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Local Random Quantum Circuits are Approximate Polynomial-Designs


We prove that local random quantum circuits acting on n qubits composed of O(t 10 n 2) many nearest neighbor two-qubit gates form an approximate unitary t-design. Previously it was unknown whether random quantum circuits were a t-design for any t >  3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are ∞-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove that circuits of depth O(t 10 n) constitute a quantum t-copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov, Arad, Landau, and Vazirani. We give applications of the results to cryptography, equilibration of closed quantum dynamics, and the generation of topological order. In particular we show the following pseudo-randomness property of generic quantum circuits: Almost every circuit U of size O(n k) on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size O(n (k-9)/11) that are given oracle access to U.

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  1. 1.

    Abeyesinghe A., Devetak I., Hayden P., Winter A.: The mother of all protocols: restructuring quantum information’s family tree. Proc. R. Soc. A 465(2108), 2537–2563 (2009) arXiv:quant-ph/0606225

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Hayden P., Shor P.W., Leung D.W., Winter A.J.: Randomizing quantum states: constructions and applications. Commun. Math. Phys. 250, 371–391 (2004) arXiv:quant-ph/0307104

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Sen, P.: Random measurement bases, quantum state distinction and applications to the hidden subgroup problem. In: IEEE Conference on Computational omplexity, pp. 274–287 (2006). arXiv:quant-ph/0512085

  4. 4.

    Knill, E.: Approximation by quantum circuits (1995). arXiv:quant-ph/9508006

  5. 5.

    Tóth G., García-Ripoll J.J.: Efficient algorithm for multiqudit twirling for ensemble quantum computation. Phys. Rev. A 75, 042311 (2007) arXiv:quant-ph/0609052

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Dankert C., Cleve R., Emerson J., Livine E.: Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009) arXiv:quant-ph/0606161

    ADS  Article  Google Scholar 

  7. 7.

    Gross, D., Audenaert, K., Eisert, J.: Evenly distributed unitaries: on the structure of unitary designs. J. Math. Phys. 48, 052104 (2007). arXiv:quant-ph/0611002

  8. 8.

    Emerson, J., Livine, E., Lloyd, S.: Convergence conditions for random quantum circuits. Phys. Rev. A 72, 060302 (2005). arXiv:quant-ph/0503210

  9. 9.

    Oliveira, R., Dahlsten, O.C.O., Plenio, M.B.: Efficient generation of generic entanglement. Phys. Rev. Lett. 98, 130502 (2007). arXiv:quant-ph/0605126

  10. 10.

    Dahlsten O.C.O., Oliveira R., Plenio M.B.: The emergence of typical entanglement in two-party random processes. J. Phys. A: Math. Theor. 40(28), 8081–8108 (2007) arXiv:quant-ph/0701125

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Harrow A., Low R.: Random quantum circuits are approximate 2-designs. Commun. Math. Phys. 291, 257–302 (2009) arXiv:0802.1919

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Diniz I., Jonathan D.: Comment on “Random Quantum Circuits are Approximate 2-designs”. Commun. Math. Phys. 304, 281–293 (2011) arXiv:1006.4202

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Arnaud L., Braun D.: Efficiency of producing random unitary matrices with quantum circuits. Phys. Rev. A 78, 062329 (2008) arXiv:0807.0775

    ADS  Article  Google Scholar 

  14. 14.

    Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. JHEP 0709, 120 (2007) arXiv:0708.4025

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Žnidarič M.: Exact convergence times for generation of random bipartite entanglement. Phys. Rev. A 78, 032324 (2008) arXiv:0809.0554

    ADS  Article  Google Scholar 

  16. 16.

    Harrow, A.W., Low, R.A.: Efficient quantum tensor product expanders and k-designs. In: Proceedings of APPROX-RANDOM, volume 5687 of LNCS, pp. 548–561. Springer (2009). arXiv:0811.2597

  17. 17.

    Roy A., Scott A.J.: Unitary designs and codes. Des. Codes Cryptogr. 53, 13–31 (2009) arXiv:0809.3813

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Brown W.G., Viola L.: Convergence rates for arbitrary statistical moments of random quantum circuits. Phys. Rev. Lett. 104, 250501 (2010). arXiv:0910.0913

    ADS  Article  Google Scholar 

  19. 19.

    Brandao F.G.S.L., Horodecki M.: Exponential quantum speed-ups are generic. Quantum Inf. Comput. 13, 0901–0924 (2013) arXiv:1010.3654

    MathSciNet  Google Scholar 

  20. 20.

    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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Hallgren, S., Harrow, A.W.: Superpolynomial speedups based on almost any quantum circuit. In: ICALP, volume 5125, pp. 782–795 (2008). arXiv:0805.0007

  22. 22.

    Low R.A.: Large deviation bounds for k-designs. Proc. R. Soc. A: Math. Phys. Eng. Sci. 465(2111), 3289–3308 (2009) arXiv:0903.5236

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Brodsky A., Hoory S.: Simple permutations mix even better. Random Struct. Algorithms 32, 274–289 (2008) arXiv:math/0411098

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kaplan E., Naor M., Reingold O.: Derandomized constructions of k-wise (almost) independent permutations. Algorithmica 55, 113–133 (2009) (ECCC TR06-002)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Hastings M.B., Harrow A.W.: Classical and quantum tensor product expanders. Quantum Inf. Comput. 9(3&4), 336–360 (2009) arXiv:0804.0011

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Low, R.A.: Pseudo-randomness and Learning in Quantum Computation. PhD thesis, University of Bristol (2010). arXiv:1006.5227

  27. 27.

    Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. AMS (2002)

  28. 28.

    Bourgain, J., Gamburd, A.: A spectral gap theorem in SU(d) (2011). arXiv:1108.6264

  29. 29.

    Shende, V.V., Bullock, S.S., Markov, I.L.: Synthesis of quantum-logic circuits. IEEE Trans. Comput.- Aided Design Integr. Circuits Syst. 25(6), 1000–1010 (2006). arXiv:quant-ph/0406176

  30. 30.

    Kassabov M.: Symmetric groups and expanders. Electron. Res. Announc. Am. Math. Soc. 11, 47–56 (2005) arXiv:math/0503204

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Hoory S., Magen A., Myers S., Rackoff C.: Simple permutations mix well. Theor. Comput. Sci. 348(2), 251–261 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Zatloukal, K.: Improved bounds for k-tensor product expanders (2012) (in preparation)

  33. 33.

    Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Bremner M.J., Mora C., Winter A.: Are random pure states useful for quantum computation?. Phys. Rev. Lett. 102, 190502 (2009) arXiv:00812.3001

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Gross D., Flammia S.T., Eisert J.: Most quantum states are too entangled to be useful as computational resources. Phys. Rev. Lett. 102, 190501 (2009) arXiv:0810.4331

    ADS  MathSciNet  Article  Google Scholar 

  36. 36.

    Aaronson, S.: The ten most annoying questions in quantum computing (2006)

  37. 37.

    Goldstein S., Lebowitz J.L., Tumulka R., Zanghi N.: Long-time behavior of macroscopic quantum systems: commentary accompanying the english translation of john von neumann’s 1929 article on the quantum ergodic theorem. Eur. Phys. J. 35, 173 (2010)

    Google Scholar 

  38. 38.

    Linden N., Popescu S., Short A.J., Winter A.: Quantum mechanical evolution towards thermal equilibrium. Phys. Rev. E 79, 061103 (2009) arXiv:0812.2385

    ADS  MathSciNet  Article  Google Scholar 

  39. 39.

    Cramer M., Eisert J.: A quantum central limit theorem for non-equilibrium systems: exact local relaxation of correlated states. New J. Phys. 12(5), 055020 (2010) arXiv:0911.2475

    ADS  Article  Google Scholar 

  40. 40.

    Gogolin C., Müller M.P., Eisert J.: Absence of thermalization in nonintegrable systems. Phys. Rev. Lett. 106, 040401 (2011) arXiv:1009.2493

    ADS  Article  Google Scholar 

  41. 41.

    Reimann P.: Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403 (2008) arXiv:0810.3092

    ADS  Article  Google Scholar 

  42. 42.

    Vinayak, Znidaric, M.: Subsystem dynamics under random Hamiltonian evolution. J. Phys. A: Math. Theor. 45, 125204 (2012).arXiv:1107.6035. doi:10.1088/1751-8113/45/12/125204

  43. 43.

    Masanes L., Roncaglia A.J., Acin A.: The complexity of energy eigenstates as a mechanism for equilibration. Phys. Rev. E 87, 032137 (2013) arXiv:1108.0374

    ADS  Article  Google Scholar 

  44. 44.

    Brandão F.G.S.L., Ćwikliński P., Horodecki M., Horodecki P., Korbicz J., Mozrzymas M.: Convergence to equilibrium under a random hamiltonian. Phys. Rev. E 86, 031101 (2012) arXiv:1108.2985

    ADS  Article  Google Scholar 

  45. 45.

    Cramer M.: Thermalization under randomized local hamiltonians. New J. Phys. 14(5), 053051 (2012) arXiv:1112.5295

    ADS  Article  Google Scholar 

  46. 46.

    von Neumann J.: Beweis des ergodensatzes und des h-theorems in der neuen mechanik. Zeitschrift für Physik 57(1–2), 30–70 (1929)

    ADS  Article  MATH  Google Scholar 

  47. 47.

    Goldstein S., Lebowitz J.L., Tumulka R, Zanghì N: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  48. 48.

    Trotzky S., Chen Y.-A., Flesch A., McCulloch I.P., Schollwöck U., Eisert J., Bloch I.: Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional bose gas. Nat. Phys. 8(4), 325–330 (2012)

    Article  Google Scholar 

  49. 49.

    Bañuls M.C., Cirac J.I., Hastings M.B.: Strong and weak thermalization of infinite nonintegrable quantum systems. Phys. Rev. Lett. 106, 050405 (2011)

    ADS  Article  Google Scholar 

  50. 50.

    Susskind L.: Computational complexity and black hole horizons. Fortschr. Phys. 64(1), 24–43 (2016) arXiv:1402.5674

    MathSciNet  Article  MATH  Google Scholar 

  51. 51.

    Wen X.-G.: Topological orders in rigid states. Int. J. Mod. Phys. 239, 050401 (1990)

    MathSciNet  Google Scholar 

  52. 52.

    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 1 (2003) arXiv:quant-ph/9707021

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. 53.

    Nayak C., Simon S.H., Stern A., Freedman M., Sarma S.D.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083 (2008) arXiv:0707.1889

    ADS  MathSciNet  Article  MATH  Google Scholar 

  54. 54.

    Bravyi S., Hastings M.B., Verstraete F.: Lieb–Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 97, 050401 (2006) arXiv:quant-ph/0603121

    ADS  Article  Google Scholar 

  55. 55.

    Chen X., Gu Z.-G., Wen X.-G.: Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Phys. Rev. B 82, 1083 (2010) arXiv:1004.3835

    Google Scholar 

  56. 56.

    Hastings M.B.: Topological order at non-zero temperature. Phys. Rev. Lett. 107, 210501 (2011) arXiv:1106.6026

    ADS  Article  Google Scholar 

  57. 57.

    Sekino Y., Susskind L.: Fast scramblers. J. High Energy Phys. 10, 065 (2008) arXiv:0808.2096

    ADS  Article  Google Scholar 

  58. 58.

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

  59. 59.

    Nachtergaele B.: The spectral gap for some spin chains with discrete symmetry breaking. Commun. Math. Phys. 175, 565–606 (1996) arXiv:cond-mat/9410110

    ADS  MathSciNet  Article  MATH  Google Scholar 

  60. 60.

    Fannes M., Nachtergaele B., Werner R.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  61. 61.

    Perez-Garcia D., Verstraete F., Wolf M.M., Cirac J.I.: Matrix product state representations. Quantum Inf. Comput. 7(5&6), 401–430 (2007) arXiv:quant-ph/0608197

    MathSciNet  MATH  Google Scholar 

  62. 62.

    Bubley, R., Dyer, M.E.: Path coupling: a technique for proving rapid mixing in Markov chains. In: Proceedings of the 38th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 223–231 (1997)

  63. 63.

    Oliveira R.I.: On the convergence to equilibrium of Kac’s random walk on matrices. Ann. Appl. Probab. 19(3), 1200–1231 (2009) arXiv:0705.2253

    MathSciNet  Article  MATH  Google Scholar 

  64. 64.

    Aharonov D., Arad I., Vazirani U., Landau Z.: The detectability lemma and its applications to quantum Hamiltonian complexity. New J. Phys. 13(11), 113043 (2011) arXiv:1011.3445

    ADS  Article  Google Scholar 

  65. 65.

    Damgard, I.B., Fehr, S., Salvail, L., Schaffner, C.: Cryptography in the bounded quantum-storage model. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’05, pp. 449–458, Washington, DC, USA, 2005. IEEE Computer Society. arXiv:quant-ph/0508222.

  66. 66.

    Barenco A., Bennett C.H., Cleve R., DiVincenzo D.P., Margolus N., Shor P., Sleator T., Smolin J.A., Weinfurter H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995) arXiv:quant-ph/9503016

    ADS  Article  Google Scholar 

  67. 67.

    Goodman, R., Wallach, N.R.: Representations and Invariants of the Classical Groups. Cambridge University Press, Cambridge (1998).

  68. 68.

    Christandl, M.: The structure of bipartite quantum states: insights from group theory and cryptography. PhD thesis, University of Cambridge (2006). arXiv:quant-ph/0604183

  69. 69.

    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. 148, 9–12 (1963)

  70. 70.

    Szarek T.: Feller processes on nonlocally compact spaces. Ann. Probab. 34(5), 1849–1863 (2006) arXiv:math/0512221

    MathSciNet  Article  MATH  Google Scholar 

  71. 71.

    Szarek, T.: (2011) (private communication)

  72. 72.

    Harrow, A.W.: The church of the symmetric subspace (2013). arXiv:1308.6595

  73. 73.

    Bhatia, R.: Matrix Analysis, volume 169. Springer Science & Business Media, Berlin (1997)

  74. 74.

    Ambainis A., Bouda J., Winter A.: Nonmalleable encryption of quantum information. J. Math. Phys. 50(4), 042106 (2009) arXiv:0808.0353

    ADS  MathSciNet  Article  MATH  Google Scholar 

  75. 75.

    Aharonov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. In: Proceedings of the thirtieth annual ACM symposium on Theory of computing, pp. 20–30. ACM (1998). arXiv:quant-ph/9806029

  76. 76.

    Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces. Lecture notes in mathematics. Springer, Berlin (1986)

  77. 77.

    Bernstein, E., Vazirani, U.: Quantum complexity theory. In: Proceedings of the 25th Annual ACM Symposium on the Theory of Computation (STOC), pp. 11–20. ACM Press, El Paso, Texas (1993)

  78. 78.

    Ledoux, M.: The Concentration of Measure Phenomenon. AMS Monographs, Providence (2001)

  79. 79.

    Kastoryano M.J., Temme K.: Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys. 54, 052202 (2013) arXiv:1207.3261

    ADS  MathSciNet  Article  MATH  Google Scholar 

  80. 80.

    Stolz, G.: An introduction to the mathematics of anderson localization. Contemp. Math. 552 (2011). arXiv:1104.2317

  81. 81.

    Harrow, A.W.: Applications of coherent classical communication and Schur duality to quantum information theory. PhD thesis, M.I.T., Cambridge, MA (2005). arXiv:quant-ph/0512255

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Correspondence to Fernando G. S. L. Brandão.

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Brandão, F.G.S.L., Harrow, A.W. & Horodecki, M. Local Random Quantum Circuits are Approximate Polynomial-Designs. Commun. Math. Phys. 346, 397–434 (2016).

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