Skip to main content
Log in

Mixing Properties of Stochastic Quantum Hamiltonians

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Brown W., Viola L.: Convergence rates for arbitrary statistical moments of random quantum circuits. Phys. Rev. Lett. 104, 250501 (2010)

    Article  ADS  Google Scholar 

  2. Brown, W., Fawzi O.: Scrambling speed of random quantum circuits. arXiv:1210.6644 (2012)

  3. Brandao F.G.S.L., Horodecki M.: Exponential quantum speed-ups are generic. Quantum Inf. Comput. 13, 0901 (2013)

    MathSciNet  Google Scholar 

  4. Brown, W., Fawzi, O.: Decoupling with random quantum circuits. Commun. Math. Phys. 340, 867–900 (2015)

  5. Oliveira R., Dahlsten O.C.O., Plenio M.B.: Efficient generation of generic entanglement. Phys. Rev. Lett. 98, 130502 (2007)

    Article  ADS  Google Scholar 

  6. Brown, W., Fawzi, O.: Short random circuits define good quantum error correcting codes. In: Proceedings of the ISIT, pp. 346 (2013)

  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)

    Article  ADS  Google Scholar 

  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)

  9. Lashkari N., Stanford D., Hastings M., Osborne T.J., Hayden P.: Towards the fast scrambling conjecture. JHEP 2013, 22 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harrow A.W., Low R.A.: Random quantum circuits are approximate 2-designs. Commun. Math. Phys. 291, 257 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Bouten L., van Handel R.: Discrete approximation of quantum stochastic models. J. Math. Phys. 49, 102109 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Weinstein Y.S., Brown W.G., Viola L.: Parameters of pseudo-random quantum circuits. Phys. Rev. A 78, 052332 (2008)

    Article  ADS  Google Scholar 

  14. Belton A.C.R., Gnacik M., Lindsay J.M.: The convergence of unitary quantum random walks. Lancaster EPrints (2014). http://eprints.lancs.ac.uk/69293/. Accessed on 18 July 2017

  15. Gross D, Audenaert K.M.R., Eisert J.: Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48, 052104 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Levin D.A., Peres Y., Wilmer E.L.: Markov Chains and Mixing Times. American Mathematical Society, New York (2008)

    Book  Google Scholar 

  17. Aldous D., Diaconis P.: Shuffling cards and stopping time. Am. Math. Soc. Mon. 93(5), 333–348 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dunjko V., Briegel H.J.: Quantum mixing of Markov chains for special distributions. New. J. Phys. 17, 073004 (2015)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  20. Ohliger M., Nesme V., Eisert J.: Efficient and feasible state tomography of quantum many-body systems. New J. Phys. 15, 015024 (2013)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  22. Szehr O., Dupuis F., Tomamichel M., Renner R.: Decoupling with unitary approximate two-design. New J. Phys. 15, 053022 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  23. Horodecki M., Oppenheim J., Winter A.: Quantum state merging and negative information. Commun. Math. Phys. 269, 107 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  25. Buscemi F.: Private quantum decoupling and secure disposal of information. New J. Phys. 11, 123002 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  26. Gogolin C., Eisert J.: Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems. Rep. Prog. Phys. 79, 56001 (2016)

    Article  ADS  Google Scholar 

  27. Banks T., Fischler W., Shenker S., Susskind L.: m theory as a matrix model: a conjecture. Phys. Rev. D 55, 5112 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Maldacena J.: The large n limit of super-conformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 213 (1998)

    Article  Google Scholar 

  29. Sekino Y., Susskind L.: Fast scramblers. JHEP 10, 65 (2008)

    Article  ADS  Google Scholar 

  30. Asplund C.T., Berenstein D., Trancanelli D.: Evidence for fast thermalization in the plane-wave matrix model. Phys. Rev. Lett. 107, 171602 (2011)

    Article  ADS  Google Scholar 

  31. Hübener R., Sekino Y., Eisert J.: Equilibration in low-dimensional quantum matrix models. JHEP 2015, 166 (2015)

    Article  MathSciNet  Google Scholar 

  32. Bourgain J., Gamburd A.: A spectral gap theorem in su (d). J. Eur. Math. Soc. 14(5), 1455–1511 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Benoist, Y., de Saxcé, N.: A spectral gap theorem in simple Lie groups. Invent. Math. 205, 337–361 (2016)

  34. Montroll E.W., Weiss G.H.: Random walks on lattices ii. J. Math. Phys. 6, 167–181 (1965)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Weiss G.H.: Aspects and applications of the random walk. J. Stat. Phys. 79(1), 497–500 (1995)

    Google Scholar 

  36. Zaburdaev, V., Denisov, S., Hanggi, P.: Perturbation spreading in many-particle systems: a random walk approach. Phys. Rev. Lett. 106(18), 180601 (2011)

  37. Schulz, J.H.P., Barkai, E.: Fluctuations around equilibrium laws in ergodic continuous-time random walks. Phys. Rev. E 91, 062129 (2015)

  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)

  39. Watrous J.: Semidefinite programs for completely bounded norms. Theor. Comput. 5, 11 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Low, R.: Pseudo-randomness and learning in quantum computation. Ph.D. thesis, university of Bristol (2010)

  41. Collins B., Sniady P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Dyson, F.J.: The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75, 486–502 (1949)

  43. Dollard J.D., Friedman C.N.: Product integrals and the Schrödinger equation. J. Math. Phys. 18, 1598 (1977)

    Article  ADS  MATH  Google Scholar 

  44. Ito S.: Brownian motions in a topological group and in its covering group. Rend. Circ. Mat. Palermo 1, 40–48 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tsirelson, B.: Unitary Brownian motions are linearisable. arXiv:math/9806112 (1988)

  46. Liao M.: Lévy Processes in Lie Groups, vol. 162. Cambridge university press, Cambridge (2004)

    Book  Google Scholar 

  47. McKean H.P.: Stochastic Integrals. Academic Press, London (1969)

    MATH  Google Scholar 

  48. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, 2 edn, vol. 2. Cambridge Mathematical Library, Cambridge (2000)

  49. Diniz, I.T., Jonathan, D.: Comment on the paper “random quantum circuits are approximate 2-designs”. Commun. Math. Phys. 304, 281–293 (2011)

  50. Diaconis P., Shahshahani M.: Generating a random permutation with random transpositions. Probab. Theory Relat. Fields 57(2), 159–179 (1981)

    MathSciNet  MATH  Google Scholar 

  51. Choi, M.: Completely positive linear maps on complex matrices. Linear Algebra. Appl. 10, 285–290 (1975)

  52. Dupuis, F., Berta, M., Wullschleger, J., Renner, R.: One-shot decoupling. Commun. Math. Phys. 328, 251–284 (2014)

  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)

  54. Verstraete F., Wolf M.M., Cirac J.I.: Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5(9), 633 (2009)

    Article  Google Scholar 

  55. Kastoryano M.J., Wolf M.M., Eisert J: Precisely timing dissipative quantum information processing. Phys. Rev. Lett. 110, 110501 (2013)

    Article  ADS  Google Scholar 

  56. Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. JHEP 2007, 120 (2007)

    Article  MathSciNet  Google Scholar 

  57. Lloyd S., Preskill J.: Unitarity of black hole evaporation in final-state projection models. JHEP 2014, 1 (2014)

    Google Scholar 

  58. Georgi H.: Lie Algebras in Particle Physics. 2nd edn. Westview Press, Boulder (1999)

    Google Scholar 

  59. Roberts D.A., Yoshida B.: Chaos and complexity by design. JHEP 2017, 121 (2017)

    Article  MathSciNet  Google Scholar 

  60. Sattinger D.H., Weaver O.L.: Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  61. Fulton W., Harris J.: Representation Theory: A First Course. Springer, Heidelberg (1991)

    MATH  Google Scholar 

  62. Eisert J., Felbinger T., Papadopoulos P., Plenio M.B., Wilkens M.: Classical information and distillable entanglement. Phys. Rev. Lett. 84, 1611 (2000)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. Eisert.

Additional information

Communicated by M. M. Wolf

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Onorati, E., Buerschaper, O., Kliesch, M. et al. Mixing Properties of Stochastic Quantum Hamiltonians. Commun. Math. Phys. 355, 905–947 (2017). https://doi.org/10.1007/s00220-017-2950-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2950-6

Navigation