Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Quantum Computational Complexity

  • John Watrous
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_428

Definition of the Subject

The inherent difficulty, or hardness, of computational problems is a fundamental concept in computationalcomplexity theory. Hardness is typically formalized in terms of the resources required by different models of computation to solve problems, such asthe number of steps of a deterministic Turing machine. A variety of models and resources are often considered, including deterministic,nondeterministic and probabilistic models; time and space constraints; and interactions among models of differing abilities. Many interestingrelationships among these different models and resource constraints are known.

One common feature of the most commonly studied computational models and resource constraints is that they are physicallymotivated. This is quite natural, given that computers are physical devices, and to a significant extent it is their study that motivatesand directs research on computational complexity. The predominant example is the class of polynomial-time...

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

Bibliography

  1. 1.
    Aaronson S (2002) Quantum lower bound for the collision problem. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing. ACM Press, New YorkGoogle Scholar
  2. 2.
    Aaronson S (2005) Limitations of quantum advice and one-way communication. Theory Comput 1:1–28MathSciNetGoogle Scholar
  3. 3.
    Aaronson S (2006) QMA/qpoly is contained in PSPACE/poly: de-Merlinizing quantum protocols. In: Proceedings of the 21st Annual IEEE Conference on Computational Complexity. IEEE Ceomputer Society Press, Los Alamitos pp 261–273Google Scholar
  4. 4.
    Aaronson S, Kuperberg G (2007) Quantum versus classical proofs and advice. Theory Comput 3:129–157MathSciNetGoogle Scholar
  5. 5.
    Aaronson S, Shi Y (2004) Quantum lower bounds for the collision and the element distinctness problems. J ACM 51(4):595–605MathSciNetzbMATHGoogle Scholar
  6. 6.
    Adleman L (1978) Two theorems on random polynomial time. In: Proceeding of the 19th Annual IEEE Symposium on Foundations of Computer Science. IEEE Ceomputer Society Press, Los Alamitos pp 75–83Google Scholar
  7. 7.
    Adleman L, DeMarrais J, Huang M (1997) Quantum computability. SIAM J Comput 26(5):1524–1540MathSciNetzbMATHGoogle Scholar
  8. 8.
    Aharonov D, Naveh T (2002) Quantum NP – a survey. Available as arXiv.org e-Print quant-ph/0210077Google Scholar
  9. 9.
    Aharonov D, Kitaev A, Nisan N (1998) Quantum circuits with mixed states. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing. ACM Press, New York, pp 20–30Google Scholar
  10. 10.
    Aharonov D, Gottesman D, Irani S, Kempe J (2007) The power of quantum systems on a line. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science. IEEE Ceomputer Society Press, Los Alamitos pp 373–383Google Scholar
  11. 11.
    Aharonov D, van Dam W, Kempe J, Landau Z, Lloyd S, Regev O (2007) Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J Comput 37(1):166–194MathSciNetzbMATHGoogle Scholar
  12. 12.
    Allender E, Ogihara M (1996) Relationships among PL, #L, and the determinant. RAIRO – Theor Inf Appl 30:1–21MathSciNetzbMATHGoogle Scholar
  13. 13.
    Arora S, Barak B (2006) Complexity Theory: A Modern Approach. Web draft available at http://www.cs.princeton.edu/theory/complexity/
  14. 14.
    Arora S, Safra S (1998) Probabilistic checking of proofs: a new characterization of NP. J ACM 45(1):70–122MathSciNetzbMATHGoogle Scholar
  15. 15.
    Arora S, Lund C, Motwani R, Sudan M, Szegedy M (1998) Proof verification and the hardness of approximation problems. J ACM 45(3):501–555MathSciNetzbMATHGoogle Scholar
  16. 16.
    Aroya A-B, Shma A-T (2007) Quantum expanders and the quantum entropy difference problem. Available as arXiv.org e-print quant-ph/0702129Google Scholar
  17. 17.
    Babai L (1985) Trading group theory for randomness. In: Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing. ACM Press, New York pp 421–429Google Scholar
  18. 18.
    Babai L (1992) Bounded round interactive proofs in finite groups. SIAM J Discret Math 5(1):88–111MathSciNetzbMATHGoogle Scholar
  19. 19.
    Babai L, Moran S (1988) Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes. J Comput Syst Sci 36(2):254–276MathSciNetzbMATHGoogle Scholar
  20. 20.
    Babai L, Szemerédi E (1984) On the complexity of matrix group problems I. In: Proceedings of the 25th Annual IEEE Symposium on Foundations of Computer Science. IEEE Ceomputer Society Press, Los Alamitos pp 229–240Google Scholar
  21. 21.
    Babai L, Fortnow L, Lund C (1991) Non-deterministic exponential time has two-prover interactive protocols. Comput Complex 1(1):3–40MathSciNetzbMATHGoogle Scholar
  22. 22.
    Beigel R, Reingold N, Spielman D (1995) PP is closed under intersection. J Comput Syst Sci 50(2):191–202MathSciNetzbMATHGoogle Scholar
  23. 23.
    Beigi S, Shor P (2007) On the complexity of computing zero-error and Holevo capacity of quantum channels. Available as arXiv.org e-Print 0709.2090Google Scholar
  24. 24.
    Bell J (1964) On the Einstein-Podolsky-Rosen paradox. Phys 1(3):195–200Google Scholar
  25. 25.
    Bellare M, Goldreich O, Sudan M (1998) Free bits, PCPs, and non-approximability —towards tight results. SIAM J Comput 27(3):804–915MathSciNetzbMATHGoogle Scholar
  26. 26.
    Bennett CH (1973) Logical reversibility of computation. IBM J Res Dev 17:525–532zbMATHGoogle Scholar
  27. 27.
    Bennett CH, Bernstein E, Brassard G, Vazirani U (1997) Strengths and weaknesses of quantum computing. SIAM J Comput 26(5):1510–1523MathSciNetzbMATHGoogle Scholar
  28. 28.
    Bera D, Green F, Homer S (2007) Small depth quantum circuits. ACM SIGACT News 38(2):35–50Google Scholar
  29. 29.
    Bernstein E, Vazirani U (1993) Quantum complexity theory (preliminary abstract). In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing. ACM Press, New York pp 11–20Google Scholar
  30. 30.
    Bernstein E, Vazirani U (1997) Quantum complexity theory. SIAM J Comput 26(5):1411–1473MathSciNetzbMATHGoogle Scholar
  31. 31.
    Borodin A (1977) On relating time and space to size and depth. SIAM J Comput 6:733–744MathSciNetzbMATHGoogle Scholar
  32. 32.
    Borodin A, Cook S, Pippenger N (1983) Parallel computation for well-endowed rings and space-bounded probabilistic machines. Inf Control 58:113–136MathSciNetzbMATHGoogle Scholar
  33. 33.
    Brassard G (2003) Quantum communication complexity. Found Phys 33(11):1593–1616MathSciNetzbMATHGoogle Scholar
  34. 34.
    Cleve R (2000) An introduction to quantum complexity theory. In: Macchiavello C, Palma GM, Zeilinger A (eds) Collected Papers on Quantum Computation and Quantum Information Theory. World Scientific. Singapore pp 103–127Google Scholar
  35. 35.
    Cleve R, Watrous J (2000) Fast parallel circuits for the quantum Fourier transform. In: Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science. pp 526–536Google Scholar
  36. 36.
    Cleve R, Høyer P, Toner B, Watrous J (2004) Consequences and limits of nonlocal strategies. In: Proceedings of the 19th Annual IEEE Conference on Computational Complexity. pp 236–249Google Scholar
  37. 37.
    Cleve R, Slofstra W, Unger F, Upadhyay S (2007) Perfect parallel repetition theorem for quantum XOR proof systems. In: Proceedings of the 22nd Annual IEEE Conference on Computational Complexity. pp 109–114Google Scholar
  38. 38.
    Cook S (1972) The complexity of theorem proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing. ACM Press, New York pp 151–158Google Scholar
  39. 39.
    de Wolf R (2002) Quantum communication and complexity. Theor Comput Sci 287(1):337–353zbMATHGoogle Scholar
  40. 40.
    Deutsch D (1985) Quantum theory, the Church–Turing principle and the universal quantum computer. Proc Roy Soc Lond A 400:97–117MathSciNetADSzbMATHGoogle Scholar
  41. 41.
    Deutsch D (1989) Quantum computational networks. Proc Roy Soc Lond A 425:73–90MathSciNetADSzbMATHGoogle Scholar
  42. 42.
    Dinur I (2007) The PCP theorem by gap amplification. J ACM 54(3)MathSciNetGoogle Scholar
  43. 43.
    Du D-Z, Ko K-I (2000) Theory of Computational Complexity. Wiley, New YorkzbMATHGoogle Scholar
  44. 44.
    Even S, Selman A, Yacobi Y (1984) The complexity of promise problems with applications to public-key cryptography. Inf Control 61:159–173MathSciNetzbMATHGoogle Scholar
  45. 45.
    Feige U, Kilian J (1997) Making games short. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing. ACM Press, New York pp 506–516Google Scholar
  46. 46.
    Feige U, Lovász L (1992) Two-prover one-round proof systems: their power and their problems. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing. ACM Press, New York pp 733–744Google Scholar
  47. 47.
    Fenner S, Fortnow L, Kurtz S (1994) Gap-definable counting classes. J Comput Syst Sci 48:116–148MathSciNetzbMATHGoogle Scholar
  48. 48.
    Fenner S, Green F Homer S, Zhang Y (2005) Bounds on the power of constant-depth quantum circuits. In: Proceedings of the 15th International Symposium on Fundamentals of Computation Theory. Lect Notes Comput Sci 3623:44–55Google Scholar
  49. 49.
    Feynman R (1983) Simulating physics with computers. Int J Theor Phys 21(6/7):467–488MathSciNetGoogle Scholar
  50. 50.
    Fortnow L (1997) Counting complexity. In: Hemaspaandra L, Selman A (eds) Complexity Theory Retrospective II. Springer, New York pp 81–107Google Scholar
  51. 51.
    Fortnow L, Rogers J (1999) Complexity limitations on quantum computation. J Comput Syst Sci 59(2):240–252MathSciNetzbMATHGoogle Scholar
  52. 52.
    Goldreich O (2005) On promise problems (a survey in memory of Shimon Even [1935–2004]). Electronic Colloquium on Computational Complexity; Report TR05-018Google Scholar
  53. 53.
    Goldreich O, Vadhan S (1999) Comparing entropies in statistical zero-knowledge with applications to the structure of SZK. In: Proceedings of the 14th Annual IEEE Conference on Computational Complexity. pp 54–73Google Scholar
  54. 54.
    Goldwasser S, Sipser M (1989) Private coins versus public coins in interactive proof systems. In: Micali S (ed) Randomness and Computation, vol 5 of Advances in Computing Research. JAI Press, Greenwich, Conn pp 73–90Google Scholar
  55. 55.
    Goldwasser S, Micali S, Rackoff C (1985) The knowledge complexity of interactive proof systems. In: Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing. ACM Press, New York pp 291–304Google Scholar
  56. 56.
    Goldwasser S, Micali S, Rackoff C (1989) The knowledge complexity of interactive proof systems. SIAM J Comput 18(1):186–208MathSciNetzbMATHGoogle Scholar
  57. 57.
    Gutoski G, Watrous J (2007) Toward a general theory of quantum games. In: Proceedings of the 39th ACM Symposium on Theory of Computing. ACM Press, New York pp 565–574Google Scholar
  58. 58.
    Håstad J (2001) Some optimal inapproximability results. J ACM 48(4):798–859Google Scholar
  59. 59.
    Janzing D, Wocjan P, Beth T (2005) Non-identity-check is QMA-complete. Int J Quantum Inf 3(2):463–473zbMATHGoogle Scholar
  60. 60.
    Kaye P, Laflamme R, Mosca M (2007) An introduction to quantum computing. Oxford University Press, OxfordzbMATHGoogle Scholar
  61. 61.
    Kempe J, Kitaev A, Regev O (2006) The complexity of the local Hamiltonian problem. SIAM J Comput 35(5):1070–1097MathSciNetzbMATHGoogle Scholar
  62. 62.
    Kempe J, Kobayashi H, Matsumoto K, Toner B, Vidick T (2007) Entangled games are hard to approximate. Proceedings ot the 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos 2008Google Scholar
  63. 63.
    Kempe J, Regev O, Toner B (2007) The unique games conjecture with entangled provers is false. Proceedings ot the 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos 2008Google Scholar
  64. 64.
    Kitaev A (1997) Quantum computations: algorithms and error correction. Russ Math Surv 52(6):1191–1249MathSciNetzbMATHGoogle Scholar
  65. 65.
    Kitaev A (1999) Quantum NP. Talk at AQIP'99: Second Workshop on Algorithms in Quantum Information Processing. DePaul University, ChicagoGoogle Scholar
  66. 66.
    Kitaev A, Watrous J (2000) Parallelization, amplification, and exponential time simulation of quantum interactive proof system. In: Proceedings of the 32nd ACM Symposium on Theory of Computing. ACM Press, New York pp 608–617Google Scholar
  67. 67.
    Kitaev A, Shen A, Vyalyi M (2002) Classical and Quantum Computation. Graduate Studies in Mathematics, vol 47. American Mathematical Society, ProvidenceGoogle Scholar
  68. 68.
    Knill E (1995) Approximation by quantum circuits. Technical Report LAUR-95-2225, Los Alamos National Laboratory. Available as arXiv.org e-Print quant-ph/9508006Google Scholar
  69. 69.
    Knill E (1996) Quantum randomness and nondeterminism. Technical Report LAUR-96-2186, Los Alamos National Laboratory. Available as arXiv.org e-Print quant-ph/9610012Google Scholar
  70. 70.
    Kobayashi H, Matsumoto K (2003) Quantum multi-prover interactive proof systems with limited prior entanglement. J Comput Syst Sci 66(3):429–450MathSciNetzbMATHGoogle Scholar
  71. 71.
    Kobayashi H, Matsumoto K, Yamakami T (2003) Quantum Merlin–Arthur proof systems: Are multiple Merlins more helpful to Arthur? In: Proceedings of the 14th Annual International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol 2906. Springer, Berlin Google Scholar
  72. 72.
    Levin L (1973) Universal search problems. Probl Inf Transm 9(3):265–266 (English translation)Google Scholar
  73. 73.
    Liu Y-K (2006) Consistency of local density matrices is QMA-complete. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Springer, Berlin; Lect Notes Comput Sci 4110:438–449Google Scholar
  74. 74.
    Lloyd S (2000) Ultimate physical limits to computation. Nature 406:1047–1054Google Scholar
  75. 75.
    Lund C, Fortnow L, Karloff H, Nisan N (1992) Algebraic methods for interactive proof systems. J ACM 39(4):859–868MathSciNetzbMATHGoogle Scholar
  76. 76.
    Marriott C, Watrous J (2005) Quantum Arthur-Merlin games. Comput Complex 14(2):122–152MathSciNetGoogle Scholar
  77. 77.
    Moore C, Nilsson M (2002) Parallel quantum computation and quantum codes. SIAM J Comput 31(3):799–815MathSciNetGoogle Scholar
  78. 78.
    Moore G (1965) Cramming more components onto integrated circuits. Electron 38(8):82–85Google Scholar
  79. 79.
    Nielsen MA, Chuang IL (2000) Quantum Computation and Quantum Information. Cambridge University Press, CambridezbMATHGoogle Scholar
  80. 80.
    Nishimura H, Yamakami T (2004) Polynomial time quantum computation with advice. Inf Process Lett 90(4):195–204MathSciNetzbMATHGoogle Scholar
  81. 81.
    Oliveira R, Terhal B (2005) The complexity of quantum spin systems on a two-dimensional square lattice. Available as arXiv.org e-Print quant-ph/0504050Google Scholar
  82. 82.
    Papadimitriou C (1994) Computational Complexity. Addison-Wesley, Readig, MasszbMATHGoogle Scholar
  83. 83.
    Raz R (2005) Quantum information and the PCP theorem. In: 46th Annual IEEE Symposium on Foundations of Computer Science. pp 459–468Google Scholar
  84. 84.
    Rosgen B (2008) Distinguishing short quantum computations. Proceedings of the 25th Annual Symposium on theoretical Aspects of Computer Science, IBFI, Schloss Dagstuhl, pp 597–608Google Scholar
  85. 85.
    Rosgen B, Watrous J (2005) On the hardness of distinguishing mixed-state quantum computations. In: Proceedings of the 20th Annual Conference on Computational. pp 344–354Google Scholar
  86. 86.
    Sahai A, Vadhan S (2003) A complete promise problem for statistical zero-knowledge. J ACM 50(2):196–249MathSciNetGoogle Scholar
  87. 87.
    Shamir A (1992) IP \( { = } \) PSPACE. J ACM 39(4):869–877MathSciNetzbMATHGoogle Scholar
  88. 88.
    Shor P (1994) Algorithms for quantum computation: discrete logarithms ande factoring. In: Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science. pp 124–134Google Scholar
  89. 89.
    Shor P (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Comput 26(5):1484–1509MathSciNetzbMATHGoogle Scholar
  90. 90.
    Stinespring WF (1955) Positive functions on C *-algebras. Proc Am Math Soc 6(2):211–216MathSciNetzbMATHGoogle Scholar
  91. 91.
    Toda S (1991) PP is as hard as the polynomial-time hierarchy. SIAM J Comput 20(5):865–887MathSciNetzbMATHGoogle Scholar
  92. 92.
    Toffoli T (1980) Reversible computing. Technical Report MIT/LCS/TM-151, Laboratory for Computer Science, Massachusetts Institute of TechnologyGoogle Scholar
  93. 93.
    Vadhan S (2007) The complexity of zero knowledge. In: 27th International Conference on Foundations of Software Technology and Theoretical Computer Science. Lect Notes Comput Sci 4855:52–70MathSciNetGoogle Scholar
  94. 94.
    Valiant L (1979) The complexity of computing the permanent. Theor Comput Sci 8:189–201MathSciNetzbMATHGoogle Scholar
  95. 95.
    Watrous J (1999) Space-bounded quantum complexity. J Comput Syst Sci 59(2):281–326MathSciNetzbMATHGoogle Scholar
  96. 96.
    Watrous J (2000) Succinct quantum proofs for properties of finite groups. In: Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science. pp 537–546Google Scholar
  97. 97.
    Watrous J (2002) Limits on the power of quantum statistical zero-knowledge. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science. pp 459–468Google Scholar
  98. 98.
    Watrous J (2003) On thecomplexity of simulating space-bounded quantum computations. Comput Complex12:48–84MathSciNetzbMATHGoogle Scholar
  99. 99.
    Watrous J (2003) PSPACE has constant-round quantum interactive proof systems. Theor Comput Sci 292(3):575–588MathSciNetzbMATHGoogle Scholar
  100. 100.
    Watrous J (2006) Zero-knowledge against quantum attacks. In: Proceedings of the 38th ACM Symposium on Theory of Computing. ACM Press, New York pp 296–305Google Scholar
  101. 101.
    Wehner S (2006) Entanglement in interactive proof systems with binary answers. In: Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science. Lect Notes Comput Sci 3884:162–171MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • John Watrous
    • 1
  1. 1.Institute for Quantum Computing and School of Computer ScienceUniversity of WaterlooWaterlooCanada