Abstract

Determining the worst-case uncertainty added by a quantum circuit is shown to be computationally intractable. This is the problem of detecting when a quantum channel implemented as a circuit is close to a linear isometry, and it is shown to be complete for the complexity class QMA of verifiable quantum computation. The main idea is to relate the problem of detecting when a channel is close to an isometry to the problem of determining how mixed the output of the channel can be when the input is a pure state.

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References

  1. 1.
    Aharonov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. In: 30th ACM Symposium on the Theory of Computing, pp. 20–30 (1998)Google Scholar
  2. 2.
    Beigi, S., Shor, P.W.: On the complexity of computing zero-error and Holevo capacity of quantum channels (2007), arXiv:0709.2090v3 [quant-ph] Google Scholar
  3. 3.
    Buhrman, H., Cleve, R., Watrous, J., de Wolf, R.: Quantum fingerprinting. Physical Review Letters 87(16), 167902 (2001)CrossRefGoogle Scholar
  4. 4.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra and its Applications 10(3), 285–290 (1975)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ekert, A.K., Alves, C.M., Oi, D.K., Horodecki, M., Horodecki, P., Kwek, L.C.: Direct estimations of linear and nonlinear functionals of a quantum state. Physical Review Letters 88(21), 217901 (2002)CrossRefGoogle Scholar
  6. 6.
    Janzing, D., Wocjan, P., Beth, T.: “Non-identity-check” is QMA-complete. International Journal of Quantum Information 3(3), 463–473 (2005)CrossRefMATHGoogle Scholar
  7. 7.
    Ji, Z., Wu, X.: Non-identity check remains QMA-complete for short circuits (2009), arXiv:0906.5416 [quant-ph] Google Scholar
  8. 8.
    Kempe, J., Kitaev, A., Regev, O.: The complexity of the local Hamiltonian problem. SIAM Journal on Computing 35(5), 1070–1097 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kitaev, A.Y.: Quantum NP. Talk at the 2nd Workshop on Algorithms in Quantum Information Processing (AQIP), DePaul University (1999)Google Scholar
  10. 10.
    Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Graduate Studies in Mathematics, vol. 47. American Mathematical Society, Providence (2002)MATHGoogle Scholar
  11. 11.
    Knill, E.: Quantum randomness and nondeterminism. Tech. Rep. LAUR-96-2186, Los Alamos National Laboratory (1996)Google Scholar
  12. 12.
    Liu, Y.K.: Consistency of local density matrices is QMA-complete. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 438–449. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Liu, Y.-K., Christandl, M., Verstraete, F.: Quantum computational complexity of the N-representability problem: QMA complete. Physical Review Letters 98(11), 110503 (2007)CrossRefGoogle Scholar
  14. 14.
    Marriott, C., Watrous, J.: Quantum Arthur-Merlin games. Computational Complexity 14(2), 122–152 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  16. 16.
    Schuch, N., Cirac, I., Verstraete, F.: Computational difficulty of finding matrix product ground states. Physical Review Letters 100(25), 250501 (2008)CrossRefGoogle Scholar
  17. 17.
    Schuch, N., Verstraete, F.: Computational complexity of interacting electrons and fundamental limitations of density functional theory. Nature Physics 5(10), 732–735 (2009)CrossRefGoogle Scholar
  18. 18.
    Watrous, J.: Succinct quantum proofs for properties of finite groups. In: 41st IEEE Symposium on Foundations of Computer Science, pp. 537–546 (2000)Google Scholar
  19. 19.
    Wei, T.-C., Mosca, M., Nayak, A.: Interacting boson problems can be QMA hard. Physical Review Letters 104(4), 40501 (2010)CrossRefGoogle Scholar
  20. 20.
    Zanardi, P., Lidar, D.A.: Purity and state fidelity of quantum channels. Physical Review A 70(1), 012315 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bill Rosgen
    • 1
  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingapore

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