Quantum Computing Quantum Computing be found at: be found at: - - PDF document

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Quantum Quantum Computing ? Computing ?

Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr.

• Dept. of Comp. Sci. & Electrical Engineering
• Dept. of Comp. Sci. & Electrical Engineering

University of Maryland Baltimore County University of Maryland Baltimore County Baltimore, MD 21250 Baltimore, MD 21250 Email: Email: Lomonaco@UMBC.EDU Lomonaco@UMBC.EDU WebPage: WebPage: http://www.csee.umbc.edu/~lomonaco http://www.csee.umbc.edu/~lomonaco

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Quantum Computing Quantum Computing

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

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

Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr.

• Dept. of Comp. Sci. & Electrical Engineering
• Dept. of Comp. Sci. & Electrical Engineering

University of Maryland Baltimore County University of Maryland Baltimore County Baltimore, MD 21250 Baltimore, MD 21250 Email: Email: Lomonaco@UMBC.EDU Lomonaco@UMBC.EDU WebPage: WebPage: http://www.csee.umbc.edu/~lomonaco http://www.csee.umbc.edu/~lomonaco

A A Rosetta Stone Rosetta Stone for for Quantum Computation Quantum Computation

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Adami, Barencol, Benioff, Bennett, Brassard, Calderbank, Chen, Crepeau, Deutsch, DiVincenzo, Ekert, Einstein, Feynman, Grover, Heisenberg, Jozsa, Knill, Laflamme, Lloyd, Peres, Popescu, Preskill, Podolsky,Rosen, Schumacher, Shannon, Shor,Simon, Sloane, Schrodinger, Townsend, Unruh, von Neumann, Vazirani, Wootters, Yao, Zeh, Zurek Lomonaco, & many more

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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 computation quantum computation, in AMS PSAPM/58, , in AMS PSAPM/58, (2002), pages 3 (2002), pages 3 – – 65. 65.

Quantum Computation and Information Quantum Computation and Information, Samuel J. Samuel J. Lomonaco, Jr. and Howard E. Brandt (editors), Lomonaco, Jr. and Howard E. Brandt (editors), AMS AMS CONM/305, (2002). CONM/305, (2002).

? ? ? Why ? ? ? ? ? ? Why ? ? ?

Quantum Quantum Computation Computation

• Limits of small scale integration

Limits of small scale integration technology to be reached 2010 technology to be reached 2010-

• 2020

2020

• No Longer !

No Longer ! Moore’s Law Moore’s Law, i.e., , i.e., every year, double the computing power every year, double the computing power at half the price. at half the price. No Longer ! No Longer !

• A whole new industry will be built around

A whole new industry will be built around the new & emerging quantum technology the new & emerging quantum technology

Collision Collision Course Course

Quantum Quantum Computation Computation Multi Multi-

• Disciplinary

Disciplinary Math Comp Sci EE Physics

The The Classical Classical World World

Classical Classical Shannon Shannon Bit Bit 0 or 1 Decisive Individual

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Copying Machine

Out In Classical Classical Bits Bits Can Can Be Be Copied Copied

The The Quantum Quantum World World

Introducing Introducing the the Qubit Qubit

? ? ? ? ? ?

Quantum Bit Quantum Bit Qubit Qubit

Indecisive Individual

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

• f Qubits
• f Qubits

Example Example 1. A spin A spin-

• particle

particle

1 2

Spin Up Spin Down

1 1

Quantum Representations Quantum Representations

• f Qubits (Cont.)
• f Qubits (Cont.)

Example Example 2. Polarization States of a Photon Polarization States of a Photon

1 = 0 = ↔ = ↔ 1 = 0 =

• r

, ,

H =

Where does a Qubit live ? Where does a Qubit live ?

Home

• Def. A Hilbert Space is a vector

space over together with an inner product such that

H

• ,

: − − − − × → H H H H

• The elements of will be called

The elements of will be called kets kets, and , and will be denoted by will be denoted by label

H

1) &

1 2 1 2

, , , u u v u v u v + = + = +

1 2 1 2

, , , vu u v u vu + = + = +

2)

, , u v u v λ λ λ λ =

3)

, , u v v u =

4) Cauchy seq in ,

∀

1 2

, , u u … H lim

n n

u

→∞ →∞

∈ H

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A A Qubit Qubit is a is a quantum quantum system system whose whose state state is is represented by a represented by a Ket Ket lying in a 2 lying in a 2-

• D Hilbert

D Hilbert Space Space H

1

1 α α α α = + = + Superposition of States Superposition of States

A typical Qubit is ??? where

2 2 1

1 α α α α + = + =

The above Qubit is in a Superposition Superposition of states and It is simultaneously both and !!!

1 1

Amplitudes

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

1

1 α α α α + = + =

Observer Qubit

i

Kets as Column Vectors over Kets as Column Vectors over

Let be a 2-D Hilbert space with orthonormal basis

H

0 , 1

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

1   =     1 1   =    

and And in general, we have

1 1 1 a a b a b b ψ       = + = + = + =             Tensor Product of Hilbert Spaces Tensor Product of Hilbert Spaces

The tensor product of two Hilbert spaces and is the “simplest” Hilbert space such that the map is bilinear, i.e., such that

H K

( ) ( )

, h k h k ⊗

• ×

→ × → ⊗ H K H K H K

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2 1 2 1 2

h h k h k h k h k k h k h k h k h k λ λ λ λ + ⊗ + ⊗ = ⊗ + ⊗   ⊗ + ⊗ + = ⊗ + ⊗   ⊗ = ⊗ = ⊗ 

We define the action of on as

( ) ( ) ( ) ( )

h k h k h k λ λ λ λ λ ⊗ ⊗ ⊗ ⊗ = ⊗

• ⊗

H K H K In other words, In other words,

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

⊗ H K H K

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2 1 2 1 2

, h h k h k h k h k k h k h k h k h k h k λ λ λ λ λ λ + ⊗ + ⊗ = ⊗ + ⊗   ⊗ + ⊗ + = ⊗ + ⊗   ⊗ = ⊗ = ⊗ ⊗ ∀ ∈ 

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

and

11 12 21 22

a a A a a   =    

11 12 21 22

b b B b b   =    

The Kronecker(tensor) product is defined as:

1 1 1 2 1 1 1 2 1 1 1 2 2 1 2 2 2 1 2 2 1 1 1 2 1 1 1 2 2 1 2 2 2 1 2 2 2 1 2 2

b b b b a a b b b b A B b b b b a 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 b a b a b a b a b a b a b a b a b a b a b a b a b       =      

So … So … 1 01 0 1 1 1 1 1 1 1     = = = = ⊗ = ⊗                           = = = =                   i i

Representing Integers in Quantum Computation Representing Integers in Quantum Computation Let be a 2-D Hilbert space with orthonormal basis

2

H 0 , 1

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

1 2 n−

= ⊗ = ⊗ H H H H 00 , 0 01 , 0 10 , 0 11 , 1 11

• …

where we are using the notational convention

1 2 1 0 1 2 1 n n n n

b b b b b b b b

− − − − − −

= ⊗ = ⊗ ⊗ ⊗ ⊗

• Representing Integers in Quantum Computation

Representing Integers in Quantum Computation So in the 2n-D Hilbert space with induced

• rthonormal basis

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

• …

we represent the integer with binary expansion

m

1

2 , 1,

n j j j j

m m m

• r

j

− =

= = = = ∀

∑

as the ket

1 2 1 n n

m m m m m

− − − −

=

• For example,

23 010111 =

Indexing Convention for Matrices Indexing Convention for Matrices The indices of matrices start at 0, not 1. For example, in

2 2 2

⊗ ⊗ ⊗ ⊗ H H H H H

1 2 1 3 5 101 1 1 4 1 5 6 7 index index index index index index index index ← = ← =    ← = ← =   ← = ← =    ← = ← =         = = = = ⊗ ⊗ =       ← = ← =          ← = ← =    ← = ← =  ← = ← =   The Qubit Village The Qubit Village Qubitville Qubitville Kets Kets Each in Each in

1 2

, ,

n

Ψ Ψ Ψ Ψ Ψ

• 1

2

, , ,

n

H H H H H

• The Qubits in Qubit Village collectively live in

1 2 1 n n j j =

⊗ ⊗ ⊗ ⊗ ⊗ = ⊗ H H H H H H

• The populace of Qubit Village is

1 2 1 n n j j

Populace

=

= Ψ = Ψ ⊗ Ψ ⊗ Ψ ⊗ ⊗ Ψ ∈ ⊗ H

• Other names for the populace of Qubit Village

1 2 1 2 n n

Populace = Ψ = Ψ Ψ Ψ = Ψ Ψ Ψ

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Massive Parallelism Massive Parallelism

• Example. For , let

Then

1,2, , j n = …

1 2

j

+ Ψ = Ψ =

1 2 1

1 2

n n j=

+   Ψ Ψ Ψ Ψ Ψ =    

⊗

• Therefore, the n-qubit register contains

n-bit binary numbers simultaneously ! all

2 1

1 2

n

n a

a

− =

  =    ∑ ( ) ( )( ) ( ) ( ) ( ) 1 1 1 1 2

n

  = + = + + +    

• (

) ( )

1 00 00 1 11 1 2

n

  = + = + + +     … … … …

But But ! ! ! ! ! !

1 2 n

Ψ Ψ Ψ Ψ Ψ

• a

Observer Observer U

• 1

Ψ

Activities in Quantum Village Activities in Quantum Village All activities in Quantum Village are All activities in Quantum Village are Unitary Unitary transformations transformations At time At time t=0 t=0 At time At time t=1 t=1

H H

U

T T

U U I UU = = = =

Ψ

where a where a unitary unitary transformation is one such that transformation is one such that

Measurement Measurement

Connecting Connecting Quantum Village Quantum Village to the to the Classical World Classical World

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

Measurement Measurement

Measurement Measurement Group of Friendly Physicists Group of Friendly Physicists Group of Group of Angry

Angry Physicists

Physicists Observables Observables

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

??? ???

Observables = Hermitian Operators Observables = Hermitian Operators

A

→

O

H H H H

T A A

= O O O O

where where

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, and let , and let denote the corresponding denote the corresponding eigenvalues eigenvalues Let be the Let be the eigenkets eigenkets of

• f

Observables (Cont.) Observables (Cont.) What does our observer actually What does our observer actually

• bserve ?
• bserve ?

??? ???

i

ϕ

A

O

A i i i

a ϕ ϕ ϕ ϕ = O

i

a

, 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

• bserves the eigenvalue , and
• bserves the eigenvalue , and

The state of an n The state of an n-

• Qubit register can

Qubit register can be written in the eigenket basis as be written in the eigenket basis as

i i iα ϕ

α ϕ Ψ = Ψ = ∑

i

a

2 i i

p α =

i

ϕ

Example: Pauli Spin Matrices Example: Pauli Spin Matrices

Consider the following observables, called the Consider the following observables, called the Pauli Spin Pauli Spin matrices matrices:

1 2 3

0 1 1 , , 1 0 1 i i σ σ σ σ σ       = = = = =       − − − −      

which can readily be checked to be Hermitian. which can readily be checked to be Hermitian. E.g., E.g.,

* † 2 2 T T

i i i i i i σ σ σ σ − − − −       = = = = = =       −      

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

Eigenvalue Eigenvalue

( ) ( )

1 / 2 +

( ) ( )

1 / 2 i +

( ) ( )

1 / 2 −

( ) ( )

1 / 2 i −

1

1

σ

2

σ

3

σ

1 +

1 −

Measurement Example Measurement Example Consider a 2 Consider a 2-

• D quantum system in state

D quantum system in state , where , where

1 a b ψ = + = +

2 2

1 a b + = + = What happens if we measure w.r.t. observable ? What happens if we measure w.r.t. observable ?

1

σ

First express in terms of the eigenket basis of First express in terms of the eigenket basis of

1

σ ψ

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

1

σ

Possibility0 Possibility0 Eigenvalue Eigenvalue is meas. is meas.

1 a = + = + ( ) ( )

1 / 2 ψ +

• 2

Prob / 2 a b = + = + 1 1 2 2 2 2 a b a b ψ + − + − + − + −         = + = +                

ψ

Possibility1 Possibility1 Eigenvalue Eigenvalue is meas. is meas.

2

Prob / 2 a b = − = −

1

1 a = − = −

( ) ( )

1 / 2 ψ −

• r
• r

ψ

Important Feature of Important Feature of Quantum Mechanics Quantum Mechanics

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

We cannot completely We cannot completely control the outcome of control the outcome of quantum measurement quantum measurement

Copying Machine

Out In

Cloning Cloning The No The No-

• Theorem

Theorem Dieks, Wootters, Zurek Dieks, Wootters, Zurek

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The No Cloning Theorem The No Cloning Theorem Definition Definition. Let be a Hilbert space. Then Let be a Hilbert space. Then a a quantum

quantum replicator replicator consists of an

consists of an auxiliary Hilbert space , a fixed state auxiliary Hilbert space , a fixed state (called the (called the initial

initial state state of

• f

the replicator), and a unitary transformation the replicator), and a unitary transformation

A

H

H

# A

ψ ∈ H

:

A A

U ⊗ ⊗ ⊗ ⊗ → ⊗ ⊗ H H H H H H H H

states , where states , where (called the (called the replicator replicator state state after replication of after replication of ) depends on . ) depends on . such that, for some fixed state , such that, for some fixed state ,

blank ∈ H

# a

U a blank a a ψ ψ ψ ψ = a ∈ H

a A

ψ ∈ H a a

for for all all The No Cloning Theorem The No Cloning Theorem

• Cloning is:

Cloning is: ( ) ( ) ( ) ( )( ) ( )

# @

1 1 1 a b blank a b a b ψ ψ ψ ψ + + + + +

• Cloning is

Cloning is NOT NOT: ( ) ( ) ( ) ( )

# @

1 00 11 a b blank a b ψ ψ ψ ψ + + + +

• The No Cloning Theorem

The No Cloning Theorem

• Cloning

Cloning is inherently is inherently non non-

• linear

linear

• Quantum mechanics

Quantum mechanics is inherently is inherently linear linear

• Ergo

Ergo, , quantum replicators quantum replicators do not exist do not exist

Key Idea Key Idea

Introduction Introduction to to Quantum Entanglement Quantum Entanglement

A Illustration of the A Illustration of the

• f Quantum Mechanics
• f Quantum Mechanics

Weirdness Weirdness Qubits Qubits

• Not Entangled

Not Entangled

• Separate

Separate

• Entangled

Entangled

• Not Separate

Not Separate ! !

2 1 1 ⊗ − ⊗ ⊗ − ⊗

⊗

⊗

U

Unitary Unitary Transf Transf

Entangled Entangled Observing Entangled Qubits Observing Entangled Qubits

2 1 1 ⊗ − ⊗ − ⊗

Observe Only Observe Only the Blue Qubit the Blue Qubit

1 ⊗ 1 ⊗

• No Longer Entangled

No Longer Entangled

• Separate Identity

Separate Identity Whoosh ! Whoosh !

SLIDE 9

9

Hidden Variable Theory Hidden Variable Theory vs

vs Quantum Mechanics

Quantum Mechanics EPR Pair EPR Pair 2 1 1 ⊗ − ⊗ − ⊗

Einstein Einstein Podolsky Podolsky Rosen Rosen

Bah ! Humbug ! Bah ! Humbug !

Something is Missing from Quantum Mechanics. Something is Missing from Quantum Mechanics. There Must Exist Hidden Variables There Must Exist Hidden Variables Hidden Variable Theory Hidden Variable Theory vs

vs Quantum Mechanics

Quantum Mechanics EPR Pair EPR Pair 2 1 1 ⊗ − ⊗ − ⊗

Einstein Einstein Podolsky Podolsky Rosen Rosen

Bah ! Humbug ! Bah ! Humbug !

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

Bell Inequalities Bell Inequalities Aspect Experiment Aspect Experiment

Score So Far Score So Far

• HVT Score = 0

HVT Score = 0

• QM Score = 1

QM Score = 1

Why did Why did Einstein Einstein Podolsky Podolsky Rosen Rosen Object So Object So Vehemently ? Vehemently ?

Forces of Nature Are Local Interactions Forces 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 light

Propagate no faster than the speed of light c

• Strength drops off with distance

Strength drops off with distance Spacelike Distance Spacelike Distance

Hello ! Hello ! Can’t Hear Can’t Hear You !! ?? You !! ??

1

P

2

P ( ) ( )

, , , x y z t ( ) ( )

, , , X Y Z T Spacelike Distance Spacelike Distance

( ) ( )

1 2

, Dist P P c T t > − > −

No signal can travel between spacelike regions of space No signal can travel between spacelike regions of space

Ergo, Ergo, spacelike regions spacelike regions of space are

• f space are physically

physically independent independent, i.e., one cannot influence the other. , i.e., one cannot influence the other.

SLIDE 10

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• The forces of nature are local

The forces of nature are local interactions interactions

• Spacelike regions of space are

Spacelike regions of space are physically independent physically independent The EPR Perspective The EPR Perspective All perfectly All perfectly reasonable reasonable assumptions ! assumptions !

Alpha Centauri Alpha Centauri Earth Earth

Instantly, Instantly, Both Qubits Both Qubits Are Determined ! Are Determined !

Spacelike Distance Spacelike Distance

1 1 ( 2 0 ) / −

1 ⊗ 1 ⊗

Meaurement of EPR Pair Meaurement of EPR Pair

Blue Qubit Blue Qubit Red Qubit Red Qubit

• Meas. Blue
• Meas. Blue

Qubit Qubit

No Local Interaction ! No Local Interaction !

• No force of any kind

No force of any kind

• Not mediated by anything

Not mediated by anything

• Acts instantaneously

Acts instantaneously

• Faster than light

Faster than light

• Strength does not drop off with distance

Strength does not drop off with distance

• Full strength at any distance

Full strength at any distance Yet, still consistent with General Relativity ! Yet, still consistent with General Relativity !

Quantum Entanglement Quantum Entanglement Appears to Pinpoint the Appears to Pinpoint the

• f Quantum
• f Quantum

Mechanics Mechanics Weirdness Weirdness

Properties of Qubits Properties of Qubits

• Properties of States

Properties of States

• Qubits can exist in a superposition of

Qubits can exist in a superposition of states states

• Qubits can be entangled

Qubits can be entangled

• Actions on States

Actions on States

• Qubits “collapse” upon measurement

Qubits “collapse” upon measurement

• Qubits are transformed by unitary

Qubits are transformed by unitary transformations transformations Properties of Quantum Computer Data Properties of Quantum Computer Data Quantum Computer Instructions Quantum Computer Instructions Useful Useful for Quantum Computation for Quantum Computation

Quantum Quantum Teleportation Teleportation

An Application of Quantum Entanglement An Application of Quantum Entanglement

SLIDE 11

11

Teleportation Teleportation: Transfering an object between Transfering an object between two locations by a process of: two locations by a process of:

• Dissociation to obtain info

Dissociation to obtain info

• Scanned to extract suff. Info. to

Scanned to extract suff. Info. to recreate original recreate original

• Information Transmission

Information Transmission

• Exact replica is re

Exact replica is re-

• assembled at destination

assembled at destination

• ut of locally available material
• ut of locally available material
• Reconstruction from info

Reconstruction from info Net Net Effects Effects:

• Destruction of original object

Destruction of original object

• Creation of an exact replica at the

Creation of an exact replica at the intended destination. intended destination. Teleportation ? Teleportation ? Oxford Unabridged Dictionary Oxford Unabridged Dictionary Asked Scotty about Teleportation Asked Scotty about Teleportation Beam me up, Scotty ! Beam me up, Scotty ! Aye, Aye, Captain ! Aye, Aye, Captain ! Asked Scotty about Teleportation Asked Scotty about Teleportation Beam me up, Scotty ! Beam me up, Scotty ! 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 3

⊗ H H H H 00

Unitary Matrix

01 10 2 −

• Alice arranges for a courier to

transport entangled qubit #3 to Bob at location B.

Federation Quantum Teleportation Manual

Result

• Alice at location A shares an EPR pair

with 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

Federation Quantum Teleportation Manual

Step 2. Qubit #1 is delivered to Alice The state of all three qubits is now:

( ) ( )

1 2 3

01 10 1 2 a b   − Φ = Φ = + ∈ ⊗ ⊗     H H H H H

Unknown

SLIDE 12

12

• The Bell Basis of is

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 3 4

10 10 / 2 10 10 / 2 00 11 / 2 00 11 / 2 Ψ = Ψ = − Ψ = Ψ = + Ψ = Ψ = − Ψ = Ψ = +

1 2

⊗ H H H H A Little Algebraic Manipulation ( ) ( )

1 2 3

01 10 1 2 a b   − Φ = Φ = + ∈ ⊗ ⊗     H H 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:

Φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 3 4

1 1 2 1 1 1 a b a b a b a b   Φ = Φ = Ψ − −   + Ψ + Ψ − + + Ψ + Ψ +   + Ψ + Ψ −  

• Let be the unitary

transformation:

1 2 1 2

: U ⊗ → ⊗ → ⊗ H H H H H H H H

1

00 Ψ

• 4

11 Ψ

• 3

10 Ψ

• 2

01 Ψ

• Federation Quantum Teleportation Manual

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 H H U I ⊗ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 00 1 2 01 1 10 1 11 1 a b a b a b a b   Φ = Φ = − −   + − + − + + + + +   + − + −  

Federation Quantum Teleportation Manual

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

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 00 1 2 01 1 10 1 11 1 a b a b a b a b   Φ = Φ = − −   + − + − + + + + +   + − + −  

Step 5. (Loc. A) Send via classical communication channel the result to Bob at Loc. B

Federation Quantum Teleportation Manual

Step 6. (Loc. B) Use the two classical bits

• f received information to select a unitary

transformation of from the table: Step 7. (Loc. B) Apply the selected unitary transformation to qubit #3.

( , ) i j

U

3

H

( , ) i j

U

• Rec. Bits

Effect on Qubit #3 00 01 10 11

( , ) i j

U

1 1 a b a b − − − − +

• (0,0)

U

(0,1)

U

(1,0)

U

(1,1)

U

1 1 a b a b − + − + +

• 1

1 a b a b + + + +

• 1

1 a b a b − + − +

• (

) ( )

, i j Federation Quantum Teleportation Manual

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

1 a b +

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

SLIDE 13

13

More More Dirac Dirac Notation Notation

The Deutsch-Jozsa Algorithm Unitary transfs revisited

More Dirac Notation More Dirac Notation Let Let

( ) ( )

*

, Hom = H H H H

Hilbert Space Hilbert Space

• f morphisms
• f morphisms

from to from to

H

• We call the elements of

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

*

H

label

More Dirac Notation More Dirac Notation

There is a There is a dual dual correspondence correspondence between and between and *

H

H

Ket Ket Bra Bra There exists a bilinear map There exists a bilinear map defined by defined by which we more simpy denote by which we more simpy denote by

* ×

→ × → H H H H

• (

) ( )( ) ( )

1 2

ψ ψ ψ ψ ∈

1 2

| ψ ψ ψ ψ

†

ψ ψ ψ ψ

↔

Bra Bra-

• c

c-

• Ket

Ket

Bra’s as Row Vectors over Bra’s as Row Vectors over

• Let be a 2

Let be a 2-

• D Hilbert space with

D Hilbert space with

• rthonormal basis
• rthonormal basis 0 , 1

H

( ) ( )

*

, Hom = H H H H

0 , 1

and let and let be the corresponding dual Hilbert space with be the corresponding dual Hilbert space with corresponding dual basis corresponding dual basis

( ) ( ) ( ) ( )

and 1,0 1 0,1 = = = =

( ) ( )

1 , a b a b + = + =

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

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 by is given by and is called the and is called the adjoint adjoint

† *

↔

H H H H

( ) ( )

†

1 1 , a a b a b a b b   = + = + + =    

↔

1 2

1 1 a b c d ψ ψ = + = +   = + = + 

( ) ( )( ) ( ) ( ) ( )

1 2

| 1 1 , a b c d c a b ac bd d ψ ψ ψ ψ = + = + +   = = = = +     i

If If then the bracket product becomes then the bracket product becomes

SLIDE 14

14

as a Matrix Outerproduct as a Matrix Outerproduct

1 2

ψ ψ ψ ψ

1 2

1 1 a b c d ψ ψ = + = +   = + = + 

1 2

1 2 | ψ ψ ψ ψ

ψ ψ ψ ψ ψ ψ

→

H H H H

• If

If then is the linear transformation then is the linear transformation

1 2

ψ ψ ψ ψ

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

• uterproduct

( ) ( )

1 2

, a ac ad c d b bc bd ψ ψ ψ ψ     = = = =         i

Let be an N Let 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 at at 0, then the matrix of the linear transformation , then the matrix of the linear transformation is an is an NxN NxN 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 1 For example if For example if N=4 N=4, then , then

H 0 , 1 , , 1 N − …

m k

2 3 1       =      

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

Unitary Unitary Transformations Transformations Revisited Revisited

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

The dynamic behavior of a quantum system is determined by Schroedinger’s equation. ( ) U t ψ ψ =

Schroedinger’s Equation

i H t ψ ψ ∂ = ∂

• 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 U H H ψ

Initial State

H

Hamiltonian Dynamic State

Observable 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

• n the state space , i.e., a linear

transformation such that Ω

† T

Ω = Ω = Ω = Ω

H ( ) U t ( ) U H U H H Observable H H H

Measurement Measurement Revisited Revisited

SLIDE 15

15

Spectral Decomposition Spectral Decomposition Let be an observable, i.e., a Hermitian

• perator.

Ω

Eigenvalues λ1 λ2 … λn Eigenspaces V1 V2 … Vn Projection Ops. P1 P2 … Pn where is the projection operator corresponding to the eigenspace

:

j j

P V → H

j

V

Spectral Decomposition Theorem

Ω

1 1 2 2 n n

P P P λ λ λ λ λ = + = + + + …

Quantum Measurement Quantum Measurement In In Out Out

j

λ

j j j

P P ψ ψ ψ ψ ψ ψ =

ψ

Ω

BlackBox BlackBox MacroWorld MacroWorld Quantum Quantum World World Eigenvalue Eigenvalue Observable Observable

• Q. Sys.
• Q. Sys.

State State

• Q. Sys.
• Q. Sys.

State State

P r

j

• b

P ψ ψ ψ ψ =

j j j

P λ Ω = Ω = ∑

where where Spectral Decomposition Spectral Decomposition

Physical Physical Reality Reality Philosopher Philosopher Turf Turf

Density Operators Density Operators & Mixed Ensembles Mixed Ensembles

Skip to D-J Quit

Two Ways to Represent Quantum States Two Ways to Represent Quantum States Density Operators Density Operators

ρ

Kets Kets

ψ

&

Two Ways to Represent Quantum States Two Ways to Represent Quantum States Example

• Example. We have seen

We have seen pure ensembles pure ensembles, i.e., , i.e., pure states, such as pure states, such as Problem Problem. Certain types of quantum states Certain types of quantum states are difficult to represent in terms of kets are difficult to represent in terms of kets Ket Ket Prob Prob

ψ 1

Two Ways to Represent Quantum States Two Ways to Represent Quantum States Example

• Example. 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 ensemble mixed ensemble: where where Ket Ket Prob Prob

1

ψ

1

p

2

ψ

k

ψ

2

p

k

p

• 1

2

1

k

p p p + + + + + = …

All unit Length All unit Length & not nec. & not nec. ⊥

SLIDE 16

16

Ket Ket Prob Prob

1

ψ

1

p

2

ψ

k

ψ

2

p

k

p

• Two Ways to Represent Quantum States

Two Ways to Represent Quantum States

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 k

p p p ρ ψ ρ ψ ψ ψ ψ ψ ψ = + = + + + …

is called a is called a density operator density operator. It is a . It is a Hermitian Hermitian positive semi positive semi-

• definite operator of trace

definite operator of trace 1 1.

ρ

Mixed Mixed Ensemble Ensemble For the pure ensemble , For the pure ensemble ,

Ket Ket Prob Prob 1

ψ 1 ρ ψ ρ ψ ψ = i

Two Ways to Represent Quantum States Two Ways to Represent Quantum States If for example, If for example,

1 a b ψ = + = +

where where

2 2

1 a b + = + =

then then

( ) ( )( ) ( ) ( ) ( )

2 2

1 1 a b a b a ab a a b b ba b ρ = + = + +     = = = =          

3/8 3 /8 1 1 3 11 1 4 4 3 /8 5/8 2 2 i i i i ρ − + − +      = + = + =      −     

Two Ways to Represent Quantum States Two Ways to Represent Quantum States On the other hand, On the other hand, is the mixed ensemble is the mixed ensemble Ket Ket Prob Prob

( ) ( )

1

1 / 2 i ψ = − = −

2

1 ψ = 3 4 1 4

1

ψ

1

ψ

1

p

2

p

2 2

ψ ψ ψ ψ

Quantum Mechanics from the Two Perspectives Quantum Mechanics from the Two Perspectives

Kets Kets Density Ops Density Ops Schroed. Schroed. Eq. Eq. Unitary Unitary Evolution Evolution Observation Observation

ψ

ρ

i H t ψ ψ ∂ = ∂

• |

| A A ψ ψ ψ ψ =

( ) ( )

A trace A ρ =

[ ] [ ]

, i H t ρ ρ ∂ = ∂

• U

ψ ψ ψ ψ

• †

U U ρ ρ ρ ρ

• We now have a more powerful way of

We now have a more powerful way of representing quantum states. representing quantum states.

• Density operators are absolutely

Density operators are absolutely crucial when discussing and dealing crucial when discussing and dealing with quantum noise and quantum with quantum noise and quantum decoherence. decoherence. Density Operators Density Operators

The The Deutsch Deutsch-

• Jozsa

Jozsa Algorithm Algorithm

Quit

SLIDE 17

17

The The Hadamard Hadamard Tranformation Tranformation

( ) ( ) ( ) ( )

1 / 2 1 1 / 2 + −

• 1

1 1 1 1 2 H   =   −  

H

D-

• J Algorithm

J Algorithm

Definition

• Definition. A

. A coin coin is is fair fair (or (or balanced balanced) 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 unfair unfair (or (or constant constant) 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.

H H H H H H T T T T T T

Side1 Side1 Side2 Side2 Fair (Balanced) Fair (Balanced) Side1 Side1 Side2 Side2 Fair (Balanced) Fair (Balanced) Side1 Side1 Side2 Side2 Unfair (Constant) Unfair (Constant) Side1 Side1 Side2 Side2 Unfair (Constant) Unfair (Constant)

Observation Observation Observation

• Observation. 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 ? We represent a coin mathematically as a We represent a coin mathematically as a Boolean function: Boolean function:

{ } { }

: 0,1 0,1 f →

Side1 Side1 Side2 Side2 H T T

The Unitary Implementation of The Unitary Implementation of

( )

f

U

x y x f x y ⊕

→

H H H H

• Let be the unitary transformation

f

U

then

x

1 2 −

( ) ( )

( )

1 1 2

f x x

− −

f

U

f

Ancilla Moreover, Moreover,

1 1

f

U

H H H H H H H H

( ) ( ) ( )

(0) (1)

1 1 2

f f

  − + − + −     ( ) ( ) ( )

(0) (1)

1 1 1 2

f f

  − − − − −    

+

( ) ( ) ( )

(0) (1)

1 1 0 1 0 1 1 0 1 2 Output

f f

  − + − + − = + = + =±     i ( ) ( ) ( )

(0) (1)

1 1 0 0 1 1 1 Outpu 1 t 1 2

f f

  − − − − − = + = + =±     i

Case Case 1. is

. is fair fair, i.e., , i.e., balanced balanced

f

Case Case 1. is

. is unfair unfair, i.e., , i.e., constant constant

f

SLIDE 18

18

So … So … If we only make one observation, i.e., if we If we only make one observation, i.e., if we

• bserve the left register, then we can
• bserve 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 !