Communications in Mathematical Physics

, Volume 291, Issue 1, pp 257–302 | Cite as

Random Quantum Circuits are Approximate 2-designs

  • Aram W. Harrow
  • Richard A. Low


Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group.


Markov Chain Stationary Distribution Haar Measure Quantum Circuit Sobolev Constant 
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.


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  1. 1.
    Aaronson, S.: Quantum Copy-Protection. Talk at QIP, New Delhi, India, December 2007, available at, 2007
  2. 2.
    Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A.: The mother of all protocols: Restructuring quantum information’s family tree., 2006
  3. 3.
    Ambainis, A., Emerson, E.: Quantum t-designs: t-wise independence in the quantum world. IEEE Conference on Computational Complexity 2007,, 2007
  4. 4.
    Ambainis, A., Mosca, M., Tapp, A., de Wolf, R.: Private Quantum Channels. FOCS 2000, Washington, DC: IEEE, 2000, pp. 547–553Google Scholar
  5. 5.
    Ambainis, A., Smith, A.: Small pseudo-random families of matrices: derandomizing approximate quantum encryption. Lecture Notes in Computer Science 3122, Berlin-Heidelberg-NewYork: Springer, 2004, pp. 249–260Google Scholar
  6. 6.
    Arnold, V.I., Krylov, A.L.: Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in a complex domain. Sov. Math. Dokl. 4(1), 1962Google Scholar
  7. 7.
    Barenco A., Berthiaume A., Deutsch D., Ekert A., Jozsa R., Macchiavello C.: Stabilization of quantum computations by symmetrization. SIAM J. Comput. 26(5), 1541–1557 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Barnum, H.: Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases., 2002
  9. 9.
    Dahlsten O.C.O., Oliveira R., Plenio M.B.: The emergence of typical entanglement in two-party random processes. J. Phys. A Math. Gen. 40, 8081–8108 (2007)zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Dankert, C., Cleve, R., Emerson, J., Livine, E.: Exact and approximate unitary 2-designs: constructions and applications., 2006
  11. 11.
    Devetak I., Junge M., King C., Ruskai M.B.: Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266, 37–63 (2006)zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Diaconis P., Saloff-Coste L.: Comparison theorems for reversible markov chains. Ann. Appl. Probab. 3(3), 696–730 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Diaconis P., Saloff-Coste L.: Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6(3), 695–750 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    DiVincenzo D., Leung D., Terhal B.: Quantum data hiding. Information Theory. IEEE Transactions 48(3), 580–598 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Emerson J., Livine E., Lloyd S.: Convergence conditions for random quantum circuits. Phys. Rev. A 72, 060302 (2005)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Goodman R., Wallach N.: Representations and Invariants of the Classical Groups. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  17. 17.
    Grimmett G., Welsh D.: Probability: An Introduction. Oxford University Press, Oxford (1986)zbMATHGoogle Scholar
  18. 18.
    Gross D., Audenaert K., Eisert J.: Evenly distributed unitaries: On the structure of unitary designs. J. Math. Phys. 48, 052104 (2007)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Hallgren, S., Harrow, A.W.: Superpolynomial speedups based on almost any quantum circuit. In: Proc. 35th Intl. Colloq. on Automate Languages an Programming LCUS 5125, 2, pp. 782–795, 2008Google Scholar
  20. 20.
    Hayashi A., Hashimoto T., Horibe M.: Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A 72, 032325 (2006)CrossRefADSGoogle Scholar
  21. 21.
    Hayden P., Horodecki M., Yard J., Winter A.: A decoupling approach to the quantum capacity. Open Syst. Inf. Dyn. 15, 7–19 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. JHEP 09, 120 (2007)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Hoory, S., Brodsky, A.: Simple Permutations Mix Even Better.[math.CO] 2004
  24. 24.
    Kitaev A.Yu., Shen A.H., Vyalyi M.N.: Classical and Quantum Computation. RI Amer. Math. Soc., Providence (2002)zbMATHGoogle Scholar
  25. 25.
    Montenegro R., Tetali P.: Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1(3), 237–354 (2006)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Oliveira R., Dahlsten O.C.O., Plenio M.B.: Efficient generation of generic entanglement. Phys. Rev. Lett. 98, 130502 (2007)CrossRefADSGoogle Scholar
  27. 27.
    Paulsen V.I.: Completely Bounded Maps and Dilations. John Wiley & Sons, Inc., New York (1987)Google Scholar
  28. 28.
    Sen, P.: Random measurement bases, quantum state distinction and applications to the hidden subgroup problem. IEEE Conference on Computational Complexity 2006, 2005, pp. 274–287Google Scholar
  29. 29.
    Watrous J.: Notes on super-operator norms induced by Schatten norms. Quantum Information and Computation 5(1), 58–68 (2005)MathSciNetGoogle Scholar
  30. 30.
    Znidaric M.: Optimal two-qubit gate for generation of random bipartite entanglement. Phys. Rev. A 76, 012318 (2007)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of BristolBristolU.K.

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