quantum computing with molecular nanomagnets pres... · overview of the presentation. part 1...
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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