Communications in Mathematical Physics

, Volume 355, Issue 3, pp 905–947 | Cite as

Mixing Properties of Stochastic Quantum Hamiltonians

  • E. Onorati
  • O. Buerschaper
  • M. Kliesch
  • W. Brown
  • A. H. Werner
  • J. Eisert


Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.


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  1. 1.
    Brown W., Viola L.: Convergence rates for arbitrary statistical moments of random quantum circuits. Phys. Rev. Lett. 104, 250501 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    Brown, W., Fawzi O.: Scrambling speed of random quantum circuits. arXiv:1210.6644 (2012)
  3. 3.
    Brandao F.G.S.L., Horodecki M.: Exponential quantum speed-ups are generic. Quantum Inf. Comput. 13, 0901 (2013)MathSciNetGoogle Scholar
  4. 4.
    Brown, W., Fawzi, O.: Decoupling with random quantum circuits. Commun. Math. Phys. 340, 867–900 (2015)Google Scholar
  5. 5.
    Oliveira R., Dahlsten O.C.O., Plenio M.B.: Efficient generation of generic entanglement. Phys. Rev. Lett. 98, 130502 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    Brown, W., Fawzi, O.: Short random circuits define good quantum error correcting codes. In: Proceedings of the ISIT, pp. 346 (2013)Google Scholar
  7. 7.
    Brandao F.G.S.L., Cwiklinski P., Horodecki M., Horodecki P., Korbicz J., Mozrzymas M.: Convergence to equilibrium under a random Hamiltonian. Phys. Rev. E 86, 031101 (2012)ADSCrossRefGoogle Scholar
  8. 8.
    Hallgren, S., Harrow, A.W.: Superpolynomial speedups based on almost any quantum circuit. In: Proc. of the 35th Int. Coll. Aut. Lang. Prog. LNCS, vol. 5125, p. 782 (2008)Google Scholar
  9. 9.
    Lashkari N., Stanford D., Hastings M., Osborne T.J., Hayden P.: Towards the fast scrambling conjecture. JHEP 2013, 22 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Harrow A.W., Low R.A.: Random quantum circuits are approximate 2-designs. Commun. Math. Phys. 291, 257 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bouten L., van Handel R.: Discrete approximation of quantum stochastic models. J. Math. Phys. 49, 102109 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Brandao F.G.S.L., Harrow A.W., Harrow A.W.: Local random quantum circuits are approximate polynomial-designs. Commun. Math. Phys. 346(2), 397–434 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Weinstein Y.S., Brown W.G., Viola L.: Parameters of pseudo-random quantum circuits. Phys. Rev. A 78, 052332 (2008)ADSCrossRefGoogle Scholar
  14. 14.
    Belton A.C.R., Gnacik M., Lindsay J.M.: The convergence of unitary quantum random walks. Lancaster EPrints (2014). Accessed on 18 July 2017
  15. 15.
    Gross D, Audenaert K.M.R., Eisert J.: Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48, 052104 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Levin D.A., Peres Y., Wilmer E.L.: Markov Chains and Mixing Times. American Mathematical Society, New York (2008)CrossRefGoogle Scholar
  17. 17.
    Aldous D., Diaconis P.: Shuffling cards and stopping time. Am. Math. Soc. Mon. 93(5), 333–348 (1986)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dunjko V., Briegel H.J.: Quantum mixing of Markov chains for special distributions. New. J. Phys. 17, 073004 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    Kabanava M., Kueng R., Rauhut H., Terstiege U.: Stable low-rank matrix recovery via null space properties. Inf. Inference 5(4), 405–441 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ohliger M., Nesme V., Eisert J.: Efficient and feasible state tomography of quantum many-body systems. New J. Phys. 15, 015024 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    Knill E., Leibfried D., Reichle R., Britton J., Blakestad R.B., Jost J.D., Langer C., Ozeri R., Seidelin S., Wineland D.J.: Randomized benchmarking of quantum gates. Phys. Rev. A 77, 12307 (2008)ADSCrossRefMATHGoogle Scholar
  22. 22.
    Szehr O., Dupuis F., Tomamichel M., Renner R.: Decoupling with unitary approximate two-design. New J. Phys. 15, 053022 (2013)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Horodecki M., Oppenheim J., Winter A.: Quantum state merging and negative information. Commun. Math. Phys. 269, 107 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Bennett C.H., Devetak I., Harrow A.W., Shor P.W., Winter A.: Quantum reverse Shannon theorem. IEEE Trans. Inf. Theory 60(5), 2926–2959 (2014)CrossRefMATHGoogle Scholar
  25. 25.
    Buscemi F.: Private quantum decoupling and secure disposal of information. New J. Phys. 11, 123002 (2009)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Gogolin C., Eisert J.: Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems. Rep. Prog. Phys. 79, 56001 (2016)ADSCrossRefGoogle Scholar
  27. 27.
    Banks T., Fischler W., Shenker S., Susskind L.: m theory as a matrix model: a conjecture. Phys. Rev. D 55, 5112 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Maldacena J.: The large n limit of super-conformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 213 (1998)CrossRefGoogle Scholar
  29. 29.
    Sekino Y., Susskind L.: Fast scramblers. JHEP 10, 65 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    Asplund C.T., Berenstein D., Trancanelli D.: Evidence for fast thermalization in the plane-wave matrix model. Phys. Rev. Lett. 107, 171602 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    Hübener R., Sekino Y., Eisert J.: Equilibration in low-dimensional quantum matrix models. JHEP 2015, 166 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Bourgain J., Gamburd A.: A spectral gap theorem in su (d). J. Eur. Math. Soc. 14(5), 1455–1511 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Benoist, Y., de Saxcé, N.: A spectral gap theorem in simple Lie groups. Invent. Math. 205, 337–361 (2016)Google Scholar
  34. 34.
    Montroll E.W., Weiss G.H.: Random walks on lattices ii. J. Math. Phys. 6, 167–181 (1965)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Weiss G.H.: Aspects and applications of the random walk. J. Stat. Phys. 79(1), 497–500 (1995)Google Scholar
  36. 36.
    Zaburdaev, V., Denisov, S., Hanggi, P.: Perturbation spreading in many-particle systems: a random walk approach. Phys. Rev. Lett. 106(18), 180601 (2011)Google Scholar
  37. 37.
    Schulz, J.H.P., Barkai, E.: Fluctuations around equilibrium laws in ergodic continuous-time random walks. Phys. Rev. E 91, 062129 (2015)Google Scholar
  38. 38.
    Chaudhuri, P., Gao, Y., Berthier, L., Kilfoil, M., Kob, W.: A random walk description of the heterogeneous glassy dynamics of attracting colloids. J. Phys. Cond. Matter 20, 244126 (2008)Google Scholar
  39. 39.
    Watrous J.: Semidefinite programs for completely bounded norms. Theor. Comput. 5, 11 (2009)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Low, R.: Pseudo-randomness and learning in quantum computation. Ph.D. thesis, university of Bristol (2010)Google Scholar
  41. 41.
    Collins B., Sniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Dyson, F.J.: The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75, 486–502 (1949)Google Scholar
  43. 43.
    Dollard J.D., Friedman C.N.: Product integrals and the Schrödinger equation. J. Math. Phys. 18, 1598 (1977)ADSCrossRefMATHGoogle Scholar
  44. 44.
    Ito S.: Brownian motions in a topological group and in its covering group. Rend. Circ. Mat. Palermo 1, 40–48 (1952)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Tsirelson, B.: Unitary Brownian motions are linearisable. arXiv:math/9806112 (1988)
  46. 46.
    Liao M.: Lévy Processes in Lie Groups, vol. 162. Cambridge university press, Cambridge (2004)CrossRefGoogle Scholar
  47. 47.
    McKean H.P.: Stochastic Integrals. Academic Press, London (1969)MATHGoogle Scholar
  48. 48.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, 2 edn, vol. 2. Cambridge Mathematical Library, Cambridge (2000)Google Scholar
  49. 49.
    Diniz, I.T., Jonathan, D.: Comment on the paper “random quantum circuits are approximate 2-designs”. Commun. Math. Phys. 304, 281–293 (2011)Google Scholar
  50. 50.
    Diaconis P., Shahshahani M.: Generating a random permutation with random transpositions. Probab. Theory Relat. Fields 57(2), 159–179 (1981)MathSciNetMATHGoogle Scholar
  51. 51.
    Choi, M.: Completely positive linear maps on complex matrices. Linear Algebra. Appl. 10, 285–290 (1975)Google Scholar
  52. 52.
    Dupuis, F., Berta, M., Wullschleger, J., Renner, R.: One-shot decoupling. Commun. Math. Phys. 328, 251–284 (2014)Google Scholar
  53. 53.
    Diehl S., Micheli A., Kantian A., Kraus B., Buechler H.P., Zoller P.: Quantum states and phases in driven open quantum systems with cold atoms. Nat. Phys. 4:878 (2008)Google Scholar
  54. 54.
    Verstraete F., Wolf M.M., Cirac J.I.: Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5(9), 633 (2009)CrossRefGoogle Scholar
  55. 55.
    Kastoryano M.J., Wolf M.M., Eisert J: Precisely timing dissipative quantum information processing. Phys. Rev. Lett. 110, 110501 (2013)ADSCrossRefGoogle Scholar
  56. 56.
    Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. JHEP 2007, 120 (2007)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Lloyd S., Preskill J.: Unitarity of black hole evaporation in final-state projection models. JHEP 2014, 1 (2014)Google Scholar
  58. 58.
    Georgi H.: Lie Algebras in Particle Physics. 2nd edn. Westview Press, Boulder (1999)Google Scholar
  59. 59.
    Roberts D.A., Yoshida B.: Chaos and complexity by design. JHEP 2017, 121 (2017)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Sattinger D.H., Weaver O.L.: Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  61. 61.
    Fulton W., Harris J.: Representation Theory: A First Course. Springer, Heidelberg (1991)MATHGoogle Scholar
  62. 62.
    Eisert J., Felbinger T., Papadopoulos P., Plenio M.B., Wilkens M.: Classical information and distillable entanglement. Phys. Rev. Lett. 84, 1611 (2000)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • E. Onorati
    • 1
  • O. Buerschaper
    • 1
  • M. Kliesch
    • 1
    • 2
  • W. Brown
    • 1
  • A. H. Werner
    • 1
    • 3
  • J. Eisert
    • 1
  1. 1.Dahlem Center for Complex Quantum SystemsFreie Universität BerlinBerlinGermany
  2. 2.Institute of Theoretical Physics and Astrophysics, National Quantum Information CentreUniversity of GdańskGdańskPoland
  3. 3.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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