Communications in Mathematical Physics

, Volume 340, Issue 3, pp 867–900 | Cite as

Decoupling with Random Quantum Circuits

  • Winton Brown
  • Omar FawziEmail author


Decoupling has become a central concept in quantum information theory, with applications including proving coding theorems, randomness extraction and the study of conditions for reaching thermal equilibrium. However, our understanding of the dynamics that lead to decoupling is limited. In fact, the only families of transformations that are known to lead to decoupling are (approximate) unitary two-designs, i.e., measures over the unitary group that behave like the Haar measure as far as the first two moments are concerned. Such families include for example random quantum circuits with O(n 2) gates, where n is the number of qubits in the system under consideration. In fact, all known constructions of decoupling circuits use Ω(n 2) gates. Here, we prove that random quantum circuits with O(n log2 n) gates satisfy an essentially optimal decoupling theorem. In addition, these circuits can be implemented in depth O(log3 n). This proves that decoupling can happen in a time that scales polylogarithmically in the number of particles in the system, provided all the particles are allowed to interact. Our proof does not proceed by showing that such circuits are approximate two-designs in the usual sense, but rather we directly analyze the decoupling property.


Black Hole Markov Chain Haar Measure Geometric Distribution Quantum Circuit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abeyesinghe A., Devetak I., Hayden P., Winter A.: The mother of all protocols: Restructuring quantum information’s family tree. Proc. Roy. Soc. A Math. Phys. 465, 2537 (2009) arXiv:quant-ph/0606225 zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Berta, M.: Single-shot quantum state merging (2009). arXiv:0912.4495
  3. 3.
    Brown, W., Fawzi, O.: Short random circuits define good quantum error correcting codes. In: Proc. IEEE ISIT, pp. 346–350 (2013). arXiv:1312.7646
  4. 4.
    Berta, M., Fawzi, O., Wehner, S.: Quantum to classical randomness extractors. In: Proc. CRYPTO, LNCS, vol. 7417, pp. 776–793 (2012). arXiv:1111.2026
  5. 5.
    Bhatia R.: Matrix Analysis. Springer, Berlin (1997)CrossRefGoogle Scholar
  6. 6.
    Brandao, F.G.S.L, Harrow, A.W., Horodecki, M.: Local random quantum circuits are approximate polynomial-designs (2012). arXiv:1208.0692
  7. 7.
    Brown, W., Poulin, D.: Approximate designs need not scramble (2015, in preparation)Google Scholar
  8. 8.
    Brown W.G., Viola L.: Convergence rates for arbitrary statistical moments of random quantum circuits. Phys. Rev. Lett. 104, 250501 (2010) arXiv:0910.0913 CrossRefADSGoogle Scholar
  9. 9.
    Cleve, R., Leung, D., Liu, L., Wang, C.: Near-linear constructions of exact unitary 2-designs (2015). arXiv:1501.04592
  10. 10.
    Dupuis F., Berta M., Wullschleger J., Renner R.: One-shot decoupling. Commun. Math. Phys. 328(1), 251–284 (2014) arXiv:1012.6044 zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Dankert C., Cleve R., Emerson J., Livine E.: Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80(1), 12304 (2009) arXiv:quant-ph/0606161 CrossRefADSGoogle Scholar
  12. 12.
    Dupuis F., Fawzi O., Wehner S.: Entanglement sampling and applications. IEEE Trans. Inform. Theory 61(2), 1093–1112 (2015) arXiv:1305.1316 MathSciNetCrossRefGoogle Scholar
  13. 13.
    del Rio L., Åberg J., Renner R., Dahlsten O., Vedral V.: The thermodynamic meaning of negative entropy. Nature 474(7349), 61–63 (2011)CrossRefGoogle Scholar
  14. 14.
    del Rio, L., Hutter, A., Renner, R., Wehner, S.: Relative thermalization (2014). arXiv:1401.7997
  15. 15.
    Dupuis, F.: The decoupling approach to quantum information theory. PhD thesis, Université de Montreal (2010). arXiv:1004.1641
  16. 16.
    Emerson J., Livine E., Lloyd S.: Convergence conditions for random quantum circuits. Phys. Rev. A 72(6), 060302 (2005) arXiv:quant-ph/0503210 MathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Emerson J., Weinstein Y.S., Saraceno M., Lloyd S., Cory D.G.: Pseudo-random unitary operators for quantum information processing. Science 302(5653), 2098–2100 (2003)zbMATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Harrow A., Low R.: Random quantum circuits are approximate 2-designs. Commun. Math. Phys. 291, 257–302 (2009) arXiv:0802.1919 zbMATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006) arXiv:quant-ph/0407049 zbMATHMathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Horodecki M., Oppenheim J., Winter A.: Partial quantum information. Nature 436, 673–676 (2005) arXiv:quant-ph/0505062 CrossRefADSGoogle Scholar
  21. 21.
    Horodecki M., Oppenheim J., Winter A.: Quantum state merging and negative information. Commun. Math. Phys. 269, 107 (2006) arXiv:quant-ph/0512247 MathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Hayden, P., Preskill, J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys., 120 (2007). arXiv:0708.4025
  23. 23.
    Hamma A., Santra S., Zanardi P.: Quantum entanglement in random physical states. Phys. Rev. Lett. 109, 040502 (2012) arXiv:1109.4391 CrossRefADSGoogle Scholar
  24. 24.
    Hutter, A.: Understanding Equipartition and Thermalization from Decoupling. (2011)
  25. 25.
    Hutter A., Wehner S.: Dependence of a quantum-mechanical system on its own initial state and the initial state of the environment it interacts with. Phys. Rev. A 87, 012121 (2013) arXiv:1111.3080 CrossRefADSGoogle Scholar
  26. 26.
    Low, R.A.: Pseudo-randomness and learning in quantum computation. PhD thesis, Bristol (2010). arXiv:1006.5227
  27. 27.
    Lloyd S., Preskill J.: Unitarity of black hole evaporation in final-state projection models. J. High Energy Phys. 08, 126 (2014)CrossRefADSGoogle Scholar
  28. 28.
    Lashkari N., Stanford D., Hastings M., Osborne T., Hayden P.: Towards the fast scrambling conjecture. J. High Energy Phys. 2013(4), 1–33 2013 (2013) arXiv:1111.6580 MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mitzenmacher M., Upfal E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  30. 30.
    Oliveira R., Dahlsten O.C.O., Plenio M.B.: Generic entanglement can be generated efficiently. Phys. Rev. Lett. 98(13), 130502 (2007) arXiv:quant-ph/0605126 CrossRefADSGoogle Scholar
  31. 31.
    Renes J., Dupuis F., Renner R.: Efficient polar coding of quantum information. Phys. Rev. Lett. 109, 050504 (2012) arXiv:1109.3195 CrossRefADSGoogle Scholar
  32. 32.
    Szehr O., Dupuis F., Tomamichel M., Renner R.: Decoupling with unitary approximate two-designs. New J. Phys. 15(5), 053022 (2013) arXiv:1109.4348 MathSciNetCrossRefADSGoogle Scholar
  33. 33.
    Sutter, D., Renes, J., Dupuis, F., Renner, R.: Efficient quantum channel coding scheme requiring no preshared entanglement. In: Proc. IEEE ISIT (2013)Google Scholar
  34. 34.
    Sekino Y., Susskind L.: Fast scramblers. J. High Energy Phys. 2008(10), 065 (2008) arXiv:0808.2096 CrossRefGoogle Scholar
  35. 35.
    Toacute;th G., García-Ripoll J.J.: Efficient algorithm for multiqudit twirling for ensemble quantum computation. Phys. Rev. A 75(4), 042311 (2007) arXiv:quant-ph/0609052 MathSciNetCrossRefADSGoogle Scholar
  36. 36.
    Vadhan, S.: Pseudorandomness. Found. Trends Theor. Comput. Sci. 7(13), 1–336 (2011).
  37. 37.
    Wilde, M., Renes, J.: Quantum polar codes for arbitrary channels. In: Proc. IEEE ISIT, pp. 334–338 (2012)Google Scholar
  38. 38.
    Žnidarič M.: Exact convergence times for generation of random bipartite entanglement. Phys. Rev. A 78(3), 032324 (2008) arXiv:0809.0554 CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Département de PhysiqueUniversité de SherbrookeSherbrookeCanada
  2. 2.Institute for Theoretical PhysicsETH ZürichZurichSwitzerland
  3. 3.LIP, UMR 5668 ENS Lyon-CNRS-UCBL-INRIAUniversite de LyonLyon France

Personalised recommendations