Quantum computing withmolecular nanomagnets

Spring school on

NANOMAGNETISM and SPINTRONICS

Cargese, Corsica May 2005

M. AffronteINFM - S3 National Research Center

on nanoStructures and Biosystems at SurfacesModena, Italy.

Overview of the presentation.Part 1• Back in time…• Classical bit & qubit• Quantum algorithms• Quantum hardware• Read out.

Part 2•Recipe: the DiVincenzo criteria•Grover’s algorithm with Mn12

•AF spin clusters•Cr7Ni molecular ring as qubits.

Motivations:• Miniaturization: Moore’s law

Gordon Moore“every 1.5 years complexity doubles”

†

a

b

Ê

Ë Á

ˆ

¯ ˜

†

fa

b

Ê

Ë Á

ˆ

¯ ˜

Quantum gate

Q

abstract notion of programmable machine:the Turing machine.

Can any algorithm process be simulated efficiently using a Turingmachine?

Efficient algorithm is one that runs in time polynomial in size of theproblem solved.

Alan Turing (1912-1954)

R. Feynman (‘80s).

… there seemed to be essential difficulties insimulating quantum mechanical systems

He suggested to build computers basedon the principles of quantum mechanics

Universal Quantum computer• 1985 D. Deutsch suggested that a universal quantum

computer is sufficient to efficiently simulate an arbitraryphysical system.

Parallelism Computer- Quantum system

Processor - Physical systemComputation - Motion

Input -Initial stateRules -Law of motion

Output -Final state

classical circuits

bit 0, 1

the first transistor made of Germanium0.5”

logic Boolean gates:

Universal set of gates:

Truth table

Bit & qubit

Atomsatoms of rubidium (ENS experiments) or berylium (NIST experiments)

Quantum algorithms.Quantum mechanics: algebra of s=1/2Logic gates.Quantum Algorithms:

Loss-DiVincenzo, Universal setShor’s,Grover’s,

Representation of qubitDirac’s notation of two-level system

†

0 , 1

†

y = a 0 + b 1 ¨ Æ æ a

b

Ê

Ë Á

ˆ

¯ ˜

Bloch sphere

†

y = cos J2

0 + eif sin J2

1†

y y = a2

+ b2

=1

Spin 1/2

x

y

z

Pauli matrices

is a perfect qubit!

†

↑

†

Ø

†

H = gmBS ⋅ B

S =12

r s

s x =0 11 0

Ê

Ë Á

ˆ

¯ ˜ ;s y =

0 -ii 0

Ê

Ë Á

ˆ

¯ ˜ ;s z =

1 00 -1

Ê

Ë Á

ˆ

¯ ˜ ;

Multiple qubit

†

00 ; 01 ; 10 ; 11

†

y = a00 00 +a01 01 +a10 10 +a11 11

…

†

x1x2...xn

xi = 0,1Hilbert space 2n

†

y =

a1

a2

...an

Ê

Ë

Á Á Á Á

ˆ

¯

˜ ˜ ˜ ˜

Motion & Quantum gates

Schrödinger equation

†

ihd y

dt= H(t)y

y(t) = e-

iH (t )h y(0)

Unitary transformation

†

y(t) = U y(0)U +U = I

Examples of Quantum gates:one qubit gates

†

I =1 00 1

È

Î Í

˘

˚ ˙

X =0 11 0

È

Î Í

˘

˚ ˙

S =1 00 i

È

Î Í

˘

˚ ˙

H =12

1 11 -1È

Î Í

˘

˚ ˙

Unity

NOT

Phase

Hadamard

truth table

†

a 0 + b 1 Iæ Æ æ a 0 + b 1a 0 + b 1 Xæ Æ æ b 0 + a 1

a 0 + b 1 Sæ Æ æ a 0 + ib 1

a 0 + b 1 Hæ Æ æ a0 + 1

2+ b

0 + 12

two-qubit gates

†

A

†

A

†

B

†

A ⊕ B

†

UCN =

1 0 0 00 1 0 00 0 0 10 0 1 0

È

Î

Í Í Í Í

˘

˚

˙ ˙ ˙ ˙

truth table†

11

†

10

†

01

†

00

†

00

†

01

†

10

†

11

C-Not

Universal set of quantum gatesAny unitary operators can be approximated by acombination of few one-qubit and two-qubit gates

Possible universal sets:

•Phase+Hadamard+CNOT+p/8 gates• General rotation of one-qubit+CNOT

Loss-DiVincenzo scheme

one-qubit gate

two-qubit gate

read-out

D. Loss, P. DiVincenzo Phys. Rev.A57, 120 (1998)Nanotechnology 16, R27 (2005)

universal quantum gates

Shor’s algorithm• 1994 P. Shor demonstrates that the problem of finding prime

factors of an integer can be efficiently solved on a quantumcomputer.

Efficiency: classical ˜exp(n1/3) ; quantum ˜n2logn

†

j Æ1N

e2pijk / N

k=0

N -1

 kQuantum Fourier transform

discrete log

factoring

order finding

Grover’s algorithm• 1995 L. Grover showed that the problem of

conducting a search through some unstructuratedsearch space can be sped up on a quantum computer.

Efficiency: classical ˜N ; quantum ˜√N

Quantum search

StatisticsNP problems

†

x p oracle-operatoræ Æ æ æ æ æ x p ⊕ f (x)

f (x) =01

Ï Ì Ó oracle operator

Quantum hardware

cold ion trapssingle photonsnucleiquantum dotsJosephson Junctions…

Cold ion trap

J.I. Cirac and P. Zoller PRL 74, 4091 (1995)

Single Photons

Q.A. Turchette etal. PRL 75, 4710 (1995)

Quantum Phase gate

NMR:

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

factoring 15 in prime numbers!

Quantum dots

See L. KouwenhovenS. Tarucha

Josephson Junctions

Mc Dermott, SCIENCE VOL 307 25 FEBRUARY 2005

environment

decoherence

qubitnoise

Read out: measuring qubits.

Operators Mm are associated to measurements of quantum properties

Projection measurement of a qubit in the computational basis:

†

M0 = 0 0M1 = 1 1

†

M0 y = a 0M1 y = b1

entanglement

When two systems, of which we know the states by their respectiverepresentatives, enter into temporary physical interaction due to known forcesbetween them, and when after a time of mutual influence the systems separateagain, then they can no longer be described in the same way as beforeSchrödinger

†

00 + 112

Bell state or EPR pair(Einstein, Podolsky, Rosen)

A measurement of the second qubitalways gives the same result as thefirst

Quantum probability

• average in time in single devices• “one shot” measurement, average over ensemble

measurements on ensemble:molecular magnets

W. Wernsdorfer R. Sessoli, Science 284, 133 (1999)

Thomas, L.!; Lionti, F.!; Ballou, R.!; Gatteschi, D.!; Sessoli, R.!; Barbara,B.,!Nature, 1996, 383, 145-147.

Sangregorio, C.!; Ohm T.!; Paulsen C.!; Sessoli R.!; Gatteschi D. Phys. Rev. Lett., 1997, 78, 4645-4648.

Magnetic Resonance Force Microscopy

D. Rugar et al. Nature 430, 329 (2004)

Single-shot read-out of an individual electronspin in a quantum dot

J.M. Elzerman et al. Nature 430, 431 (2004)

Spin Polarized-STM

Wiesendanger, Science 288, 1805 (2000); Science 298, 577 (2002); Science 292, 2053 (2001)

useful readings and web-sighting.• M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum

Information, (Cambridge University Press, Cambridge, 2000).

• http://www.qubit.org/• http://www.theory.caltech.edu/people/preskill/ph229/• http://qubit.damtp.cam.ac.uk/• http://www.quiprocone.org/Protected/DD_lectures.htm• http://www.weizmann.ac.il/condmat/heiblum.html