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Computational Complexity pp 2303–2333Cite as

Quantum Algorithms

  • Reference work entry

Article Outline

Glossary

Definition of the Subject

Introduction and Overview

Early Quantum Algorithms

Factoring, Discrete Logarithms and the Abelian Hidden Subgroup Problem

Algorithms Based on Amplitude Amplification

Simulation of Quantum Mechanical Systems

Generalizations of the Abelian Hidden Subgroup Problem

Quantum Walk Algorithms

Adiabatic Algorithms

Topological Algorithms

Quantum Algorithms for Quantum Tasks

Future Directions

Bibliography

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Notes

  1. 1.

    For any two elements \( { g,h } \) of a group, we define their commutator, denoted \( { [g,h] } \) to be \( { [g,h] = g^{-1} h^{-1} g h } \), and for any two subgroups \( { H, K \leq G } \) we define \( { [H,K] } \) to be the (normal) subgroup of G generated by all the commutators \( { [h,k] } \) where \( { h \in H, k \in K } \).

Abbreviations

Quantum circuit model :

One of the standard and most commonly used models of quantum computation which generalizes the classical model of acyclic circuits and closely models most of the proposed physical implementations of quantum computers.When studying algorithms for a problem with an infinite number of possible inputs, one usually restricts attention to uniform families of circuits, which are families of circuits in which the circuit for inputs of size n can be generated efficiently as a function of n. For example, one might require that there is a classical Turing machine that can generate the nth circuit in time polynomial in n.

Black box model :

A model of computation where the input to the problem includes a “black‐box” that can be applied (equivalently, an “oracle ” that can be “queried”). This is the only way to extract information from the black-box. For example, the black-box could accept inputs \( { j \in\{0,1\}^n } \) and output a value \( { X_j \in \{0,1\} } \). In this particular case, we can think of the black-box as a means for querying the bits of the string \( { \mathbf{X} = X_1 X_2 X_3 \ldots X_{2^n} } \). In the black-box model, one usually measures complexity in terms of the number of applications of the black-box.

Computational complexity :

When referring to an algorithm, the computational complexity (often just called the complexity) is a measure of the resources used by the algorithm (which we can also refer to as the cost of the algorithm) usually measured as a function of the size of the input to the algorithm. The complexity for input size n is taken to be the cost of the algorithm on a worst-case input of size n to the problem. This is also referred to as worst-case complexity. When referring to a problem, the computational complexity is the minimum amount of resources required by any algorithm to solve the problem. See Quantum Computational Complexity for an overview.

Query complexity :

When referring to a black-box algorithm, the query complexity is the number of applications of the black-box or oracle used by the algorithm. When referring to a black-box problem, the query complexity is the minimum number of applications of the black-box required by any algorithm to solve the problem.

Bibliography

  1. Aaronson S (2003) Algorithms for Boolean function query properties.SIAM J Comput32:1140–1157

    Article  MathSciNet  MATH  Google Scholar 

  2. Aaronson S, Ambainis A (2003) Quantum search of spatial regions. In: Proceedings ofthe 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, Washington, DC,pp 200–209

    Google Scholar 

  3. Abrams D, Lloyd S (1999) Quantum algorithm providing exponential speed increasefor finding eigenvalues and eigenvectors.Phys Rev Lett 83:5162–5165

    Article  Google Scholar 

  4. Aharonov D, Regev O (2005) Lattice problems in NP intersect coNP. J ACM52:749–765

    Article  MathSciNet  Google Scholar 

  5. Aharonov D, Ta-Shma A (2007) Adiabatic quantum state generation. SIAM J Comput37:47–82

    Article  MathSciNet  MATH  Google Scholar 

  6. Aharonov D, Ambainis A, Kempe J, Vazirani U (2001) Quantum walks on graphs. In:Proceedings of ACM Symposium on Theory of Computation (STOC'01). ACM, New York, pp 50–59

    Chapter  Google Scholar 

  7. Aharonov D, van Dam W, Kempe J, Landau Z, Lloyd S, Regev O (2004) Adiabaticauantum computation is equivalent to standard quantum computation. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science(FOCS'04). IEEE Computer Society, Washington, DC, pp 42–51

    Chapter  Google Scholar 

  8. Aharonov D, Jones V, Landau Z (2006) A polynomial quantum algorithm forapproximating the Jones polynomial. In: Proceedings of the thirty‐eighth annual ACM symposium on Theory of computing (STOC). ACM, New York,pp 427–436

    Chapter  Google Scholar 

  9. Aharonov D, Gottesman D, Irani S, Kempe J (2007) The power of quantum systems on a line. In: Proc. 48th IEEE Symposium on the Foundations of Computer Science (FOCS). IEEE Computer Society, Washington, DC,pp 373–383

    Chapter  Google Scholar 

  10. Aharonov D, Arad I, Eban E, Landau Z Polynomial quantum algorithms for additiveapproximations of the Potts model and other points of the Tutte plane. quant-ph/0702008

    Google Scholar 

  11. Alagic G, Moore C, Russell A (2007) Quantum algorithms for Simon's problem overgeneral groups. In: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms (SODA). Society for Industrial and Applied Mathematics,Philadelphia, pp 1217–1224

    Google Scholar 

  12. Ambainis A (2002) Quantum lower bounds by quantum arguments. J Comput SystSci 64:750–767

    Article  MathSciNet  MATH  Google Scholar 

  13. Ambainis A (2003) Quantum walks and their algorithmic applications. Int JQuantum Inf 1:507–518

    Article  MATH  Google Scholar 

  14. Ambainis A (2004) Quantum search algorithms. SIGACT News. 35(2):22–35

    Article  Google Scholar 

  15. Ambainis A (2004) Quantum walk algorithm for element distinctness. In:Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS'04). IEEE Computer Society, Washington, DC,pp 22–31

    Chapter  Google Scholar 

  16. Ambainis A, Spalek R (2006) Quantum algorithms for matching and networkflows. In: Proceedings of STACS'06. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, pp 172–183

    Google Scholar 

  17. Ambainis A, Bach E, Nayak A, Vishwanath A, Watrous J (2001)One‐dimensional quantum walks. In: Proceedings of the 33rd ACM Symposium on Theory of Computing. ACM, New York,pp 37–49

    Google Scholar 

  18. Ambainis A, Childs A, Reichardt B, Spalek R, Zhang S (2007) Any AND-OR formulaof size N can be evaluated in time \( { N^{1/2 + o(1)} }\) on a quantum computer. In: 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07). IEEE ComputerSociety, Washington, DC, pp 363–372

    Chapter  Google Scholar 

  19. Bacon D, Childs A, van Dam W (2005) From optimal measurement to efficientquantum algorithms for the hidden subgroup problem over semidirect product groups. In: Proc 46th IEEE Symposium on Foundations of Computer Science (FOCS2005). IEEE Computer Society, Washington, DC, pp 469–478

    Chapter  Google Scholar 

  20. Bacon D, Chuang I, Harrow A (2006) Efficient quantum circuits for Schur andClebsch-Gordan transforms. Phys Rev Lett 97:170502

    Article  MathSciNet  Google Scholar 

  21. Beals R (1997) Quantum computation of Fourier transforms over symmetricgroups. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC). ACM, New York, pp 48–53

    Google Scholar 

  22. Beals R, Buhrman H, Cleve R, Mosca M, deWolf R (2001) Quantum lower bounds bypolynomials. J ACM 48:778–797

    Article  MathSciNet  MATH  Google Scholar 

  23. Benioff P (1982) Quantum mechanical models of Turing machines that dissipate noenergy. Phys Rev Lett 48(23):1581–1585

    Article  MathSciNet  Google Scholar 

  24. Bennett C (1988) Notes on the history of reversible computation by CharlesBennett. IBM J Res Dev 32(1):16–23

    Article  Google Scholar 

  25. Bennett C, Bernstein E, Brassard G, Vazirani U (1997) Strengths and weaknessesof quantum computing. SIAM J Comput 26:1510–1523

    Article  MathSciNet  MATH  Google Scholar 

  26. Bennett C, Harrow A, Lloyd S (2006) Universal quantum data compression viagentle tomography. Phys Rev A 73:032336

    Article  Google Scholar 

  27. Bennett CH, Brassard G (1984) Quantum cryptography: Public-key distribution andcoin tossing. In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, India. IEEE Computer Society, Washington, DC,pp 175–179

    Google Scholar 

  28. Bernstein E, Vazirani U (1997) Quantum complexity theory. SIAM J Comput26:1411–1473

    Article  MathSciNet  MATH  Google Scholar 

  29. Berry DW, Ahokas G, Cleve R, Sanders BC (2007) Efficient quantum algorithms forsimulating sparse Hamiltonians. Comm Math Phys 270:359

    Article  MathSciNet  MATH  Google Scholar 

  30. Berthiaume A, Brassard G (1992) The quantum challenge to structural complexitytheory. In: Proc 7th Conference Structure Complexity Theory, IEEE Comp Soc Press, pp 132–137

    Google Scholar 

  31. Berthiaume A, Brassard G (1994) Oracle quantum computing.J Modern Opt41(12):2521–2535

    Article  MathSciNet  MATH  Google Scholar 

  32. Boneh D, Lipton R (1995) Quantum cryptanalysis of hidden linear functions(Extended Abstract). In: Proceedings of 15th Annual International Cryptology Conference (CRYPTO'95). Springer, London, pp 424–437

    Google Scholar 

  33. Bordewich M, Freedman M, Lovasz L, Welsh DJA (2005) Approximate counting andquantum computation. Comb Probab Comput 14:737–754

    Article  MathSciNet  MATH  Google Scholar 

  34. Boyer M, Brassard G, Høyer P, Tapp A (1998) Tight bounds on quantum searching.Fortschr Phys 56(5-5):493–505

    Google Scholar 

  35. Brassard G, Høyer P (1997) An exact quantum polynomial-time algorithm forSimon's problem. In: Proc of Fifth Israeli Symposium on Theory of Computing and Systems (ISTCS'97). IEEE Computer Society, Washington, DC,pp 12–23

    Google Scholar 

  36. Brassard G, Høyer P, Tapp A (1997) Cryptology column–quantum algorithm forthe collision problem. ACM SIGACT News 28:14–19

    Google Scholar 

  37. Brassard G, Høyer P, Tapp A (1998) Quantum Counting. In: Proceedings of theICALP'98. Lecture notes in computer science. Springer, Berlin, pp 1820–1831

    Google Scholar 

  38. Brassard G, Høyer P, Mosca M, Tapp A (2002) Quantum amplitude amplification andestimation. In: Quantum Computation and Quantum Information Science. AMS Contemp Math Ser 35:53–74

    Google Scholar 

  39. Brown M (2003) Classical cryptosystems in a quantum setting. Master Thesis,University of Waterloo

    Google Scholar 

  40. Buhrman H, Špalek B (2006) Quantum verification of matrix products. In:Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). IEEE Computer Society, Washington, DC,pp 880–889

    Google Scholar 

  41. Buhrman H, Fortnow L, Newman I, Röhrig H (2003) Quantum property testing. In:Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, Philadelphia,pp 480–488

    Google Scholar 

  42. Buhrman H, Dürr C, Heiligman M, Høyer P, Magniez F, Santha M, de Wolf R (2005)Quantum algorithms for element distinctness. SIAM J Comput 34:1324–1330

    Google Scholar 

  43. Byrnes T, Yamamoto Y (2006) Simulating lattice gauge theories on a quantumcomputer. Phys Rev A 73:022328

    Article  Google Scholar 

  44. Cheung K, Mosca M (2001) Decomposing finite Abelian groups. Quantum Inf Comput1(2):26–32

    MathSciNet  MATH  Google Scholar 

  45. Childs A (2002) Quantum information processing in continuous time. Ph D thesis, MIT

    Google Scholar 

  46. Childs A, van Dam W (2007) Quantum algorithm for a generalized hidden shiftproblem. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM, New York,pp 1225–1232

    Google Scholar 

  47. Childs A, Farhi E, Gutmann S (2002) An example of the difference between quantumand classical random walks. Quantum Inf Process 1:35–43

    Article  MathSciNet  Google Scholar 

  48. Childs A, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman D (2003) Exponentialalgorithmic speedup by a quantum walk. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing. IEEE Computer Society, Washington,DC

    Google Scholar 

  49. Childs A, Landahl A, Parrilo P (2007) Improved quantum algorithms for theordered search problem via semidefinite programming.Phys Rev A 75:032335

    Article  Google Scholar 

  50. Childs AM, Cleve R, Jordan SP, Yeung D Discrete-query quantum algorithm for NANDtrees. quant-ph/0702160

    Google Scholar 

  51. Childs A, Schulman L, Vazirani U (2007) Quantum algorithms for hidden nonlinearstructures. In: Proceeding of 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007). IEEE Computer Society, Washington, DC

    Google Scholar 

  52. Cleve R (2000) An introduction to quantum complexity theory.In: MacchiavelloC, Palma GM, Zeilinger A (eds) Collected Papers on Quantum Computation and Quantum Information Theory. World Scientific, Singapore,pp 103–127

    Google Scholar 

  53. Cleve R (2000) The query complexity of order‐finding. In: IEEE Conference onComputational Complexity. IEEE Computer Society, Washington, DC, p 54

    Chapter  Google Scholar 

  54. Cleve R, Ekert A, Macchiavello C, Mosca M (1998) Quantum algorithms revisited.Proc Royal Soc Lond A 454:339–354

    Article  MathSciNet  MATH  Google Scholar 

  55. Cleve R, Gavinsky D, Yeung D (2008) Quantum algorithms for evaluating Min-Max trees.In: Proceedings of TQC 2008. Lecture Notes in Computer Science, vol 5106. Springer, Berlin

    Google Scholar 

  56. van Dam W, Seroussi G Efficient quantum algorithms for estimating Gauss sums.quant-ph/0207131

    Google Scholar 

  57. van Dam W, Shparlinski I (2008) Classical and quantum algorithms for exponentialcongruences. In: Proceedings of TQC 2008. Lecture Notes in Computer Science, vol 5106. Springer, Berlin, pp 1–10

    Google Scholar 

  58. van Dam W, Mosca M, Vazirani U (2001) How powerful is adiabatic quantumcomputation? In: Proceedings 46th IEEE Symposium on Foundations of Computer Science (FOCS'01). IEEE Computer Society, Washington, DC,pp 279–287

    Google Scholar 

  59. van Dam W, Hallgren S, Ip L (2003) Quantum algorithms for some hidden shiftproblems. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA'03). Society for Industrial and Applied Mathematics, Philadelphia,pp 489–498

    Google Scholar 

  60. van Dam W, D'Ariano GM, Ekert A, Macchiavello C, Mosca M (2007) Generaloptimized schemes for phase estimation.Phys Rev Lett 98:090501

    Article  Google Scholar 

  61. van Dam W, D'Ariano GM, Ekert A, Macchiavello C, Mosca M (2007) Optimal phaseestimation in quantum networks.J Phys A: Math Theor 40:7971–7984

    Article  MATH  Google Scholar 

  62. Das A, Chakrabarti BK (2008) Quantum annealing and analog quantumcomputation. Rev Mod Phys 80:1061

    Article  MathSciNet  MATH  Google Scholar 

  63. Decker T, Draisma J, Wocjan P (2009) Efficient quantum algorithm for identifying hiddenpolynomials. Quantum Inf Comput (to appear)

    Google Scholar 

  64. Deutsch D (1985) Quantum theory, the Church–Turing principle and theuniversal quantum computer. Proc Royal Soc Lond A 400:97–117

    Article  MathSciNet  MATH  Google Scholar 

  65. Deutsch D, Jozsa R (1992) Rapid solutions of problems by quantum computation.Proc Royal Soc Lond A 439:553–558

    Article  MathSciNet  MATH  Google Scholar 

  66. Dörn S, Thierauf T (2007) The quantum query complexity of algebraicproperties. In: Proceedings of the 16th International Symposium on Fundamentals of Computation Theory (FCT). Lecture Notes in Computer Science, vol 4639. Springer, Berlin, pp 250–260

    Google Scholar 

  67. Durr C, Heiligman M, Høyer P, Mhalla M (2004) Quantum query complexity of somegraph problems. In: Proceedings of 31st International Colloquium on Automata, Languages, and Programming (ICALP'04). Lecture Notes in Computer Science,vol 3142. Springer, Berlin pp 481–493

    Google Scholar 

  68. Ettinger JM (1998) On noncommutative hidden subgroups. Lecture atAQIP'98. http://www.brics.dk/~salvail/aqip/abstlist.html#upload-paper.10480.1

  69. Ettinger JM Quantum time‐frequencytransforms. quant-ph/0005134

    Google Scholar 

  70. Ettinger M, Høyer P (2000) On quantum algorithms for noncommutative hiddensubgroups.Adv Appl Math 25(3):239–251

    Google Scholar 

  71. Ettinger M, Høyer P, Knill E (2004) The quantum query complexity of the hiddensubgroup problem is polynomial. Inf Process Lett 91:43–48

    Google Scholar 

  72. Farhi E, Gutmann S (1998) Quantum computation and decision trees.Phys Rev A58:915–928

    Article  MathSciNet  Google Scholar 

  73. Farhi E, Gutmann S An analog analogue of a digital quantum computation.quant-ph/9612026

    Google Scholar 

  74. Farhi E, Goldstone J, Gutmann S, Sipser M (2000) Quantum computation byadiabatic evolution.quant-ph/0001106

    Google Scholar 

  75. Farhi E, Goldstone J, Gutmann S, Lapan J, Lundgren A, Preda D (2001) A quantumadiabatic evolution algorithm applied to random instances of an NP.Science 20 April:472

    Google Scholar 

  76. Farhi E, Goldstone J, Gutmann S A quantum algorithm for the Hamiltonian NANDtree. quant-ph/0702144v2

    Google Scholar 

  77. Farhi E, Goldstone J, Gutmann S Quantum adiabatic evolution algorithms withdifferent paths. quant-ph/0208135

    Google Scholar 

  78. Fenner S, Zhang Y (2005) Quantum algorithms for a set of group theoreticproblems. In: Proceedings of the Ninth IC-EATCS Italian Conference on Theoretical Computer Science.Lecture Notes in Computer Science, vol 3701. Springer,Berlin, pp 215–227

    Google Scholar 

  79. Feynman R (1982) Simulating physics with computers.Int J Theor Phys21(6,7):467–488

    Article  MathSciNet  Google Scholar 

  80. Freedman M (1998) P/NP, and the quantum field computer. Proc Natl Acad Sci95(1):98–101

    Article  MathSciNet  MATH  Google Scholar 

  81. Freedman M, Kitaev A, Wang Z (2002) Simulation of topological Field theories byquantum computers.Comm Math Phys 227(3):587–603

    Article  MathSciNet  MATH  Google Scholar 

  82. Friedl K, Ivanyos G, Magniez F, Santha M, Sen P (2003) Hidden translation andorbit coset in quantum computing. In: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing (STOC'03). IEEE Computer Society,Washington, DC, pp 1–9

    Chapter  Google Scholar 

  83. Friedl K, Ivanyos G, Santha M (2005) Efficient testing of groups. In: Proceedingsof the Thirty-seventh Annual ACM Symposium on Theory of Computing (STOC). IEEE Computer Society, Washington, DC, pp 157–166

    Chapter  Google Scholar 

  84. Furrow B (2008) A panoply of quantum algorithms. Quantum Inf Comput8(8–9):0834–0859

    MathSciNet  Google Scholar 

  85. Geraci J, Lidar D (2008) On the exact evaluation of certain instances of thepotts Partition function by quantum computers. Comm Math Phys 279:735

    Article  MathSciNet  MATH  Google Scholar 

  86. Gerhardt H, Watrous J (2003) Continuous-time quantum walks on the symmetricgroup.6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, (APPROX) and 7th International Workshop onRandomization and Approximation Techniques in Computer Science, (RANDOM), pp 290–301

    Google Scholar 

  87. Grigni M, Schulman L, Vazirani M, Vazirani U (2001) Quantum mechanicalalgorithms for the nonabelian hidden subgroup problem. In: Proceedings of the thirty-third annual ACM symposium on Theory of computing (SODA'03),pp 68–74

    Google Scholar 

  88. Gottesman D, Chuang I Quantum digitalsignatures. quant-ph/0105032

    Google Scholar 

  89. Grigoriev D (1997) Testing shift‐equivalence of polynomials bydeterministic, probabilistic and quantum machines.Theor Comput Sci 180:217–228

    Article  MathSciNet  MATH  Google Scholar 

  90. Grover L (1996) A fast quantum mechanical algorithm for database search. In:Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC 1996), pp 212–219

    Google Scholar 

  91. Grover L (1998) A framework for fast quantum mechanical algorithms. In: Proceedingsof the Thirtieth Annual ACM Symposium on Theory of Computing (STOC), pp 53–62

    Google Scholar 

  92. Hales L, Hallgren S (2000) An Improved Quantum Fourier Transform Algorithm andApplications. FOCS 2000, pp 515–525

    Google Scholar 

  93. Hallgren S (2002) Polynomial-time quantum algorithms for Pell's equation and theprincipal ideal problem. STOC 2002, pp 653–658

    Google Scholar 

  94. Hallgren S (2005) Fast quantum algorithms for computing the unit group and classgroup of a number field. In: Proceedings of the 37th ACM Symposium on Theory of Computing (STOC 2005), pp 468–474

    Google Scholar 

  95. Hallgren S (2007) Polynomial-time quantum algorithms for Pell's equation and theprincipal ideal problem. J ACM 54(1):653–658

    MathSciNet  Google Scholar 

  96. Hallgren S, Russell A, Ta-Shma A (2000) Normal subgroup reconstruction andquantum computation using group representations. In: Proceedings of the Thirty‐Second Annual ACM Symposium on Theory of Computing,pp 627–635

    Google Scholar 

  97. Hallgren S, Moore C, Roetteler M, Russel A, Sen P (2006) Limitations of quantum coset states forgraph isomorphism. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing. IEEE Computer Society, Washington, DC

    Google Scholar 

  98. Hassidim A, Ben-Or M (2008) The Bayesian learner is optimal for noisy binarysearch (and pretty good for quantum as well). In: Proceedings of the 49th Symposium on Foundations of Computer Science. IEEE Computer Society, Washington,DC

    Google Scholar 

  99. Hausladen P, Wootters WK (1994) A pretty good measurement for distinguishingquantum. states.J Mod Opt 41:2385

    Article  MathSciNet  MATH  Google Scholar 

  100. Helstrom CW (1976) Quantum detection and estimation theory.Academic Press, NewYork

    Google Scholar 

  101. Hogg T (2000) Quantum search heuristics.Phys Rev A61:052311

    Article  Google Scholar 

  102. Holevo AS (1982) Probabilistic and statistical aspects of quantum theory.NorthHolland, Amsterdam

    MATH  Google Scholar 

  103. Høyer P Efficient quantum transforms. quant-ph/9702028

    Google Scholar 

  104. Høyer P (1999) Conjugated operators in quantum algorithms.Phys Rev A59(5):3280–3289

    Google Scholar 

  105. Høyer P (2001) Introduction to recent quantum algorithms. In: Proceedings of 26thInternational Symposium on Mathematical Foundations of Computer Science (MFCS'01).Lecture Notes in Computer Science, vol 2136. Springer, Berlin,pp 62–73

    Google Scholar 

  106. Høyer P, Dürr C A quantum algorithm for finding the minimum. quant-ph/9607014

    Google Scholar 

  107. Høyer P, Neerbek J, Shi Y (2002) Quantum complexities of ordered searching,sorting, and element distinctness.Algorithmica 34(4):429–448

    Google Scholar 

  108. Høyer P, Mosca M, de Wolf R (2003) Quantum search on bounded-error inputs. In:Proceedings of the Thirtieth International Colloquium on Automata, Languages and Programming (ICALP'03),pp 291–299

    Google Scholar 

  109. Inui Y, Le Gall F (2008) Quantum property testing of group solvability. In:Proceedings of the 8th Latin American Symposium (LATIN'08: Theoretical Informatics). Lecture Notes in Computer Science, vol 4957. Springer,Berlin, pp 772–783

    Google Scholar 

  110. Ivanyos G, Magniez F, Santha M (2003) Efficient quantum algorithms for someinstances of the non‐Abelian hidden subgroup problem.Int J Found Comput Sci 14(5):723–739

    Article  MathSciNet  MATH  Google Scholar 

  111. Ivanyos G, Sanselme L, Santha M (2007) An efficient quantum algorithm for theHidden subgroup problem in extraspecial groups.24th STACS. Lecture Notes in Computer Science, vol 4393. Springer, Berlin,pp 586–597

    Google Scholar 

  112. Ivanyos G, Sanselme L, Santha M (2008) An efficient quantum algorithm for the hiddensubgroup problem in nil-2 groups. In: Proceedings of the 8th Latin American Symposium (LATIN'08: Theoretical Informatics). Lecture Notes in ComputerScience, vol 4957. Springer, Berlin, pp 759–771

    Google Scholar 

  113. Ioannou L (2002) Continuous-time quantum algorithms: Searching and adiabaticcomputation.Master Math Thesis, University of Waterloo

    Google Scholar 

  114. Jaeger F, Vertigan DL, Welsh DJA (1990) On the Computational complexity of theJones and Tutte polynomials. Math Proc Cambridge Phil Soc 108(1):5–53

    MathSciNet  Google Scholar 

  115. Jansen S, Ruskai M, Seiler R (2007) Bounds for the adiabatic approximationwith applications to quantum computation.J Math Phys 48:102111

    Article  MathSciNet  Google Scholar 

  116. Jozsa R (1998) Quantum algorithms and the Fourier transform. Proc Royal SocLond A 454:323–337

    Article  MathSciNet  MATH  Google Scholar 

  117. Jozsa R (2003) Notes on Hallgren's efficient quantum algorithm for solving Pell'sequation. quant-ph/0302134

    Google Scholar 

  118. Jozsa R (2006) An introduction to measurement based quantum computation.NATOScience Series, III: Computer and Systems Sciences.Quantum Information Processing - From Theory to Experiment, vol 199,pp 137–158

    Google Scholar 

  119. Kassal I, Jordan S, Love P, Mohseni M, Aspuru-Guzik A (2008) Quantumalgorithms for the simulation of chemical dynamics.arXiv:0801.2986

  120. Kauffman L (2001) Knots and Physics. World Scientific, Singapore

    MATH  Google Scholar 

  121. Kaye P (2005) Optimized quantum implementation of elliptic curve arithmeticover binary fields.Quantum Inf Comput 5(6):474–491

    MathSciNet  MATH  Google Scholar 

  122. Kaye P, Laflamme R, Mosca M (2007) An introduction to quantum computation.Oxford University Press, Oxford

    Google Scholar 

  123. Kempe J (2003) Quantum random walks - an introductory overview.Contemp Phys44(4):307–327

    Article  MathSciNet  Google Scholar 

  124. Kempe J (2007) Approaches to quantum error correction.Quantum Decoherence,Poincare Seminar 2005.Progress in Mathematical Physics. Birkhäuser, Basel, pp 85–123

    Google Scholar 

  125. Kempe J, Shalev A (2005) The hidden subgroup problem and permutation grouptheory. In: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms (SODA'05), pp 1118–1125

    Google Scholar 

  126. Kitaev A (1995) Quantum measurements and the Abelian stabilizer problem.quant-ph/9511026

    Google Scholar 

  127. Kitaev A (1996) Quantum measurements and the Abelian stabilizer problem.Electronic Colloquium on Computational Complexity (ECCC), p 3

    Google Scholar 

  128. Kitaev A (1997) Quantum computations: Algorithms and error correction.RussMath Surv 52(6):1191–1249

    Article  MathSciNet  MATH  Google Scholar 

  129. Kitaev A, Shen A, Vyalvi M (2002) Classical and quantum computation.AmericanMathematical Society

    Google Scholar 

  130. Kuperberg G (2005) A subexponential-time quantum algorithm for the dihedralhidden subgroup problem.SIAM J Comput 35:170–188

    Article  MathSciNet  MATH  Google Scholar 

  131. Lloyd S (1996) Universal quantum simulators.Science273:1073–1078

    Article  MathSciNet  MATH  Google Scholar 

  132. Lomonaco S, Kauffman L (2006) Topological quantum computing and the Jonespolynomial. In: Proc SPIE 6244

    Google Scholar 

  133. Magniez F, Nayak A (2007) Quantum complexity of testing group commutativity.Algorithmica 48(3):221–232

    Article  MathSciNet  MATH  Google Scholar 

  134. Magniez F, Nayak A, Roland J, Santha M (2007) Search via quantum walk.39thACM Symposium on Theory of Computing (STOC), pp 575–584

    Google Scholar 

  135. Magniez F, Santha M, Szegedy M (2007) Quantum algorithms for the triangleproblem.SIAM J Comput 37(2):413–424

    Article  MathSciNet  MATH  Google Scholar 

  136. Manin Y (2000) Classical computing, quantum computing, and Shor's factoringalgorithm.Séminaire Bourbaki, 41 (1998-1999), 862, pp 375–404.The appendix translates an excerpt about quantum computing from a 1980 paper(in Russian)

    Google Scholar 

  137. Menezes A, van Oorschot P, Vanstone S (1996) Handbook of Applied Cryptography.CRC Press, Boca Raton

    Book  Google Scholar 

  138. Moore C, Russell A (2007) For distinguishing conjugate hidden dubgroups, the prettygood measurement is as hood as it gets. Quantum Inf Comput 7(8):752–765

    MathSciNet  MATH  Google Scholar 

  139. Moore C, Rockmore D, Russell A, Schulman L (2004) The power of basis selectionin fourier sampling: hidden subgroup problems in affine groups. In: Proceedings of the fifteenth annual ACM-SIAM symposium on discrete algorithms (SODA'04),pp 1113–1122

    Google Scholar 

  140. Moore C, Rockmore D, Russell A (2006) Generic Quantum Fourier Transforms.ACMTrans Algorithms 2:707–723

    Article  MathSciNet  Google Scholar 

  141. Moore C, Russell A, Sniady P (2007) On the impossibility of a quantum sievealgorithm for graph isomorphism: unconditional results. In: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing (STOC),pp 536–545

    Google Scholar 

  142. Moore C, Russell A, Vazirani U A classical one-way function to confoundquantum adversaries. quant-ph/0701115

    Google Scholar 

  143. Mosca M (1999) Quantum computer algorithms.Ph D thesis,Oxford

    Google Scholar 

  144. Mosca M (2001) Counting by quantum eigenvalue estimation.Theor Comput Sci264:139–153

    Article  MathSciNet  MATH  Google Scholar 

  145. Mosca M, Ekert A (1998) The hidden subgroup problem and eigenvalue estimationon a quantum computer. In: Proceedings 1st NASA International Conference on Quantum Computing & Quantum Communications.Lecture Notes in ComputerScience, vol 1509. Springer, Berlin, pp 174–188

    Google Scholar 

  146. Mosca M, Zalka C (2004) Exact quantum Fourier transforms and discretelogarithm algorithms.Int J Quantum Inf 2(1):91–100

    Article  MATH  Google Scholar 

  147. Motwani R, Raghavan P (1995) Randomized Algorithms. Cambridge UniversityPress, Cambridge

    Book  MATH  Google Scholar 

  148. Nayak A, Wu F (1999) The quantum query complexity of approximating the medianand related statistics. In: Proceedings of the Thirty-first Annual ACM Symposium on Theory of Computing (STOC),pp 384–393

    Google Scholar 

  149. Nielsen M, Chuang I (2000) Quantum computation and quantum information.Cambridge University Press, Cambridge

    MATH  Google Scholar 

  150. Proos J, Zalka C (2003) Shor's discrete logarithm quantum algorithm forelliptic curves.Quantum Inf Comput 3:317–344

    MathSciNet  MATH  Google Scholar 

  151. Püschel M, Rötteler M, Beth T (1999) Fast quantum Fourier transforms for a class of non‐Abelian groups. In: Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error‐CorrectingCodes, pp 148–159

    Google Scholar 

  152. Radhakrishnan J, Roetteler M, Sen P (2005) On the power of random bases inFourier sampling: Hidden subgroup problem in the Heisenberg group. In: Proceedings of the 32nd International Colloquium on Automata, Languages andProgramming (ICALP), pp 1399–1411

    Google Scholar 

  153. Ramesh H, Vinay V (2003) String matching in \( { \tilde{O}(\sqrt{n} + \sqrt{m}) } \) quantum time.J Discret Algorithms 1:103–110

    Google Scholar 

  154. Regev O (2004) Quantum computation and lattice problems.SIAM J Comput33:738–760

    Article  MathSciNet  MATH  Google Scholar 

  155. Reichardt B, Spalek R Span-program-based quantum algorithm for evaluatingformulas. In: Proceedings of the fortieth annual ACM symposium on Theory of computing (STOC 2008), pp 103–112

    Google Scholar 

  156. Roetteler M, Beth T Polynomial-time solution to the hidden subgroup problemfor a class of non‐Abelian groups. quant-ph/9812070

    Google Scholar 

  157. Roland J, Cerf N (2002) Quantum search by local adiabatic evolution.Phys RevA 65(4):042308

    Article  MathSciNet  Google Scholar 

  158. Rudolph T, Grover L (2004) How significant are the known collision and elementdistinctness quantum algorithms? Quantum Inf Comput 4:201–206

    MathSciNet  MATH  Google Scholar 

  159. Russell A, Shparlinski I (2004) Classical and quantum function reconstructionvia character evaluation. J Complex 20:404–422

    Article  MathSciNet  MATH  Google Scholar 

  160. Santha M (2008) Quantum walk based search algorithms. In: Proceedings of the5th Annual International Conference on Theory and Applications of Models of Computation (TAMC 2008). Lecture Notes in Computer Science,vol 4978. Springer, Berlin, pp 31–46

    Google Scholar 

  161. Sasaki M, Carlini A, Jozsa R (2001) Quantum template matching.Phys Rev A64:022317

    Article  Google Scholar 

  162. Schmidt A, Vollmer U (2005) Polynomial time quantum algorithm for thecomputation of the unit group of a number field. In: Proceedings of the thirty‐seventh annual ACM symposium on Theory of computing,pp 475–480

    Google Scholar 

  163. Schumacher B (1995) Quantum coding.Phys Rev A51:2738–2747

    Article  MathSciNet  Google Scholar 

  164. Severini S (2003) On the digraph of a unitary matrix.SIAM J Matrix Anal Appl(SIMAX) 25(1):295–300

    Article  MathSciNet  MATH  Google Scholar 

  165. Shor P (1994) Algorithms for quantum computation: discrete logarithms andfactoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp 124–134

    Google Scholar 

  166. Shor P (1997) Polynomial-time algorithms for prime factorization and discretelogarithms on a quantum computer.SIAM J Comput 26:1484–1509

    Article  MathSciNet  MATH  Google Scholar 

  167. Simon D (1994) On the power of quantum computation. In: Proceedings of the 35thIEEE Symposium on the Foundations of Computer Science (FOCS), pp 116–123

    Google Scholar 

  168. Simon D (1997) On the power of quantum computation.SIAM J Comput26:1474–1483

    Article  MathSciNet  MATH  Google Scholar 

  169. Sokal A (2005) The multivariate Tutte polynomial (alias Potts model) forgraphs and matroids.Surveys in Combinatorics. Cambridge University Press, Cambridge, pp 173–226

    Google Scholar 

  170. Somma R, Ortiz G, Gubernatis JE, Knill E, Laflamme R (2002) Simulatingphysical phenomena by quantum networks.Phys Rev A 65:042323

    Article  Google Scholar 

  171. Szegedy M (2004) Quantum speed-Up of Markov chain based algorithms. In:Proceedings of the 45th IEEE Symposium on the Foundations of Computer Science (FOCS), pp 32–41

    Google Scholar 

  172. Terhal B (1999) Quantum algorithms and quantum entanglement. Ph D thesis,University of Amsterdam

    Google Scholar 

  173. Vazirani U (1998) On the power of quantum computation.Phil Trans Royal SocLond A 356:1759–1768

    Article  Google Scholar 

  174. Vazirani U (1997) Berkeley Lecture Notes. Lecture 8http://www.cs.berkeley.edu/~vazirani/qc.html

  175. Watrous J (2001) Quantum algorithms for solvable groups. In: Proceedings of thethirty-third annual ACM symposium on Theory of computing (STOC), pp 60–67

    Google Scholar 

  176. Welsh DJA (1993) Complexity: knots, colourings and counting.CambridgeUniversity Press, Cambridge

    Book  MATH  Google Scholar 

  177. Wiesner S (1983) Conjugate coding.Sigact News15(1):78–88

    Article  Google Scholar 

  178. Wiesner S (1996) Simulations of many-body quantum systems by a quantumcomputer. quant-ph/9603028

    Google Scholar 

  179. Wocjan P, Yard J (2008) The Jones polynomial: quantum algorithms andapplications in quantum complexity theory.Quantum Inf Comput 8(1–2):147–180

    MathSciNet  MATH  Google Scholar 

  180. Zalka C (1998) Efficient simulation of quantum systems by quantum computers.Proc Roy Soc Lond A 454:313–322

    Article  MATH  Google Scholar 

  181. Zalka C (2000) Using Grover's quantum algorithm for searching actualdatabases. Phys Rev A 62(5):052305

    Article  MathSciNet  Google Scholar 

  182. Zhang S (2006) New upper and lower bounds for randomized and quantum localsearch. In: Proceedings of the thirty‐eighth annual ACM symposium on Theory of computing (STOC), pp 634–643

    Google Scholar 

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Authors and Affiliations

  1. Institute for Quantum Computing and Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

    Michele Mosca

  2. St. Jerome's University, Waterloo, Canada

    Michele Mosca

  3. Perimeter Institute for Theoretical Physics, Waterloo, Canada

    Michele Mosca

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  1. Michele Mosca
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Mosca, M. (2012). Quantum Algorithms. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_144

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