Qu

antum

What is

In

formation

and

T

echnology

?

Prof. Ivan H. Deutsch Dept. of Physics and Astronomy University of New Mexico

Second Biannual Student Summer Retreat of the SQuInT Network

St. John’s College, Santa Fe, New Mexico June 25, 2001 – June 29, 2001

Outline

• What is

classical information

?

• What is

quantum information

?

• What is

quantum computing

?

• How would we

implement

these ideas?

References: • M. A. Nielsen and I. Chuang,

Quantum computation and quantum information”

(Cambridge Press, 2000).

• Prof. Preskill’s notes: http://www.theory.caltech.edu/people/preskill/ph229/ •

Introduction to Quantum Computation and Information

, H-K Lo, S. Popescu, T. Spliller eds., (World Scientific, 1998).

• Special Issue of Physical Implementations: Fortschritte der Physik

48

2000.

U I O

Q Z W

Quantifying Information

3 bit message:

{000, 001, 010, 011, 100, 101, 110, 111}

With no

prior knowledge

:

Information

 log 2 (# of possibities) = log 2 (8) = 3 bits

With

a priori knowledge (probabilistic)

:

Probability 0 =

p

Probability 1 = 1-

p

• “Typical word” of length

N

has:

Np

0’s and

N

(1-

p

) 1’s • #

typ

  

N



Np N

!

(

Np

)!



N

(1 

p

)  !

• Information in a typical word = log 2 (#

typ

)

Shanon Information

Shanon Information =

Average information/ letter

H

(

p

)  log 2 (#

typ

)  

p

log 2 (

p

)  (1 

N p

)log 2  1 

p



Generally, given alphabet:

{

a j

|

j

 1,..,

n

}with probability

p j H

(

A

)  

n



j

 1

p j

log 2 (

p j

)

=

Entropy

 log 2 (

p j

)  Information in

a j

0 

H

(

A

)  log 2 (

n

)

Mutual Information

Suppose:

Alice send message

A

Bob receives message with probability

B p(A)

with probability

p(B)

How much information can Alice communicate to Bob?

Degree of correlation:

p

(

A

;

B

) 

p

(

A

)

p

(

B

) 

p

(

A

,

B

) Joint probability

I

(

A

;

B

)   log(

p

(

A

;

B

)) 

I

(

B

;

A

)

• Information Bob gets from Alice • Information common to

A

and

B

Quantum Information

Information encoded in a

quantum state



Qubit:

Two-level quantum system

0   cos  q / 2) 0 q  0 , 1  “Logical Basis” f 

e i

f sin  q / 2  1

Bloch sphere

1 Tremendous amount of information

encoded

in 

e

n

( q , f ) N S  Given finite precision to specify q,f Alice can specify

many

different 

Inaccessible Information

• Alice encodes 2 “ classical bit” in a qubit Message 0 1 1 2 2 3 3 0 1 2 3 • Bob decodes through Stern-Gerlach apparatus.

1 2 0 3

Quantum information

cannot be read

1

Holevo Bound

Alice cannot send more than one bit of information to Bob per qubit!

I

(

A

;

B

)  1bit

 

No Cloning

        e.g. copy the unknown qubit onto an “ancilla” qubit • Transformation on basis states 0 0  0 0 1 0

• Linearity

  0   1  0   0 0   1 0  0 0   1 1    0     1 1 1    0   1   0   

Quantum Information cannot be copied!

Information Gain Disturbance

Attempt to copy “distinguishing information” into an ancilla.



u

 

v



u

  If

u

  

u



v

          0  1 

v v



v v

  Any attempt to distinguish between two

non-orthogonal

necessarily results in a disturbance of the states states

   

Composite Systems

Classical

• One bit:

A

={

a| a=

0 or 1} • Two bits:

A



B

 {(

a

,

b

) |

a

,

b



product • N

bits: Space has 2

N

Quantum

0 or1} configurations.

Cartesian

• One qubit:

A

   ,  

c

0 0 

c

1 1 ,

c

0 ,

c

1 finite precision complex numbers • Two qubits:

Tensor product A



B



AB



c

00 0,0 

c

01 0,1 

c

10 1, 0 

c

11 1,1  

• N N

qubits: Space has states, with

x

bits for each

c

.

  

Entanglement

(Pure) states

N

-qubits are generally “

entangled

”

   1   2   

N

Nonclassical correlations

e.g. Two qubits, Bell’s Inequalities Alice receives a random bit 0 1  (  ) 0   0

A

 1

B

 1

A

 0 1

B

 / 2 Bob receives a random bit 

•

Quantum-Information cannot be read.

•

Quantum-Information cannot be copied.

•

Nonorthogonal states cannot be distinguished.

•

Exponential growth in

inaccessible

information.

•

Quantum correlations cannot be used for communicating classical information

•

Measurement is

irreversible - collapse of the wave function

.

Quantum Information as a

Information-Gain/Disturbance:

• Quantum - Secret key distribution

Tensor Structure of Composites (Entanglement)

• Quantum dense coding - Sending two classical bits with one classical bit plus EPR.

• Quantum teleportation - Communicating a qubit •

  

Elements of Quantum Computation



in



out

Take advantage of exponentially large state space

• Quantum Register: quantum state of the system • Quantum Logic Gates: unitary transformation on subsystems • Error-tempering: • Measurement: correct/suppress errors read out classical information

  

Quantum Parallelism

(Deutsch)

E.g. 3-qubit “quantum register”

0 bits of classical input information.

1 0 2 3 4 5 6 7 1 0 2 3 4 5 6 7 3 bits of classical output information.

    

Quantum Computing: Quantum Control in



in

 

out



Many - body State

Complexity:

Asymptotic Behavior Given n bits to specify the state, how do the resources scale as n

 

Resources:

• Time • Energy • Space •

“

Easy

”

Polynomial in

n • “

Hard

”

Exponential in

n

    

Quantum Logic Gates

• Single qubit:

U

 0  3 

i

 0  1,

c i



i

,  

i

 1,2,3 

c i

2  1 Pauli

Rotation on Bloch sphere

NOT: Hadamard

:

 1

H

  0  1 1 0     1   3 2 •

Two qubit:

“Entangling unitary”

0 1   1 0 0 1     0 0   1 1  / 2  / 2 Controlled NOT: Control 0 0  0 0 0 1 1 0   0 1 1 1

x y



x x

 Target

y

Flip the state of the target bit conditonal on the state of the control bit  3   1 1 1  1 0   0  2 1   0  0 0  2 1 1

  

Universality

• Logic Gates: Basic building blocks

U

:H  

a i

1

i

2

i N



i

1  

i

2   

i N

Unitary acts on combinations of qubits 

R

e

n

( q );

cNOT



Single qubit rotation Two-qubit entangling gate

Entangling Gate:

ˆ 12  ˆ 1  ˆ 2

Efficient algorithms: # of gates not exponential in N



Quantum Circuits

0 0 0 1 1 0 1 1   0 0 0 1  1 0   1 1

CNOT from CPHASE

1 1

(I)

1 0  1 2

(II)

1 1 0

(III)

0  1 2

(IV)

Deutsch’s Problem

Coupling Between Qubits

• Entanglement

Coupling To External Drive

• Unitary evolution

The Tao of Quantum Computation

Coherence

Coupling to the Environment

Errors

Physical Implementations

Atomic-Molecular Optical Systems (Gas Phase)

• Ion Traps • Cavity

QED

• Neutral Atom Traps • Linear/Nonlinear Optics

Condensed Matter (Liquid or Solid Phase)

• Semiconductors (electronics) • Nuclear/Electron Magnetic Resonance (liquid, solid, spintronics) • Superconductors (flux or charge qubits) • Electrons floating on liquid helium

• Ion Trap + + + • Strong Qubit Coupling: Coulomb Repulsion • Strong Coupling to Environment - Technical Noise

• Coherent photon exchange • Spontaneous emission + + -

Cavity QED:

• Interactions turned “on” and “off” • Real photon exchange • Strong-coupling regime (enhance coherence) Optical Lattice: + + + + + + • Virtual photon exchange • Near-field interaction dominates + + + + + + -

       

Ideal:



in

Reality:



in

Measurement

$



out



out

   

j

ˆ

j

† 

j

ˆ

j

ˆ

j

  1 ˆ †

j

Summary

• Information Processing constrained by

physical laws

.

• Quantum Information:

Information-gain/disturbance.

Exponential growth of state space. Entanglement.

• Quantum Computation - asymptotic savings of

physical resources

.

• Physical Implementation -

Quantum Control of Many-body System

!

Classical Physics Digital: Particles Analog: Waves Quantum Physics