Fundamental gravitational limitations to quantum computing

1
Fundamental gravitational limitations to quantum
computing

• Rafael A. Porto
• (Carnegie Mellon U. University of the Republic,
  Uruguay.)
• In collaboration with
• Rodolfo Gambini Jorge Pullin

2
Outline

• Limits on QC from standard QM
• Fundamental limits on space-time measurements
• Relational time and decoherence from QG
• Fundamental limits on QC revised
• Conclusions

3
Limits on QC from standard QM(S. Lloyd Science
406, 1047 (2000) )

• Margolus-Levitin Theorem
• Quantum Gates Unitary Evolution among
  orthogonal states
• To perform an elementary logical operation in
  time
• requires an average amount of energy
• As a consequence, a system with average energy E
  can
• perform a maximum of n op/s.
• For an ultimate laptop (1Kg, 1 liter) this
  bound turns out to
• be op/s, independently of being
  parallel or serial.

4

• Limits of memory/entropy (Bekenstein, Lloyd)
• For a 1Kg, 1 liter computer, the maximum entropy
  can be
• estimated to be, L
• For a more realistic computer, L
  bits.
• Error bounds
• If is the probability of being erroneous,
  the maximum
• error rate is given by . In the other hand,
• the maximum rate it can tolerate is
  
• (error correction can not go faster than speed of
  light. R typical size)

5
Fundamental limits on space-time measurements(Y.
Ng Annals N.Y.Acad.Sci. 755, 579 (1995))

• Basic QM and GR principles.
• More accurate clocks more mass
  (Wigner)
• Prevent gravitational collapse
• Ultimate accuracy
• Black holes saturate this bound as the most
  accurate clocks.
• Think of a BH as a (dumped) oscillator.

6
Relational time and decoherence from QG (
R. Gambini, RAP, J. Pullin, New Journal Physics
6, 45 (2004), Phys. Rev. Lett. 94, 240401,
(2004) )

• Conditional probabilities between physical
  observables
• By considering semiclassical states of the time
  variable we
• can obtain and approximated Schroedinger
  evolution,
• with,
• For an optimal clock,

7
When the evolution of a quantum system is
described by a real clock a similar equation was
obtained by phenomenological arguments (open
systems, thermal fluctuations, etc) Milburn
Phys. Rev. A44 5401 (1991) Eguzquiza et al.
Phys.Rev A59 3236 (1999) Bonifacio Nuovo
Cimento114B 473 (1999) This effect has been
observed in the Rabi oscillations describing the
exchange of excitations between atoms and fields.
Meekhof et al. Phys.Rev.Lett. 76, 1796
(1996) Still orders of magnitude away from QG
effects
8

• As we stated evolution is no longer unitary and
  states do
• not completely evolve into orthogonal states
  according to ML.
• For an initial state , we
  will have
• and therefore,
• For a NOT gate we will have after a time

9
Fundamental limits on QC revised(R. Gambini,
RAP, J. Pullin )

• The extension of ML theorem is state-dependent.
• The bound is however saturated when the QC is in
• serial mode (All its resources (E,L) are used
  per logical
• operation.) fast step rate
• Decoherence effect for
• 1Kg computer in serial mode
• remarkably large
• A QC can not
  utilize all its resources.
• However, an ultimate laptop has a degree of
  parallelization
• ( ) of the order of

10

• The difference with serial mode is that now
  energy is
• redistributed amongst parallel qubits and
  the energy
• per gate goes down to
• Similarly to what happens in the serial case
• An ultimate laptop can not utilize all its
  mass-energy
• resources without running into an error crash.
• The new bound for the number of operations per
  second
• turns out to be
• This expression is general for a QC of L bits and
  size R operating with a given dp. The numerical
  estimates was obtained from Lloyds values. For
  dp1, n lt

11

• If one is interested in miniaturization, one may
  wish to
• consider BH as QC (Lloyd, Ng).
• In this case Bekenstein bound applies and a
  similar
• calculation leads to
• For a BH of mass M. For a 1kg BH we have again
• approx the same bound as before.
• Finally let us add if one wished to consider a
  more realistic
• (Avogadro) computer the bound is also a few order
  of
• magnitude stringent to that of Lloyd.

12
Conclusions

• Quantum computing faces the fundamental limits of
  Nature.
• Based on QG ideas (a fully quantum relational
  notion of
• time and Heisenberg-like uncertainties in time
• measurements) a modification of standard QM is
• introduced, and a fundamental decoherence effect
  found,
• which provides a new path for phenomenological
• applications as well as providing new conceptual
  hints (BH
• information paradox). Macroscopic quantum
  effects, such
• as QC, are amongst the promising probes. As an
  example,
• here it was shown that QG put more stringent
  constraints
• in the maximum number of operations per second a
  QC
• can achieve than standard QM. The quantum
  character of
• time might end up tested at home rather than in
  the skies.