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Communications in Mathematical Physics

, Volume 350, Issue 2, pp 603–637 | Cite as

Thermalization Time Bounds for Pauli Stabilizer Hamiltonians

  • Kristan Temme
Article

Abstract

We prove a general lower bound to the spectral gap of the Davies generator for Hamiltonians that can be written as the sum of commuting Pauli operators. These Hamiltonians, defined on the Hilbert space of N-qubits, serve as one of the most frequently considered candidates for a self-correcting quantum memory. A spectral gap bound on the Davies generator establishes an upper limit on the life time of such a quantum memory and can be used to estimate the time until the system relaxes to thermal equilibrium when brought into contact with a thermal heat bath. The bound can be shown to behave as \({\lambda \geq \mathcal{O}(N^{-1} \exp(-2\beta \, \overline{\epsilon}))}\), where \({\overline{\epsilon}}\) is a generalization of the well known energy barrier for logical operators. Particularly in the low temperature regime we expect this bound to provide the correct asymptotic scaling of the gap with the system size up to a factor of N −1. Furthermore, we discuss conditions and provide scenarios where this factor can be removed and a constant lower bound can be proven.

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References

  1. 1.
    Peter W.S.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), R2493 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dennis E., Kitaev A., Landahl A., Preskill J.: Topological quantum memory. J. Math. Phys. 43(9), 4452–4505 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gottesman D.: Theory of fault-tolerant quantum computation. Phys. Rev. A 57(1), 127 (1998)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Gottesman, D.: Stabilizer Codes and Quantum Error Correction. arXiv preprint arXiv:quant-ph/9705052 (1997)
  5. 5.
    Yoshida B.: Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes. Ann. Phys. 326(1), 15–95 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Xiao-Gang, W.: Quantum field theory of many-body systems from the origin of sound to an origin of light and electrons. Oxford University Press Inc., New York. ISBN 019853094, 1 (2004)Google Scholar
  7. 7.
    Nussinov Z., Ortiz G.: Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems. Phys. Rev. B 77, 064302 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Nussinov Z., Ortiz G., Cobanera E.: Effective and exact holographies from symmetries and dualities. Ann. Phys. 327(10), 2491–2521 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chesi S., Loss D., Bravyi S., Terhal B.M.: Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes. New J. Phys. 12, 025013 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Alicki R., Horodecki M., Horodecki P., Horodecki R.: On thermal stability of topological qubit in kitaev’s 4d model. Open Syst. Inf. Dyn. 17(01), 1–20 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Alicki R., Fannes M., Horodecki M.: On thermalization in kitaev’s 2d model. J. Phys. A: Math. Theor. 42(6), 065303 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Davies, E.B.: Quantum theory of open systems. IMA, London (1976)Google Scholar
  13. 13.
    Davies E.B.: Generators of dynamical semigroups. J. Funct. Anal. 34(3), 421–432 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lindblad G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119–130 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bravyi S., Terhal B.: A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. New J. Phys. 11(4), 043029 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    Landon-Cardinal O., Poulin D.: Local topological order inhibits thermal stability in 2d. Phys. Rev. Lett. 110(9), 090502 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    Haah J., Preskill J.: Logical-operator tradeoff for local quantum codes. Phys. Rev. A 86(3), 032308 (2012)ADSCrossRefGoogle Scholar
  19. 19.
    Brown, B.J., Loss, D., Pachos, J.K., Self, C.N., Wootton, J.R.: Quantum Memories at Finite Temperature. arXiv preprint arXiv:1411.6643 (2014)
  20. 20.
    Sergey B., Jeongwan H.: Quantum self-correction in the 3d cubic code model. Phys. Rev. Lett. 111(20), 200501 (2013)CrossRefGoogle Scholar
  21. 21.
    Michnicki Kamil P.: 3d topological quantum memory with a power-law energy barrier. Phys. Rev. Lett. 113(13), 130501 (2014)ADSCrossRefGoogle Scholar
  22. 22.
    Yoshida, B.: Violation of the Arrhenius Law Below the Transition Temperature. arXiv preprint arXiv:1404.0457 (2014)
  23. 23.
    Chuang I.L., Nielsen M.A.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  24. 24.
    Alicki R., Lendi K.: Quantum Dynamical Semigroups and Applications. Springer, Berlin (2007)zbMATHGoogle Scholar
  25. 25.
    Breuer H.P., Petruccione F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)zbMATHGoogle Scholar
  26. 26.
    Temme K.: Lower bounds to the spectral gap of Davies generators. J. Math. Phys. 54(12), 122110 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Temme K., Kastoryano M.J., Ruskai M.B., Wolf M.M., Verstraete F.: The \({\chi}\)2-divergence and mixing times of quantum markov processes. J. Math. Phys. 51(12), 122201–122201 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Spohn H.: Entropy production for quantum dynamical semigroups. J. Math. Phys. 19, 1227 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Weiss, U.: Quantum Dissipative Systems, vol. 10. World Scientific, Singapore (1999)Google Scholar
  30. 30.
    Kossakowski A., Frigerio A., Gorini V., Verri M.: Quantum detailed balance and KMS condition. Commun. Math. Phys. 57(2), 97–110 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sanz M., Pérez-García D., Wolf M.M., Cirac J.I.: A quantum version of wielandt’s inequality. Inf. Theory IEEE Trans. 56(9), 4668–4673 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Martinelli F.: Lectures on glauber dynamics for discrete spin models. Lect. Probab. Theory Stat. (Saint-Flour) 1997, 93–191 (1997)zbMATHGoogle Scholar
  33. 33.
    Martinelli F., Olivieri E.: Approach to equilibrium of glauber dynamics in the one phase region. Commun. Math. Phys. 161(3), 447–486 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Fuchs, C.A., De Van Graaf, J: Cryptographic distinguishability measures for quantum-mechanical states. Inf. Theory IEEE Trans. 45(4), 1216–1227 (1999)Google Scholar
  35. 35.
    Sinclair A., Jerrum M.: Approximate counting, uniform generation and rapidly mixing markov chains. Inf. Comput. 82(1), 93–133 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of markov chains. Ann. Appl. Probab. pp. 36–61 (1991)Google Scholar
  37. 37.
    Fill, J.A.: Eigenvalue bounds on convergence to stationarity for nonreversible markov chains, with an application to the exclusion process. Ann. Appl. Probab. pp. 62–87 (1991)Google Scholar
  38. 38.
    Gorini V., Kossakowski A., Frigerio A., Verri M.: Quantum detailed balance and kms condition. Commun. Math. Phys. 57, 97 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Diaconis P., Saloff-Coste et al. L.: Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6(3), 695–750 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kastoryano Michael J., Temme Kristan: Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys. 54(5), 052202 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Olkiewicz R., Zegarlinski B.: Hypercontractivity in noncommutative Lp spaces. J. Funct. Anal. 161(1), 246–285 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Temme K., Pastawski F., Kastoryano Michael J.: Hypercontractivity of quasi-free quantum semigroups. J. Phys. A: Math. Theor. 47(40), 405303 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Boman Erik G., Hendrickson B.: Support theory for preconditioning. SIAM J. Matrix Anal. Appl. 25(3), 694–717 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Chen D., Gilbert J.R.: Obtaining bounds on the two norm of a matrix from the splitting lemma. Electron. Trans. Numer. Anal. 21, 28–46 (2005)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Bhatia, R.: Matrix Analysis, vol. 169. Springer, New York (1997)Google Scholar
  47. 47.
    Sinclair A.: Improved bounds for mixing rates of markov chains and multicommodity flow. Combin. Probab. Comput. 1(04), 351–370 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Chung Fan, R.K.: Spectral Graph Theory, vol. 92. American Mathematical Soc, New York (1997)Google Scholar
  49. 49.
    Yoshida B., Chuang Isaac L.: Framework for classifying logical operators in stabilizer codes. Phys. Rev. A 81(5), 052302 (2010)ADSCrossRefGoogle Scholar
  50. 50.
    Kastoryano Michael J., Brandao Fernando G.S.L.: Quantum gibbs samplers: the commuting case. Commun. Math. Phys. 344(3), 915–957 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Peierls, R.: On ising’s model of ferromagnetism. In: Mathematical Proceedings of the Cambridge Philosophical Society, 32:477–481, 10 (1936)Google Scholar
  52. 52.
    Stark C., Pollet L., Imamoğlu A., Renner R.: Localization of toric code defects. Phys. Rev. Lett. 107(3), 030504 (2011)ADSCrossRefGoogle Scholar
  53. 53.
    Wootton James R., Pachos Jiannis K.: Bringing order through disorder: Localization of errors in topological quantum memories. Phys. Rev. Lett. 107(3), 030503 (2011)ADSCrossRefGoogle Scholar
  54. 54.
    Majewski Adam W., Olkiewicz Robert, Zegarlinski Boguslaw: Dissipative dynamics for quantum spin systems on a lattice. J. Phys. A: Math. Gen. 31(8), 2045 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute for Quantum Information and MatterCalifornia Institute of TechnologyPasadenaUSA
  2. 2.IBM TJ Watson Research CenterYorktown HeightsUSA

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