IWEPNM 2007, Kirchberg Coworkers Principles of Quantum Computing Michael Mehring Physikalisches Institut, Univ. Stuttgart, Germany A.Heidebrecht, S. Krämer,

IWEPNM 2007, Kirchberg

Coworkers

Principles of Quantum Computing

Michael MehringPhysikalisches Institut, Univ. Stuttgart, Germany

A. Heidebrecht, S. Krämer, O. Mangold J. Mende, W. Scherer

15N@C60 and 31P@C60 Cooperation

Wolfgang Harneit FU Berlin


Richard Feynman was one of the first physicists, who contemplated about quantum computing already in 1982

R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwart, J. Appl. Phys. 28, 49 (1957).


The Quantum Bit (Qubit)

Stern Gerlach Experiment (1922)


Quantumstates of a Qubit (Spin ½) The Spin Density Matrix

0

1

0

1

0

1

SuperpositionPopulation

Phase X Phase Y


The Single Qubit

, , 0 , 1Basis: or or 0 1

2 20cos sin2

12

i ie e

2 2 1; , complex

x

y

z

0

1Bloch sphere

Single qubit operations

X X

Y

Z

Y

Z

0 1

1 0

0

0

i

i

1 0

0 1

0 1NOT

0 1i

10 1

2 1

0 12

0 1

1 0

X X

?

8:35

See: Quantum Computation and Quantum Information M. A. Nielsen and I. Chuang, Cambridge University Press 2000


More Single Qubit Gates

0 10 1

2

1 0

0 ie

10 1

2ie 1

0 12

S1 0

0 i

T 4

1 0

0 ie

H1 11

1 12

Hadamard

H

0H 10 1

2

1 0

0 1

H H0 1

1 0

X X


A silicon-based nuclear spin quantum computer

B.E.Kane

Nature 393, 133 (May 1998)

NMR-Quantum-Computing a là Kane


Array of Quantum Dots with Spins

Single spin in Q-dot

Daniel Lossproposal

F. H. L. Koppens et. al. Nature, 442, 766 (2006)


Other Qubits: Atoms and ions in traps, photons, Q-dots, superconductors

0 VPhotons as flying qubits

Qalgorithm

1 H

Superconducting qubit: Nakamura et al.Spin echo with sc-qubit

Bennet, Zeilinger, Zoller, Weinfurter et al.

Two level systemFictitious spin 1/2

Ion traps: J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995)

8:40


Two Qubits

0 101

21 0 11

1

20

, , 00 01 10 , 11Basis:

Bell states (entangled states):

product state 0 1 0 1 00 01 10 11


The Greek LAOCOONand Quantum Entanglement

H1

0

Hadamard UCNOT

The EPR PairEinstein, Podolsky and Rosen,

Phys. Rev. 47, 777 (1935)

12EPR

12


Two Qubit Gates

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

01 01CNOT

11 10CNOT

Controlled NOT (CNOT)

x

CNOT

10 11CNOT

00 00CNOT

How to generate an entangled state

x

CNOT

H0

1 1

1 1

201 01 1 0 1

21 1 0CNOTH

8:45


Quantencomputing in Hilbertspace

0

1

U

8 qubits span a256 dimensional Hilbertspace

0U t

t

Quantum evolution in Hilbertspace

Quantumcomputing is: Preparation+Superposition+Entanglement+Projection


Two Qubit Algorithms

, , 00 01 10 , 11

0 11

20 1

Superdense coding: Alice Bob

Alice is allowed to send only a single qubit. Can she transmit one of four different bits of information?

Suppose Alice and Bob share the entangled state:

Alice performs one of the following single qubit operations on her qubit:

Alice sends her qubit to Bob. Bob performs a Bell state measurement.

1 1;

2 21 1

00 11 00 1

; 2

1

10 01 - 10 02

1

I Z

X iY

I, Z, X, iY

8:50



Scenario of Quantum-Teleportation

„Gentlemen beamme aboard“

Captain Kirk

source ofentangledEPR pairs

qubit 2 qubit 3

32322

1EPR

Quantum channel

Alice Bob

classical channelqubit 1

8:55


Quantum Teleportation

x

H M1

M2

X Z

Alice

Bob

Alice receives 0 1 and wants to teleport it to Bob

0 1 2 3

0 001

201 01 10 11

1 0 10 11

1 12

01 00

Initial state

After CNOT

After Hadamard 2 (next page) is followed by Alices measurements


Procedure before and after Alices measurements

2

0

1

0

1

0

01

1

0 1

1 0

0 1

1

2

1 0

0 0 10 1 0, , or 11

00 1 0

The first two qubits belong to Alice the third one to Bob

Alice performs the measurementsand finds either

Alice phones the result to Bob. Bob knows which operations he needs to perform

Alice Bob

01 1 1 00 X

10 1 0 01 Z

01 1 1 01 iY


Quantum Teleportation with Photons

Experiments performed by Bennet, DeMartini, Zeilinger, Weinfurter et al.

Jian-Wei Pan et al. Phys. Rev. Lett. 80, 3892 (1998)D. Boschi et al., Phys. Rev. Lett. 80, 1121 (1998)


There is no danger of getting beamed off(right away)

100 Million Centuries

with today‘s technology.

S. Braunstein: The complete description of a human being would require about 1032 bits. The Teleportation of this information would take about


The Deutsch Jozsa Algorithm

Mapping a single bit to a binary function f(x) with x {0,1} and f(x) {0,1}

x f00(x) f01(x) f10(x) f11(x) 0 1

0 0

0 1

1 0

1 1

constant constantbalanced balancedDeutsch:

Decision for f(x) being constant or balanced can be taken by a quantum computer in a single tep.

D. Deutsch and R. Jozsa, Proc. Roy. Soc. London A 439, 553 (1992)

9:00


Deutsch Jozsa Algorithm

Uf(x)

H

H

H

H

0

1 1

01 0 1f f

Frontside

Backside

ancilla ancilla


NMR Results of the Deutsch Jozsa Algorithm

f000

constant

f110

constant

f011

balanced

f101

balanced

J.A.Jones and M. Mosca, J. Chem. Phys. 109, 1648 (1998), I.L. Chuang et al. Nature 393, 143 (1998)

Sven Zülsdorff, master thesis, Stuttgart, 1999

M. S. Anwar et al. Phys. Rev. A 70, 032324 (2004) (parahydrogen)


Deutsch Collins Algorithm with three Qubits

Example:

x 000 001 010 011 100 101 110 111

f(x) 0 0 1 1 0 1 1 0

There are 28 = 256 functions in total!

Among these are 2 constant and 8 704

balanced functions

In the following we consider only the basic 1(constant) + 8 (balanced)functions f(x). The others follow from cyclic rotations and inversionsof the 0 and 1 bits.

H

H

HOracle

Uf(x)

H

H

0 H

0

0

out

constant000

balanced≠000


Oracle Transforms

1 0U I4

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

U

6 1 1 2 1 3 1 2 32 4z z z z z z z zU I I I I I I I I 6

1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

U

4 1 24 z zU I I

Entanglement Alert

No entanglement involved

4 0 1 0 2 0 3

1 1 1

2 2 2z z zI I I I I I

output:

A.H. Dorai and A. Kumar Paramana J. Phys. 56, 705 (2001)J. Kim, J. S. Lee and S. Lee, Phys. Rev. A 62, 0223204 (2000) O. Mangold, A. Heidebrecht and M.M. Phys. Rev. A 70, 042307 (2004)


Output of Oracle U6 = U00011110

experimental

calculated

Where is the entanglement?


Local Rotations of Bell States

3 2 6

0 1 0 2 3 2 3 2 3

2

1 1

2 4

x y

z x x y y z z

P P

I I I I I I I I I

experimental

calculated 3 2 62

1 10 0

2 20 0 0 0 0 0

0 1 0 0 0 0

1 10 0

2 2

x yP P


Find the Prime Factors (Shor Algorithm)

It is a simple task to build the product of two large prime numbers p und q. Calculating n = p x q is easy.

But: 137703491 = ?137703491 = 7919 x 17389

is extremely demanding and requires log(n) steps with arbitrary large superpolynomial

Factoring a 400 digit number would take 1010 years with today's fastest computers

P. Shor(1994): Quantum computer requires only O[(ln n)3] steps

A quantum computer based on the current fastest clock rates would factor a 400 digit number in only about 3 years.

9:05


Implementing the Shor Algorithm on a Nuclear Spin Quantumcomputer

|R1>|R2>=|x,f(x)> = |n3,n2,n1>|m4,m3,m2,m1>

We need two quantum registers R1(3 qubit) und R2(4 qubit), which contain x and f(x):

7

0

1 ,7 158

x

xx

mod

1 0,1 1,7 2,4 3,13 4,1 5,7 6,4 7,138

Superposition:

L. M. K. Vandersypen et al., Nature 414, 883 (2001)


NMR experiment factors numbers with Gauss sums

2

0

1exp 2

1

MMN

m

NA l im

M l

2

0

1cos 2

1

MMN

m

NC l m

M l

from number theory

157573 =

M. Mehring, K. Müller, W. Merkel, I. S. Averbukh, W. P. Schleich, quant-ph/0609174 v1, Phys. Rev. Lett. 98, 120502 (2007)


1H NMR implementation with phase controlled pulse sequence

phase controlled CPMG pulse sequence

M. Mehring., K. Müller, W. Merkel, I. S. Averbukh, W. P. Schleich, Phys. Rev. Lett. 98, 120502 (2007)


Proposals for Quantumcomputing with N@C60

Wolfgang Harneit, Phys. Rev. A 65, 0232222 (2002)

D. Suter and K. Lim, Phys. Rev. A. 65, 052309 (2002)

9:10


The pseudo pure initial state

(1 2)(2 8) /2109.5

4

1

31

21

6

1 14 2

0 0 0 00 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0

0 0 0 00 0 0 0

0

0 0 0 0 0

0 0 0 00 1 00 0 0 0 0 0 0 0

PPzS

I

10 4 1 2 1 21 1 1ˆ4 2 2z z z zI I I I I

7 7

15N@C60

two qubit subspace

3

2S

1

2I

7 7


Preparation and tomography of entangled statesin 15N@C60

M.M. and W. Scherer. Phys. Rev. Lett. 93, 206603-1 (2004)

W. Scherer and M. Mehring, arXiv e-print, (quant-ph/0602201), 2006

12 2 7 7 2 7 7 2

2


Phase Encoding of Entangled States

= 1 MHz

= 6 MHz

phase frequencies 1=2 MHz; 2=1 MHz 3 1 3 1

2 2 2 2

12

3 1 3 1

2 2 2 2

12

Application of phase rotations about the z-direction

1 223i

e

1 223i

e

1 223i

e

1 223i

e


Two Qubit Subsystems in 31P@C60

Planned experiments: Swap and qubit cloning, decoherence free subspaces

Cooperation with W. Harneit, FU Berlin Preliminary results (J. Mende, B. Naydenov)

Reduced symmetryLifting of degeneracies

9:10


The S-Bus Concept

M. M. and J. Mende Phys. Rev. A 73, 0520303 (2006)


Ce:CaF2 single crystal: The qubyte+1

Mims ENDOR

12Sm

12Sm

Energy

9:15


Implementation of the Collins versionof the Deutsch-algorithm

00

010100 01

UH H

M. M. and J. Mende Phys. Rev. A 73, 0520303 (2006)

01

00


Two Qubit (19F) Entanglement in the S-Bus Ce:CaF2


A. Heidebrecht

S. Krämer

O. Mangold

W. Scherer

J. Mende

Documents

IWEPNM 2007, Kirchberg Coworkers Principles of Quantum Computing Michael Mehring Physikalisches Institut, Univ. Stuttgart, Germany A.Heidebrecht, S. Krämer,