The World of Quantum The World of Quantum InformationInformation

Marianna SafronovaMarianna SafronovaDepartment of Physics and AstronomyDepartment of Physics and Astronomy

November 16, 2008November 16, 2008

School of Chemistry, Tel Aviv UniversitySchool of Chemistry, Tel Aviv University

Outline Quantum Information: fundamental principles (and how it is different from the classical one).

Bits & Qubits Quantum weirdness: entanglement, superposition &

measurement Logic gates & Quantum circuits

Cryptography & quantum information A brief introduction to quantum computing

Real world:Real world: what do we need to build a quantum computer/quantum network?

Current status & future roadmap

Why quantum information?

Information is physical!Any processing of information

is always performed by physical means

Bits of information obey laws of classical physics.

Why quantum information?

Information is physical!Any processing of information

is always performed by physical means

Bits of information obey laws of classical physics.

Why Quantum Computers?

Computer technology is making devices smaller and smaller… …reaching a point where classical

physics is no longer a suitable model for the laws of physics.

Fundamental building blocks of classical

computers:

BITS

STATE:

DefinitelyDefinitely0 or 1

Bits & QubitsFundamental building

blocks of quantum computers:

Quantum bits or

QUBITS

Basis states: and 0 1

Superposition:

0 1

Fundamental building blocks of classical

computers:

BITS

STATE:

DefinitelyDefinitely0 or 1

Bits & QubitsFundamental building

blocks of quantum computers:

Quantum bits or

QUBITS

Basis states: and 0 1

Qubit: any suitable two-level quantum system

Superposition

Bits & Qubits: primary differencesprimary differences

0 1

Measurement

Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit.

Qubit:

Bits & Qubits: primary differencesprimary differences

QOFR

Quantum calculation:number of parallel processesdue to superposition

Superposition Measurement

Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit.

Qubit: we cannot just measure and and thus determine its state! We get either or with corresponding probabilities 2and 2.

The measurement changes the state of the qubit!

Bits & Qubits: primary differencesprimary differences

0 1

0 1

2 21

Multiple qubits Hilbert space is a big place! - Carlton Caves

Multiple qubits Two bits with states 0 and 1 form four definite

states 00, 01, 10, and 11. Two qubits: can be in superposition of four

computational basis set states.

Hilbert space is a big place! - Carlton Caves

00 01 10 11

2 qubits 4 amplitudes 3 qubits 8 amplitudes 10 qubits 1024 amplitudes 20 qubits 1 048 576 amplitudes 30 qubits 1 073 741 824 amplitudes 500 qubits More amplitudes than our estimate of More amplitudes than our estimate of number of atoms in the Universe!!!number of atoms in the Universe!!!

Entanglement

0 1

2

0 1

Results of the measurement

First qubit 0 1

Second qubit 0 1

Entangledstates

Quantum cryptography

Classical cryptography

Scytale - the first known mechanical device to implement permutation of characters for cryptographic purposes

Classical cryptography

Private key cryptography

How to securely transmit a private key?

Key distribution

A central problem in cryptography: the key distribution problem.

1) Mathematics solution: public key cryptography.2) Physics solution: quantum cryptography.

Public-key cryptography relies on the computational difficulty of certain hard mathematical problems

(computational security)Quantum cryptography relies on the laws of quantum mechanics (information-theoretical security).

Quantum key distribution

Quantum mechanics: quantum bits cannot be copied or monitored.

Any attempt to do so will result in altering it that can not be corrected.

Problems Authentication Noisy channels

A quantum communication channel: physical system capable delivering quantum systems more or less intact from one place to another.

Quantum logic gates

Logic gates

Classical NOT gateQuantum NOT gate

(X gate)

A NOT A

0 1

1 0

A NOT A

The only non-trivial single bit gate

X10 01

Matrix form representation

0 1

1 0X

X

More single qubit gates

Any unitary matrix U will produce a quantum gate!

Z0 1 0 1 1 0

0 1Z

1 11

1 12H

H0 1 0 1 0 1

2 2

Hadamard gate:

Single qubit gates,two-qubit gates, three-qubit gates …

How many gates do we need to make?

Do we need three-qubit and four-qubit gates?

Where do we find such physical interactions?

Coming up with one suitable controlled interaction for physical system is already a problem!

Universality:classical computation

Only one classical gate (NAND) is needed to compute any function on bits!

A

BA AND B

A NOT A

A

BA NAND B

A B A AND B A NAND B 0 0 0 1 0 1 0 1 1 0 0 1

1 1 1 0

Universality:quantum computation

How manyHow many quantum gates do we need quantum gates do we need

to build anyto build any quantum gate?quantum gate?

Any n-qubit gate can be made from 2-qubit gates.(Since any unitary n x n matrix can be decomposed

to product of two-level matrices.)

Only one two-qubit gate is needed!

Example: CNOT gate

Quantum parallelism

I can look at bothsides of my coin ata single glance

Quantum parallelism

( ) : 0,1 0,1f x , , ( )fUx y x y f x

Ufx

y

x

f(x)

0 1

2

0

0, (0) 1, (1)

2

f f

Single circuit just evaluated f(x) for both x=0 and 1 simultaneously!

Superposition

Quantum parallelism:a major problem

So we can evaluate functions for all values of x at the same time using just one circuit!

Need only n+1 qubits to evaluate 2n values of x.

But we still get only one answer when we measure the result: it collapses to x,f(x)!!!

QOFR

Quantum parallelism:a major problem

So we can evaluate functions for all values of x at the same time using just one circuit!

Need only n+1 qubits to evaluate 2n values of x.

But we still get only one answer when we measure the result: it collapses to x,f(x)!!!

We can’t read out the result for specific value of x!

With one qubit we get either f(0) or f(1) and we don’t know which one in advance.

This is why we need quantum algorithms!

Quantum algorithms

Superposition: n qubits can represent 2n integers. Problem: if we read the outcome we lose the

superposition and we can’t know with certainty which one of the values we will obtain.

Entanglement: measurements of states of different qubits may be highly correlated.

Unique features of quantum computation

Quantum algorithms

Strategy:Strategy:

Use superposition to calculate 2n valuesof function simultaneously and

do not read out the result until a usefulresult is expected with reasonably high

probability.

Use entanglement

Current advantages of quantum computation

Shor's quantum Fourier transform provides exponential speedup over known classical algorithms.

Applications: solving discrete logarithm and factoring problems which enables a quantum computer to break public key cryptosystems such as RSA.

Quantum searching (Grover's algorithm) allows quadratic speedup over classical computers.

Simulations of quantum systems.Simulations of quantum systems.

How to factor 15?

Pick a number less then 15: 7 Calculate 7n mod 15:

Calculate

n 7n 15×A 7n mod 15 1 7 1 7 2 49 45 4 3 343 330 13

4 2401 2400 1

/ 2gcd 7 1,15R

gcd 48,15 3, gcd 50,15 5

R=4

Shor’s algorithm for N=15

Choose n such as 2n<15: n=4 Choose y: y=7 Initialize two four-qubit register Create a superposition of states of the first register Compute the function f(k)=7k mod 15 on the second register. Operate on the first register by a Fourier transform Measure the state of the first register: u=0, 4, 8, 12 are only non-zero results. Two cases give period R=4, therefore the procedure succeeds with probability 1/2 after one run.

0 0000 0000

Back to the real world:

What do we need to build a quantum computer?

Qubits which retain their properties. Scalable array of qubits.

Initialization: ability to prepare one certain state repeatedly on demand. Need continuous supply of .

Universal set of quantum gates. A system in which qubits can be made to evolve as desired.

Long relevant decoherence times.

Ability to efficiently read out the result.

0

Of course, in many cases both X and Y are beyondOf course, in many cases both X and Y are beyond

the present experimental state of the art …”the present experimental state of the art …”

Real world strategy

“…“…If X is very hard it can be If X is very hard it can be

substituted with more of Y.substituted with more of Y.

David P. DiVincenzo

The physical implementation of quantum computation.

S1/2

P1/2

D5/2

„quantumbit“

Quantum Computer(Innsbruck)

Courtesy of the R. Blatt’s Innsbruck group

Experimental proposals

Liquid state NMR

Trapped ions

Cavity QED

Trapped atoms

Solid state schemes

And other ones …

The DiVincenzo criteria

Quantum Information Processing Quantum Information Processing with with

Neutral AtomsNeutral Atoms

Quantum computing with neutral atoms

1. A scalable physical system with well characterized qubits: memory

(a) Internal atomic state qubits: ground hyperfine states of neutral trapped atoms

well characterized Very long lived!

5s1/2

F=2

F=1

87Rb: Nuclear spin I=3/2

0

16.8 GHz

MF=-2,-1,0,1,2

MF=-1,0,1

1. A scalable physical system with well characterized qubits: memory

(b) Motional qubits : quantized levels in the trapping potentialalso well characterized

1. A scalable physical system with well characterized qubits: memory

(a)Internal atomic state qubits(b) Motional qubits

Advantages: very long decoherence times!Internal states are well understood: atomic spectroscopy & atomic clocks.

1. A scalable physical system with well characterized qubits

Optical lattices: loading of one atom per sitemay be achieved using Mott insulator transition.

Scalability: the properties of optical lattice system do not change in the principal way when the size of the system is increased.

Designer lattices may be created (for example with every third site loaded).

Advantages: inherent scalability and parallelism.Potential problems: individual addressing.

2: Initialization

Internal state preparation: putting atoms in the ground hyperfine state

Very well understood (optical pumping technique is in use since 1950)

Very reliable (>0.9999 population may be achieved)

Motional states may be cooled to motional ground states (>95%)

Loading with one atom per site: Mott insulator transition and other schemes.

Zero’s may be supplied during the computation (providing individual or array addressing).

3: A universal set of quantum gates

1. Single-qubit rotations: well understood and had been carried out in atomic spectroscopy since 1940’s.

2. Two-qubit gates: none currently implemented (conditional logic was demonstrated)

Proposed interactions for two-qubit gates:

(a) Electric-dipole interactions between atoms (b) Ground-state elastic collisions(c) Magnetic dipole interactions

Only one gate proposal does not involve moving atoms (Rydberg gate).

Advantages: possible parallel operationsDisadvantages: decoherence issues during gate operations

Rydberg gate scheme

Why Rydberg gate?

Gate operations are mediated by excitation of Rydberg statesJaksch et al., Phys. Rev. Lett. 85, 2208 (2000)

Rydberg gate scheme

Do not need to move atoms!

Gate operations are mediated by excitation of Rydberg statesJaksch et al., Phys. Rev. Lett. 85, 2208 (2000)

FAST!FAST!

Local blockade of Rydberg excitations

Excitations to Rydberg states are

suppressed due to a dipole-dipole

interaction or van der Waals interaction

http://www.physics.uconn.edu/~rcote/

Rydberg gate scheme

1 2

1

0

R

Rb

5s

40p

00 00

10 0

1

1

1 11

01 10

      

Apply a series of laser pulses to realize the following logic gate:

Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000)

FAST!FAST!FAST!FAST!

Rydberg gate scheme

1 2

1

0

R

Rb

5s

40p

00 00 00

1 2 2 1

00

01 01 01 01

10 0 0 10

11 1 1 11

R R

R R

4. Long relevant decoherence times

Memory: long-lived states.

Fundamental decoherence mechanism for optically trapped qubits: photon scattering.

Decoherence during gate operations: a serious issue.

5s1/2

F=2

F=1 0

16.8 GHz

5: Reading out a result

“Quantum jump” method via cycling transitions.Advantages: standard atomic physics technique, well understood and reliable.

Experimental status

All five requirements for quantum computations are implemented.

Specifically two-qubit gates are demonstrated in several implementations.

Several-qubit systems and search algorithm with toy

data are demonstrated with NMR.

Decoherence-free subspaces, noiseless subsystems, and active control are demonstrated.

Future…

Whatever is the future of the Whatever is the future of the Quantum computation it is certainly Quantum computation it is certainly

a fascinating new fielda fascinating new field for human creativity, ingenuity and for human creativity, ingenuity and

Our strive to learn more Our strive to learn more about the Universe.about the Universe.