors overKets as Column Vectors over Let be a 2-D Hilbert space with orthonormal basis H

0 , 1

In this basis, each ket can be represented as a column vector. For example,

10

0

01

1

and

And in general, we have

0 10 1

1 0a

a b a bb

5

Tensor Product of Hilbert SpacesTensor Product of Hilbert SpacesThe tensor product of two Hilbert spaces and is the “simplest” Hilbert space such that the map

is bilinear, i.e., such that

HK

,h k h k H K H K

1 2 1 2

1 2 1 2

h h k h k h kh k k h k h kh k h k

We define the action of on as

h k h k h k H K

In other words,In other words,

is constructed in the simplest non-trivial way such that:H K

1 2 1 2

1 2 1 2

,

h h k h k h kh k k h k h kh k h k h k

Kronecker (Tensor) Product of MatricesKronecker (Tensor) Product of Matrices

and11 12

21 22

a aA

a a

11 12

21 22

b bB

b b

The Kronecker(tensor) product is defined as:1 1 1 2 1 1 1 2

1 1 1 22 1 2 2 2 1 2 2

1 1 1 2 1 1 1 22 1 2 2

2 1 2 2 2 1 2 2

b b b ba a

b b b bA B

b b b ba a

b b b b

11 11 11 12 12 11 12 12

11 21 11 22 12 21 12 22

21 11 21 12 22 11 22 12

21 21 21 22 22 21 22 22

a b a b a b a ba b a b a b a ba b a b a b a ba b a b a b a b

So …So …

1 001 0 1 0 1

0 10 0

11 1

000

01

Representing Integers in Quantum ComputationRepresenting Integers in Quantum Computation

Let be a 2-D Hilbert space with orthonormal basis

2H0 , 1

Then is a 2n-D Hilbert space with induced orthonormal basis

10 2n H H

0 00 , 0 01 , 0 10 , 0 11 , 1 11

where we are using the notational convention

1 2 1 0 1 2 1 0n n n nb b b b b b b b

Representing Integers in Quantum ComputationRepresenting Integers in Quantum Computation

So in the 2n-D Hilber space , with induced orthonormal basis

H

0 00 , 0 01 , 0 10 , 0 11 , 1 11 we represent the integer with binary expansion

m

1

02 , 0 1,n jj jj

m m m or j

as the ket1 2 1 0n nm m m m m

For example,23 010111

6

Indexing Convention for MatricesIndexing Convention for Matrices

The indices of matrices start at 0, not 1.For example, in 2 2 2 H H H

0 00 10 2

0 1 0 0 35 101

1 0 1 0 41 50 60 7

indexindexindexindexindexindexindexindex

The Qubit VillageThe Qubit Village

QubitvilleQubitvilleKetsKets

Each inEach in1 2, , n

1 2, , , nH H H• The Qubits in Qubit Village collectively live in

1 21

n

n jj

H H H H

• The populace of Qubit Village is

1 21

n

n jj

P opu lace

H

• Other names for the populace of Qubit Village

1 2 1 2n nPopulace

Massive ParallelismMassive Parallelism

Example. For , let

Then

1,2, ,j n 0 1

2j

1 21

0 12

n

nj

Therefore, the n-qubit register contains n-bit binary numbers simultaneously !

all

2 1

0

12

nn

aa

1 0 1 0 1 0 12

n

1 00 0 00 1 11 12

n

But But ! ! !! ! !1 2 n

a

ObserverObserver

U 1

Activities in Quantum VillageActivities in Quantum Village

All activities in Quantum Village are All activities in Quantum Village are UnitaryUnitarytransformationstransformations

At timeAt timet=0t=0

At timeAt timet=1t=1

HH HHU

T T

U U I UU

0

where a where a unitaryunitary transformation is one such thattransformation is one such that

MeasurementMeasurementConnectingConnecting

Quantum VillageQuantum Villageto theto the

Classical WorldClassical World

7

Another Activity in Quantum Village:Another Activity in Quantum Village:

MeasurementMeasurementAnother Activity in Quantum Village:Another Activity in Quantum Village:

MeasurementMeasurementMeasurementMeasurement

Group of Friendly PhysicistsGroup of Friendly Physicists

Another Activity in Quantum Village:Another Activity in Quantum Village:

MeasurementMeasurementMeasurementMeasurement

Group of Friendly PhysicistsGroup of Friendly PhysicistsGroup of Group of AngryAngry PhysicistsPhysicists

ObservablesObservables

What does our observer What does our observer actually observe ?actually observe ?

??????

Observables = Hermitian OperatorsObservables = Hermitian OperatorsH H

T wherewhere

, and let , and let denote the corresponding denote the corresponding eigenvalueseigenvaluesLet be the Let be the eigenketseigenkets of of

Observables (Cont.)Observables (Cont.)

What does our observer actually What does our observer actually observe ?observe ?

??????

i

i i i

i, i.e., , i.e.,

Observables (Cont.)Observables (Cont.)

What does our observer observe ?What does our observer observe ?

??????

So with probability , the observer So with probability , the observer observes the eigenvalue , andobserves the eigenvalue , and

The state of an nThe state of an n--Qubit register can Qubit register can be written in the eigenket basis asbe written in the eigenket basis as

i ii

i2

i ip

i

8

Example: Pauli Spin MatricesExample: Pauli Spin MatricesConsider the following observables, called the Consider the following observables, called the Pauli Spin Pauli Spin matricesmatrices::

1 2 3

0 1 0 1 0, ,

1 0 0 0 1i

i

which can readily be checked to be Hermitian.which can readily be checked to be Hermitian.

E.g., E.g., *

†2 2

0 0 00 0 0

T Ti i ii i i

The respective eigenvalues and eigenkets of these matrices are The respective eigenvalues and eigenkets of these matrices are listed in the table belowlisted in the table below

EigenvalueEigenvalue

0 1 / 2 0 1 / 2i 0

0 1 / 2 0 1 / 2i 1

1 23

1

1

Measurement ExampleMeasurement Example

Consider a 2Consider a 2--D quantum system in stateD quantum system in state, where , where 0 1a b 2 2 1a b

What happens if we measure w.r.t. observable ?What happens if we measure w.r.t. observable ?1First express in terms of the eigenket basis ofFirst express in terms of the eigenket basis of 1

Thus, if is observed Thus, if is observed w.r.tw.r.t. , either. , either1

First PossibilityFirst Possibility

EigenvalueEigenvalueis meas.is meas.1

0 1 / 2

2Prob / 2a b

0 1 0 12 2 2 2

a b a b

Second PossibilitySecond Possibility

EigenvalueEigenvalueis meas.is meas.

2Prob / 2a b

1 0 1 / 2

oror

Important Feature ofImportant Feature ofQuantum MechanicsQuantum Mechanics

It is important to mention that:It is important to mention that:

We cannot completelyWe cannot completelycontrol the outcome of control the outcome of quantum measurementquantum measurement

CopyingMachine

OutIn

CloningCloningThe NoThe No-- TheoremTheorem

CopyingMachine

OutIn

CloningCloningThe NoThe No-- TheoremTheorem

Dieks, Wootters, ZurekDieks, Wootters, Zurek

The No Cloning TheoremThe No Cloning Theorem

DefinitionDefinition.. Let be a Hilbert space. Then Let be a Hilbert space. Then a a quantumquantum replicatorreplicator consists of an consists of an auxiliary Hilbert space , a fixed state auxiliary Hilbert space , a fixed state

(called the (called the initialinitial statestate of of the replicator), and a unitary transformation the replicator), and a unitary transformation

AH

H

# A H

: A AU H H H H H H

states , where states , where (called the (called the replicatorreplicator statestate after replication ofafter replication of

) depends on .) depends on .

such that, for some fixed state ,such that, for some fixed state ,blank H

# aU a blank a a a H a A H

a a

forfor allall

9


•• Cloning is:Cloning is:

# @0 1 0 1 0 1a b blank a b a b

•• Cloning isCloning is NOTNOT::

# @0 1 00 11a b blank a b


•• CloningCloning is inherently is inherently nonnon--linearlinear

•• Quantum mechanicsQuantum mechanics is inherently is inherently linearlinear

•• ErgoErgo, , quantum replicatorsquantum replicators do not existdo not exist

Key IdeaKey Idea

IntroductionIntroductiontoto

Quantum EntanglementQuantum Entanglement

A Illustration of the A Illustration of the of Quantum Mechanicsof Quantum Mechanics

WeirdnessWeirdness

QubitsQubits

•• Not EntangledNot Entangled

•• SeparateSeparate

•• EntangledEntangled

•• Not Separate Not Separate !!

02

1 01 0 0

U

UnitaryUnitaryTransfTransf

Entangled Entangled

Observing Entangled QubitsObserving Entangled Qubits

02

1 01 Observe OnlyObserve Onlythe Blue Qubitthe Blue Qubit

0 1 1 0

•• No Longer EntangledNo Longer Entangled

•• Separate IdentitySeparate Identity

Whoosh !Whoosh !

Hidden Variable Theory Hidden Variable Theory vsvs Quantum MechanicsQuantum Mechanics

EPR PairEPR Pair

02

1 01

EinsteinEinsteinPodolskyPodolsky

RosenRosen

Something is Missing from Quantum Mechanics.Something is Missing from Quantum Mechanics.There Must Exist Hidden VariablesThere Must Exist Hidden Variables

10

Hidden Variable Theory Hidden Variable Theory vsvs Quantum MechanicsQuantum Mechanics

EPR PairEPR Pair

02

1 01


RosenRosen

Bah ! Humbug !Bah ! Humbug !Something is Missing from Quantum Mechanics.Something is Missing from Quantum Mechanics.

There Must Exist Hidden VariablesThere Must Exist Hidden VariablesBell InequalitiesBell Inequalities

Aspect ExperimentAspect Experiment

Score So FarScore So Far

•• HVT Score = 0HVT Score = 0

•• QM Score = 1QM Score = 1

Why didWhy did


RosenRosen

Object So Object So Vehemently ?Vehemently ?

Forces of Nature Are Local InteractionsForces of Nature Are Local Interactions

All the forces of nature (i.e., gravitational, All the forces of nature (i.e., gravitational, electromagnetic, weak, & strong forces) are electromagnetic, weak, & strong forces) are local interactions.local interactions.

•• Mediated by another entity, e.g., gravitons,Mediated by another entity, e.g., gravitons,photons, etc. photons, etc. •• Propagate no faster than the speed of lightPropagate no faster than the speed of light cc

•• Strength drops off with distanceStrength drops off with distance

Spacelike DistanceSpacelike DistanceHello !Hello ! Can’t HearCan’t HearYou !! ??You !! ??

1P 2P , , ,x y z t , , ,X Y Z T

Spacelike DistanceSpacelike Distance

1 2,Dist P P c T t

No signal can travel between No signal can travel between spacelikespacelike regions of regions of spacetimespacetime

Ergo, Ergo, spacelike regionsspacelike regions of space are of space are physically physically independentindependent, i.e., one cannot influence the other., i.e., one cannot influence the other.

•• The forces of nature are localThe forces of nature are localinteractionsinteractions

•• Spacelike regions of space areSpacelike regions of space arephysically independentphysically independent

The EPR PerspectiveThe EPR Perspective

All perfectlyAll perfectlyreasonablereasonable

assumptions !assumptions !

11

Alpha CentauriAlpha CentauriEarthEarth

Instantly,Instantly,Both QubitsBoth Qubits

Are Determined !Are Determined !

Spacelike DistanceSpacelike Distance

0 11( 20 ) /

0 1 1 0

Meaurement of EPR PairMeaurement of EPR PairBlue QubitBlue Qubit Red QubitRed Qubit

Meas. BlueMeas. BlueQubitQubit

No Local Interaction !No Local Interaction !

•• No force of any kindNo force of any kind-- Not mediated by anythingNot mediated by anything

•• Acts instantaneouslyActs instantaneously-- Faster than lightFaster than light

•• Strength does not drop off with distanceStrength does not drop off with distance-- Full strength at any distanceFull strength at any distance

Yet, still consistent with General Relativity !Yet, still consistent with General Relativity !

Quantum Entanglement Quantum Entanglement Appears to Pinpoint theAppears to Pinpoint the

of Quantum of Quantum MechanicsMechanicsWeirdnessWeirdness

•• Actions on StatesActions on StatesQuantum Computer InstructionsQuantum Computer Instructions

Properties of Qubits Properties of Qubits

•• Properties of StatesProperties of States•• QubitsQubits can exist in a superposition ofcan exist in a superposition ofstatesstates

•• QubitsQubits can be entangledcan be entangled

•• QubitsQubits “collapse” upon measurement“collapse” upon measurement

•• QubitsQubits are transformed by unitaryare transformed by unitarytransformationstransformations

Properties of Quantum Computer DataProperties of Quantum Computer Data

UsefulUsefulfor Quantum Computationfor Quantum Computation

The major obstacle to achieving the The major obstacle to achieving the promise of quantum computing ispromise of quantum computing is

But what is decoherence ?But what is decoherence ?

DecoherenceDecoherence

Decoherence Decoherence A quantum system A quantum system QQSysSys simply does not want to simply does not want to be isolated, but instead wants to entangle with be isolated, but instead wants to entangle with its environment its environment QQEnvEnv (as well as with itself)(as well as with itself)The more a The more a QQSysSys entangles with its environment entangles with its environment (and itself), the more (to one observing (and itself), the more (to one observing ONLYONLYthe the QSysQSys)) does it appear to become does it appear to become

Noisy & classically random (i.e., loses Noisy & classically random (i.e., loses coherence),coherence), and hence, uncontrollable.and hence, uncontrollable.

By this process, By this process, QQSysSys--qubitsqubits appear to the appear to the observer to be degenerating into random observer to be degenerating into random classical bitsclassical bitsWe call this phenomenonWe call this phenomenon DecoherenceDecoherence

12

QQEnvEnv

Decoherence Decoherence

QQSysSys

ObserverObserver

??????QuantumQuantum

TeleportationTeleportation

An Application of Quantum EntanglementAn Application of Quantum Entanglement

TeleportationTeleportation:: Transfering an object between Transfering an object between two locations by a process of:two locations by a process of:•• Dissociation to obtain infoDissociation to obtain info

-- Scanned to extract Scanned to extract suffsuff. Info. to. Info. torecreate originalrecreate original

•• Information TransmissionInformation Transmission

-- Exact replica is reExact replica is re--assembled at destinationassembled at destinationout of locally available materialout of locally available material

•• Reconstruction from infoReconstruction from info

NetNet EffectsEffects::•• Destruction of original objectDestruction of original object•• Creation of an exact replica at theCreation of an exact replica at theintended destination.intended destination.

Teleportation ?Teleportation ?Oxford Unabridged DictionaryOxford Unabridged Dictionary Asked Scotty about TeleportationAsked Scotty about Teleportation

Beam me up, Scotty !Beam me up, Scotty !

Asked Scotty about TeleportationAsked Scotty about Teleportation

Beam me up, Scotty !Beam me up, Scotty !

Aye, Aye, Captain !Aye, Aye, Captain !

I’m just a wee bit busy. I’m just a wee bit busy.

Federation Quantum Teleportation Manual

Step 1. (Loc. A) Preparation

• Alice at location A constructs an EPR pair of qubits (qubits #2 & #3) in 2 3H H

00 UnitaryMatrix

01 102

• Alice arranges for a courier totransport entangled qubit #3 to Bob atlocation B.

13

Federation Quantum Teleportation ManualResult

• Alice at location A shares an EPR pairwith Bob at location B

- Qubit # 2 is with Alice at location A

- Qubit # 3 is with Bob at location B

- Qubits # 2 & #3 are entangled


Step 2. Qubit #1 is delivered to Alice

The state of all three qubits is now:

1 2 3

01 100 1

2a b

H H H

Unknown

• The Bell Basis of is

1

2

3

4

10 10 / 2

10 10 / 2

00 11 / 2

00 11 / 2

1 2H H

A Little Algebraic Manipulation

1 2 3

01 100 1

2a b

H H H

• Recall that the current state of all three qubits is:

A Little Algebraic Manipulation

• Re-express in terms of the Bell Basis as:

1

2

3

4

1 0 12

0 1

1 0

1 0

a b

a b

a b

a b

• Let be the unitarytransformation:

1 2 1 2:U H H H H

1 00

4 11 3 10 2 01


Step 3. (Loc. A) Apply

to the three qubits. Thus, under , the state of all three qubits becomes:

1 2 3 1 2 3:U I H H H H H H

U I

1 00 0 12

01 0 1

10 1 0

11 1 0

a b

a b

a b

a b


Step 4. (Loc. A) Measure qubits #1 & #2

1 00 0 12

01 0 1

10 1 0

11 1 0

a b

a b

a b

a b

Step 5. (Loc. A) Send via a classicalcommunication channel the result toBob at Loc. B

Result: Two bits of information ,i j

14

Federation Quantum Teleportation ManualStep 6. (Loc. B) Use the two classical bits

of received information to select a unitary transformation of from the table:

Step 7. (Loc. B) Apply the selected unitary transformation to qubit #3.( , )i jU

3H( , )i jU

Rec. Bits Effect on Qubit #300011011

( , )i jU0 1 0 1a b a b (0,0)U

(0,1)U(1,0)U(1,1)U

0 1 0 1a b a b

1 0 0 1a b a b

1 0 0 1a b a b

,i j


Result

Qubit #3 at Loc. B now has the state that Qubit #1 originally had at Loc. A before it was disassembled, i.e.,

0 1a b

Even so, the teleported state is still unknown to Alice & Bob !

MoreMoreDiracDirac

NotationNotation

The Deutsch-Jozsa Algorithm

Unitary transfs revisited

More Dirac NotationMore Dirac Notation

LetLet * ,HomH H

Hilbert SpaceHilbert Spaceof morphismsof morphismsfrom tofrom toH

We call the elements of We call the elements of BraBra’s, and ’s, and denote them asdenote them as

*H

label

More Dirac NotationMore Dirac Notation

There is a There is a dualdual correspondencecorrespondence between and between and *H H

KetKetBraBra

There exists a bilinear mapThere exists a bilinear mapdefined bydefined by

which we more simpy denote by which we more simpy denote by

* H H 1 2

1 2|

†

BraBra--cc--KetKet

Bra’s as Row Vectors over Bra’s as Row Vectors over Let be a 2Let be a 2--D Hilbert space with D Hilbert space with orthonormal basis orthonormal basis 0 , 1

H

* ,HomH H

0 , 1

and letand let

be the corresponding dual Hilbert space with be the corresponding dual Hilbert space with corresponding dual basiscorresponding dual basis

and0 1,0 1 0,1

0 1 ,a b a b

Then with respect to this basis, we haveThen with respect to this basis, we have

15

Bra’s & Ket’s as Adjoints of One Another Bra’s & Ket’s as Adjoints of One Another

The dual correspondence The dual correspondence

is given byis given by

and is called the and is called the adjointadjoint

†*H H

†

0 1 0 1 ,a

a b a b a bb

1

2

0 10 1a bc d

1 2| 0 1 0 1

,

a b c dc

a b ac bdd

IfIf

then the bracket product becomesthen the bracket product becomes

as a Matrix Outerproductas a Matrix Outerproduct1 2

1

2

0 10 1a bc d

1 2

1 2 |

H H

IfIf

then is the linear transformation then is the linear transformation 1 2

which, when written in matrix notation, becomes the which, when written in matrix notation, becomes the matrix matrix outerproductouterproduct

1 2 ,a ac ad

c db bc bd

Let be an NLet be an N--D Hilbert space with orthonormal D Hilbert space with orthonormal basis basis

If we use the convention that If we use the convention that matrix indices begin matrix indices begin atat 00, then the matrix of the linear transformation, then the matrix of the linear transformation

is an is an NxNNxN matrix consisting of all zeroes with the matrix consisting of all zeroes with the exception of entry exception of entry (m,k)(m,k) which is which is 11

For example if For example if N=4N=4, then , then

H0 , 1 , , 1N

m k

0 0 0 00 0 0 0

2 30 0 0 10 0 0 0

Entry Entry (2,3)(2,3)

UnitaryUnitaryTransformationsTransformations

RevisitedRevisited

Dynamic Behavior of Q. Sys.Dynamic Behavior of Q. Sys.

The dynamic behavior of a quantum system is determined by Schroedinger’s equation.

0( )U t

Schroedinger’sEquation

i Ht

IN

OUT

where is time, and where is a curve in the group of unitary transformations on the state space .

t ( )U t( )U H

H

0

Initial State

H

Hamiltonian

Dynamic State

Observable

16

An observable H is just the tangent field to the curve in the group of unitary transformations on the state space .

An observable is a Hermitian operator on the state space , i.e., a linear transformation such that

† T

H

( )U t ( )U HH

Observable

MeasurementMeasurementRevisitedRevisited

Spectral DecompositionSpectral DecompositionLet be an observable, i.e., a Hermitian operator.

Eigenvalues … n

Eigenspaces V1 V2 … Vn

Projection Ops. P1 P2 … Pn

where is the projection operator corresponding to the eigenspace

:j jP VH

jV

Spectral Decomposition Theorem

1 1 2 2 n nP P P

Quantum MeasurementQuantum Measurement

InIn OutOut

j

jj

j

PP

BlackBoxBlackBox

MacroWorldMacroWorld

QuantumQuantumWorldWorld

EigenvalueEigenvalueObservableObservable

Q. Sys.Q. Sys.StateState


Pr job P

j jjP wherewhere Spectral DecompositionSpectral Decomposition

PhysicalPhysicalRealityReality

PhilosopherPhilosopherTurfTurf

Density OperatorsDensity Operators&&

Mixed EnsemblesMixed Ensembles

Skip to D-J

Quit

Two Ways to Represent Quantum StatesTwo Ways to Represent Quantum States

Density OperatorsDensity Operators

KetsKets

&&

17


ExampleExample.. We have seen We have seen pure ensemblespure ensembles, i.e., , i.e., pure states, such aspure states, such as

ProblemProblem.. Certain types of quantum states Certain types of quantum states are difficult to represent in terms of ketsare difficult to represent in terms of kets

KetKet

ProbProb1


ExampleExample.. Consider the following state for Consider the following state for which we have incomplete knowledge, called a which we have incomplete knowledge, called a mixed ensemblemixed ensemble::

wherewhere

KetKet

ProbProb11p

2 k2p kp

1 2 1kp p p

All unit LengthAll unit Length& not nec.& not nec.

KetKet

ProbProb11p

2 k2p kp


Johnny von Neumann suggested that we use the Johnny von Neumann suggested that we use the following operator to represent a state:following operator to represent a state:

1 1 1 2 2 2 k k kp p p is called a is called a density operatordensity operator. It is a . It is a Hermitian Hermitian

positive semipositive semi--definite operator of tracedefinite operator of trace 11..

MixedMixedEnsembleEnsemble

For the pure ensemble , For the pure ensemble , KetKet

ProbProb 11

1


If for example,If for example,

0 1a b wherewhere

2 2 1a b

thenthen

2

2

0 1 0 1a b a b

a abaa b

b ba b

3/8 3 /80 1 0 13 11 14 4 3 /8 5/82 2

ii ii


On the other hand, On the other hand,

is the mixed ensembleis the mixed ensembleKetKet

ProbProb

1 0 1 / 2i 2 1 34

14

1 11p 2p2 2

Quantum Mechanics from the Two PerspectivesQuantum Mechanics from the Two Perspectives

KetsKets Density OpsDensity Ops

Schroed.Schroed.Eq.Eq.

UnitaryUnitaryEvolutionEvolution

ObservationObservation

i Ht

| |A A A trace A

,i Ht

U †U U

18

Quantum MeasurementQuantum Measurement

InIn OutOut

j

j j

jj

P Ptr P

BlackBoxBlackBox

MacroWorldMacroWorld

QuantumQuantumWorldWorld

EigenvalueEigenvalueObservableObservable



Pr job tr P

j jjP wherewhere Spectral DecompositionSpectral Decomposition

PhysicalPhysicalRealityReality

PhilosopherPhilosopherTurfTurf

•• We now have a more powerful way ofWe now have a more powerful way ofrepresenting quantum states.representing quantum states.

•• Density operators are absolutelyDensity operators are absolutelycrucial when discussing and dealingcrucial when discussing and dealingwith quantum noise and quantumwith quantum noise and quantumdecoherence.decoherence.

Density OperatorsDensity Operators

Deutsch’s AlgorithmDeutsch’s Algorithm

Quit

The The HadamardHadamard TranformationTranformation

0 0 1 / 2

1 0 1 / 2

1 111 12

H

HH

Deutsch’s AlgorithmDeutsch’s Algorithm

DefinitionDefinition. A . A coincoin is is fairfair (or (or balancedbalanced) if it has ) if it has heads on one side and tails on the other side. It is heads on one side and tails on the other side. It is unfairunfair (or (or constantconstant) if either it has tails on both ) if either it has tails on both sides, or heads on both sides.sides, or heads on both sides.

HH

HHHH HH

TT TTTT

TT

Side1Side1 Side2Side2

Fair (Balanced)Fair (Balanced)


Fair (Balanced)Fair (Balanced)


Unfair (Constant)Unfair (Constant)


Unfair (Constant)Unfair (Constant)

ObservationObservation

ObservationObservation.. In the classical world, we need In the classical world, we need to observe both sides of the coin to determine to observe both sides of the coin to determine whether or not it is fair ?whether or not it is fair ?

But what about in But what about in the quantum world ?the quantum world ?

19

We represent a coin mathematically as a We represent a coin mathematically as a Boolean function: Boolean function:

: 0, 1 0, 1f

Side1Side1 Side2Side2 HH TT

The Unitary Implementation of The Unitary Implementation of

( )

fU

x y x f x yH H

Let be the unitary transformation fU

then

x0 1

2

( ) 0 112

f x x fU

f

Ancilla

Moreover, Moreover,

0

1 1fU

HH

HH

HH

HH

(0) (1)1 1 02

f f

(0) (1)1 1 12

f f

(0) (1)1 1 0 1 0 1 1 0 12

Outputf f

(0) (1)1 10 0 1 1 1Outpu 1t 12

f f

CaseCase 11. is . is fairfair, i.e., , i.e., balancedbalancedf

CaseCase 11. is . is unfairunfair, i.e., , i.e., constantconstantf

So …So …

If we only make one observation, i.e., if we If we only make one observation, i.e., if we observe the left register, then we can observe the left register, then we can determine whether or not is fair or determine whether or not is fair or unfair.unfair.

f

Weird !Weird !

the strange world abstract of quantum computing · pdf file2 lomonaco, samuel j., jr., a...

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