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Fundamental gravitational limitations to quantum computing

Fundamental gravitational limitations to quantum computing. Rafael A. Porto (Carnegie Mellon U. & University of the Republic, Uruguay.) In collaboration with Rodolfo Gambini & Jorge Pullin. Outline. Limits on QC from standard QM Fundamental limits on space-time measurements

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Fundamental gravitational limitations to quantum computing

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  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 Lloyd’s values. For dp=1, n <

  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.

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