A scheme for quantum computing with trapped electrons in vacuum

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A scheme for quantum computing with trapped
electrons in vacuum
ESF International Workshop on Quantum
Information. Logical Aspects. Porto Conte
(Sassari, Sardinia) September 23-25, 2004.

• Irene Marzoli, Giacomo Ciaramicoli, and Paolo
  Tombesi
• Stefano Mancini and David Vitali
• Dipartimento di Fisica

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J. I. Cirac P. Zoller, PRL 74, 4091 (1995)
Trapped ions
D. Jaksch et al., PRL 85, 2208 (2000)
Neutral atoms
T. Sleator H.Weinfurter, PRL 74, 4087 (1995)
Q. A. Turchette et al., PRL 75, 4710 (1995) P.
Domokos et al., PRA 52, 3554 (1995)
V.Giovannetti et al., PRA 62, 032306 (2000)
Quantum information
Cavity QED
B. E. Kane, Nature 393, 133 (1998)
Solid-state devices
For a review see J. A. Jones, LANL e-print
archive quant-ph/0009002
NMR
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Requirements forQuantum Computation
D. P. DiVincenzo, The Physical Implementation of
Quantum Computation, Fortschritte der Physik 48,
771 (2000)

• A scalable physical system with well
  characterized qubits
• The ability to initialize the state of the qubits
  to a simple fiducial state (usually the system
  ground state)
• The coherence time much longer than the gate
  operation time
• A universal set of quantum gates
• A qubit-specific measurement capability.

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Electron in a Penning trap

• Extremely high experimental accuracy and control

• Well isolated from the environment, almost
  unaffected by
• dissipative decoherence
• Extremely high experimental accuracy and control
  negligible radiative damping
• No thermal excitation below 1 K

• Electron g factor R. S. Van Dyck,Jr., et al.,
  PRL 59, 26 (1987)
• Fine structure constant

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Trapping mechanism
L. S. Brown G. Gabrielse, Rev. Mod. Phys. 58,
233 (1986)
Single electron confined with STATIC FIELDS
(less noisy than rf potentials) - a uniform
magnetic field (parallel to the z axis) - a
quadrupole electric field
Magnetic field ? radial confinement (cyclotron
motion) Quadrupole ? axial confinement (axial
motion) ? slow
circular motion in the radial plane

(magnetron motion)
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Cylindrical Penning trap
S. Peil and G. Gabrielse, Phys. Rev. Lett. 83,
1287 (1999)

• Microwave cavity J. N. Tan G. Gabrielse, PRL
  67, 3090 (1991)
• Well defined cavity modes
• Quality factor
• Cavity-induced suppression of spontaneous
  emission of synchrotron radiation.

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Hamiltonian description
The dynamics of an electron of mass m and charge
e is given by
H ( p -eA/c)2 eV - µ B
After introducing suitable annihilation and
creation operators
Three independent harmonic oscillators and a
2-level system!
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Energy level structure
with
and
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A brief summary

• Axial motion
• Decay rate
• already cooled _at_cooled to

• Cyclotron motion
• Decay rate
• (inhibited by cavity effects)
• Ground state cooling

connected to an external circuit
(

• Magnetron motion
• Decay time of order of years
• It can be cooled with sideband cooling

• Spin motion
• Decay time of order of years

Up to three qubits in the same particle Very
low dissipative decoherence
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Relativistic corrections
A. E. Kaplan, PRL 48, 138 (1982) G. Gabrielse
et al., PRL 54, 537 (1985)

• ? h?c2/mc2 frequency shift of the order of
  200 Hz
• This tells us that the anharmonicity due to the
  relativistic correction permits the use of the
  first two level as qubitn the cyclotron motion
  (even if transition degeneracy is not completely
  removed)
• State-dependent transition frequencies

Quantum information can be encoded in the
following two qubits ?0?c , ?1?c, ??? ,
???
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A new concept of trap
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Corrections

• Small anharmonicities, due to de-compensation
  electrodes, introduce the shift ?e 3eC4/(m?z)2
  
• Application of an inhomogeneous static magnetic,
  the so-called magnetic bottle, to couple spin and
  axial motion
• State-dependent transition frequencies with ?m
  ?zeB1 / (2m2c ?c ?m)
• While keeping the cyclotron oscillator in its
  ground state, we can resolve any axial transition

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Single qubit operations
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Spin one-qubit rotations
k 2

k 3
k 1
k 2
?s 3dm/2
k 0
k 1
?s dm/2
k 0
s -1
s 1
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Conditional dynamics
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Spin qubit as target
Axial qubit as target
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Two-electron gates
By adjusting the external voltage, we can vary
the axial frequencies of two electrons. When in
resonance, they are coupled by the Coulomb
interaction. Otherwise they are basically
independent.
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Perturbative approach

• Oscillation amplitude of the two electrons much
  smaller than the separation d between them

In interaction picture (IP)
with the coupling strength ????1z/2

• e ltlt 1, with e being the ratio of the Coulomb
  energy, e2/4pe0d, to the potential energy of
• the 2nd electron with respect to the 1st trap,
  mw2zd2/2

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Swapping gate
The two coupled axial oscillators can exchange a
quantum of excitation. After a time tex ? ? / (2?)
The calculated swapping time ranges from 10 ns to
60 ms, depending on the electron distance d and
trapping frequency wz. Conditional dynamics
between neighboring electrons is obtained by
combining the swapping gate with the CNOT gate
in one site.
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Decoherence sources (1)
C. Henkel et al., Appl.Phys. B 69, 379 (1999) Q.
A. Turchette et al., PRA 61, 063418 (2000)
For copper electrodes at T 80 mK, td ranges
from 0.5 s to 1 hour 107 operations within the
decoherence time!
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De-coherence sources (2)
Electronic noise ? fluctuations in the electric
field amplitude and gradient (we assume
independent noise sources for each electrode)

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For low temperature electronics with spectral
density ?10-21 V2/Hz, we are in the limit of
fault-tolerant computation with 104 coherent
operations within the decoherence time!
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Conclusions

• Scalability
• Dense coding with more qubits per site
• High clock speed
• Detection through capacitance and charge
  measurements or using resonance frequency
  techniques
• Lower de-coherence effects than in semiconductor
  quantum dots
• Benefits from techniques, like composite pulses,
  already used in NMR and ion traps
• Quantum information with atoms, ions, photons,
  and ELECTRONS!

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• We acknowledge financial support from
• EU in the 6th Framework Programme under the
  QUELE project

New
Opening Post-doc position in the frame of
new QUELE (3 years) project . PhD position for EU
non Italians (3 years Marie Curie fellowship)
For further information paolo.tombesi_at_unicam.it