Advertisement

Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 375–452 | Cite as

Quantum Algorithm Design: Techniques and Applications

  • Changpeng Shao
  • Yang Li
  • Hongbo LiEmail author
Article
First Online:

Abstract

In recent years, rapid developments of quantum computer are witnessed in both the hardware and the algorithm domains, making it necessary to have an updated review of some major techniques and applications in quantum algorithm design.

In this survey as well as tutorial article, the authors first present an overview of the development of quantum algorithms, then investigate five important techniques: Quantum phase estimation, linear combination of unitaries, quantum linear solver, Grover search, and quantum walk, together with their applications in quantum state preparation, quantum machine learning, and quantum search. In the end, the authors collect some open problems influencing the development of future quantum algorithms.

Keywords

Quantum algorithm quantum computation quantum machine learning quantum search quantum walk 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Benioff P, The computer as a physical system, Journal of Statistical Physics, 1980, 22: 563–591.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Feynman R, Simulating physics with computers, International Journal of Theoretical Physics, 1982, 21: 467–488.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Deutsch D, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. R. Soc. A, 1985, 400: 97–117.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Deutsch D, Quantum computational networks, Proc. R. Soc. A, 1989, 425: 73–90.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Yao A C-C, Quantum circuit complexity, Proc. FOCS 1993, IEEE, Palo Alto, 1993, 352–361.Google Scholar
  6. [6]
    Bernstein E and Vazirani U V, Quantum complexity theory, SIAM J. Comput., 1997, 26(5): 1411–1473.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Deutsch D and Jozsa R, Rapid solution of problems by quantum computation, Proc. R. Soc. A, 1992, 439: 553–558.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Simon D R, On the power of quantum computation, SIAM J. Comput., 1997, 26(5): 1474–1483.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Shor P W, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput., 1997, 26(5): 1484–1509.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Kitaev A Y, Quantum measurements and the abelian stabilizer problem, arXiv: quant-ph /9511026.Google Scholar
  11. [11]
    Ettinger M, Høyer P, and Knill E, The quantum query complexity of the hidden subgroup problem is polynomial, Information Processing Letters, 2004, 91(1): 43–48.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Nielsen M A and Chuang I L, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000.zbMATHGoogle Scholar
  13. [13]
    Hallgren S, Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem, Proc. STOC 2002, ACM Press, New York, 2002, 653–658.Google Scholar
  14. [14]
    Proos J and Zalka C, Shor’s discrete logarithm quantum algorithm for elliptic curves, Quantum Information and Computation, 2003, 3: 317–344.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Bacon D, Childs A M, and van Dam W, From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups, Proc. FOCS 2005, IEEE, Washington, 2005, 469–478.Google Scholar
  16. [16]
    Chi D P, Kim J S, and Lee S, Notes on the hidden subgroup problem on some semi-direct product groups, Phys. Lett. A, 2006, 359(2): 114–116.zbMATHCrossRefGoogle Scholar
  17. [17]
    Inui Y and Le Gall G, Efficient quantum algorithms for the hidden subgroup problem over a class of semi-direct product groups, Quantum Information and Computation, 2007, 7(5 & 6): 559–570.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Kuperberg G, A subexponential-time quantum algorithm for the dihedral hidden subgroup problem, SIAM J. Comput., 2005, 35(1): 170–188.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Moore C, Rockmore D, Russell A, et al., The power of basis selection in Fourier sampling: The hidden subgroup problem in affine groups, Proc. SODA 2004, SIAM, Philadelphia, 2004, 1113–1122.Google Scholar
  20. [20]
    Gavinsky D, Quantum solution to the hidden subgroup problem for poly-near-Hamiltoniangroups, Quantum Information and Computation, 2004, 4: 229–235.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Hallgren S, Russell A, and Ta-Shma A, Normal subgroup reconstruction and quantum computation using group representations, SIAM J. Comput., 2003, 32(4): 916–934.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Ivanyos G, Magniez F, and Santha M, Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem, SPAA 2001, ACM Press, New York, 2001, 263–270.Google Scholar
  23. [23]
    Grigni M, Schulman L, Vazirani M, et al., Quantum mechanical algorithms for the nonabelian hidden subgroup problem, Combinatorica, 2004, 24: 137–154.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    van Dam W, Hallgren S, and Ip L, Quantum algorithms for some hidden shift problems, SIAM J. Comput., 2006, 36(3): 763–778.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Boneh D and Lipton R J, Algorithms for black-box fields and their application to cryptography, Advances in Cryptology — CRYPTO’96, Ed. by Koblitz N, LNCS 1109, 1996, 283–297.Google Scholar
  26. [26]
    Kuwakado H and Morii M, Quantum distinguisher between the 3-round Feistel cipher and the random permutation, Proc. ISIT 2010, IEEE, Austin, TX, 2010, 2682–2685.Google Scholar
  27. [27]
    Kuwakado H and Morii M, Security on the quantum-type Even-Mansour cipher, Proc. ISITA 2012, IEEE, Honolulu, HI, 2012, 312–316.Google Scholar
  28. [28]
    Santoli T and Schaffner C, Using Simon’s algorithm to attack symmetric-key cryptographic primitives, arXiv: 1603.07856 [quant-ph].Google Scholar
  29. [29]
    Kaplan M, Leurent G, and Leverrier A, Breaking symmetric cryptosystems using quantum period finding, arXiv: 1602.05973v3 [quant-ph].Google Scholar
  30. [30]
    Ajtai M and Dwork C, A public-key cryptosystem with worst-case/average-case equivalence, Proc. STOC 1997, ACM Press, New York, 1997, 284–293.Google Scholar
  31. [31]
    Regev O, New lattice-based cryptographic constructions, Journal of the ACM, 2004, 51(6): 899–942.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Regev O, Quantum computation and lattice problems, SIAM J. Comput., 2004, 33(3): 738–760.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Galbraith S and Stolbunov A, Improved algorithm for the isogeny problem for ordinary elliptic curves, Applicable Algebra in Engineering, Communication and Computing, 2013, 24(2): 107–131.MathSciNetzbMATHGoogle Scholar
  34. [34]
    Couveignes J M, Hard Homogeneous Spaces, https://eprint.iacr.org/2006/291.pdf.Google Scholar
  35. [35]
    Rostovtsev A and Stolbunov A, Public-key cryptosystem based on isogenies, https://eprint.iacr.org/2006/145.pdf.Google Scholar
  36. [36]
    Stolbunov A, Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves, Adv. Math. Commun., 2010, 4(2): 215–235.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Childs A M, Jao D, and Soukharev V, Constructing elliptic curve isogenies in quantum subexponential time, J. Mathematical Cryptology, 2014, 8: 1–29.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Grover L K, A fast quantum mechanical algorithm for database search, Proc. STOC 1996, ACM Press, New York, 1996, 212–219.Google Scholar
  39. [39]
    Bennett C H, Bernstein E, Brassard G, et al., Strengths and weaknesses of quantum computing, SIAM J. Comput., 1997, 26(5): 1510–1523.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Zalka C, Grover’s quantum searching algorithm is optimal, Phy. Rev. A, 1999, 60: 2746–2751.CrossRefGoogle Scholar
  41. [41]
    Long G L, Grover algorithm with zero theoretical rate, Phys. Lett. A, 2001, 64: 022307.Google Scholar
  42. [42]
    Ambainis A, Quantum search algorithms, SIGACT News, 2004, 35(2): 22–35.CrossRefGoogle Scholar
  43. [43]
    Campbell E, Khurana A, and Montanaro A, Applying quantum algorithms to constraint satisfaction problems, arXiv: 1810.05582 [quant-ph].Google Scholar
  44. [44]
    Liu W Z, Zhang J F, and Long G L, A parallel quantum algorithm for the satisfiability problem, Common. Theor. Phys., 2008, 49(3): 629–630.zbMATHCrossRefGoogle Scholar
  45. [45]
    Laarhoven T, Mosca M, and van de Pol J, Solving the shortest vector problem in lattices faster using quantum search, PQCrypto 2013, Ed. by Gaborit P, LNCS 7932, Springer, Berlin, Heidelberg, 2013, 83–101, also available: arXiv: 1301.6176v1 [cs.CR].Google Scholar
  46. [46]
    Faugère J C, Horan K, Kahrobaei D, et al., Fast quantum algorithm for solving multivariate quadratic equations, arXiv: 1712.07211 [cs.CR].Google Scholar
  47. [47]
    He X Y, Sun X M, Yang G, et al., Exact quantum query complexity of weight decision problems, arXiv: 1801.05717v1 [quant-ph].Google Scholar
  48. [48]
    Le Gall F and Nishimura H, Quantum algorithms for matrix products over semirings, Chicago Journal of Theoretical Computer Science, 2017, 1: 1–25.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    Dürr C and Høyer P, A quantum algorithm for finding the minimum, arXiv: quant-ph/9607014.Google Scholar
  50. [50]
    Kowada L A B, Lavor C, Portugal R, et al., A new quantum algorithm for solving the minimum searching problem, International Journal of Quantum Information, 2008, 6(3): 427–436.zbMATHCrossRefGoogle Scholar
  51. [51]
    Brassard G, Høyer P, and Tapp A, Quantum Counting, Automata, Languages and Programming, Eds. by Larsen K G, et al., LNCS 1443, Springer, Berlin, Heidelberg, 1998, 820–831.Google Scholar
  52. [52]
    Brassard G, Høyer P, and Mosca M, Quantum amplitude amplification and estimation, Quantum Computation and Quantum Information, 2002, 305: 53–74.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [53]
    Brassard G, Høyer P, and Tapp A, Quantum cryptanalysis of hash and claw-free functions, LATIN’98: Theoretical Informatics, Eds. by Lucchesi C L and Moura A V, LNCS 1380, Springer, Berlin, Heidelberg, 1998, 163–169.CrossRefGoogle Scholar
  54. [54]
    Aaronson S and Shi Y, Quantum lower bounds for the collision and the element distinctness problems, Journal of the ACM, 2004, 51(4): 595–605.MathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    Wang X, Yao A, and Yao F, Cryptanalysis on SHA-1, http://csrc.nist.gov/groups/ST/hash/documents/Wang SHA1-New-Result.pdf.Google Scholar
  56. [56]
    Cochran M, Notes on the Wang, et al. 263 SHA-1 Differential Path, https://eprint.iacr.org/2007/474.pdf.Google Scholar
  57. [57]
    Hoffstein J, Pipher J, and Silverman J H, NTRU: A ring-based public key cryptosystem, Algorithmic Number Theory, Ed. by Buhler J P, LNCS 1423, Springer, Berlin, Heidelberg, 1998, 267–288.CrossRefGoogle Scholar
  58. [58]
    Fluhrer S, Quantum cryptanalysis of NTRU, Cryptology ePrint Archive: Report 2015/676, 2015.Google Scholar
  59. [59]
    Childs A M, Universal computation by quantum walk, Phys. Rev. Lett., 2009, 102: 180501.MathSciNetCrossRefGoogle Scholar
  60. [60]
    Magniez F, Santha M, and Szegedy M, Quantum algorithms for the triangle problem, SIAM J. Comput., 2007, 37(2): 413–424.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [61]
    Jeffery S, Kothari R, and Magniez F, Nested quantum walks with quantum data structures, Proc. SODA 2013, SIAM, Philadelphia, 2013, 1474–1485.Google Scholar
  62. [62]
    Belovs A and Reichardt B W, Span programs and quantum algorithms for st-connectivity and claw detection, European Symp. on Algorithms, Eds. by Epstein L, et al., LNCS 7501, Springer, Berlin, Heidelberg, 2012, 193–204.zbMATHGoogle Scholar
  63. [63]
    Buhrman H, Cleve R, de Wolf R, et al., Bounds for small-error and zero-error quantum algorithms, Proc. FOCS 1999, IEEE, New York, 1999, 358–368.Google Scholar
  64. [64]
    Dürr C, Heiligman M, and Høyer P, Quantum query complexity of some graph problems, SIAM J. Comput., 2006, 35(6): 1310–1328.MathSciNetzbMATHCrossRefGoogle Scholar
  65. [65]
    Ambainis A, Kempe J, and Rivosh A., Coins make quantum walks faster, Proc. SODA 2005, SIAM, Philadelphia, 2005, 1099–1108.Google Scholar
  66. [66]
    Tulsi A, Faster quantum-walk algorithm for the two-dimensional spatial search, Phys. Rev. A, 2008, 78: 012310.zbMATHCrossRefGoogle Scholar
  67. [67]
    Magniez F, Nayak A, Richter P C, et al., On the hitting times of quantum versus random walks, Algorithmica, 2012, 63(1): 91–116.MathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    Aaronson S and Ambainis A, Quantum search of spatial regions, Theory of Computing, 2005, 1: 47–79.MathSciNetzbMATHCrossRefGoogle Scholar
  69. [69]
    Childs A M, Cleve R, Jordan S P, et al., Discrete-query quantum algorithm for NAND trees, Theory of Computing, 2009, 5: 119–123.MathSciNetzbMATHCrossRefGoogle Scholar
  70. [70]
    Ambainis A, Childs A M, Reichardt B W, et al., Any AND-OR formula of size N can be evaluated in time n 1/2+o(1) on a quantum computer, SIAM J. Comput., 2010, 39(6): 2513–2530.MathSciNetzbMATHCrossRefGoogle Scholar
  71. [71]
    Reichardt B W, Faster quantum algorithm for evaluating game trees, Proc. SODA 2011, SIAM, Philadelphia, 2011, 546–559.Google Scholar
  72. [72]
    Reichardt B W, Reflections for quantum query algorithms, Proc. SODA 2011, SIAM, Philadelphia, 2011, 560–569.Google Scholar
  73. [73]
    Reichardt B W and Spalek R, Span-program-based quantum algorithm for evaluating formulas, Theory of Computing, 2012, 8: 291–319.MathSciNetzbMATHCrossRefGoogle Scholar
  74. [74]
    Childs A M and Kothari R, Quantum query complexity of minor-closed graph properties, SIAM Journal on Computing, 2012, 41(6): 1426–1450.MathSciNetzbMATHCrossRefGoogle Scholar
  75. [75]
    Le Gall F, Improved quantum algorithm for triangle finding via combinatorial arguments, Proc. FOCS 2014, IEEE, Philadelphia, 2014, 216–225.Google Scholar
  76. [76]
    Lee T, Magniez F, and Santha M, Improved quantum query algorithms for triangle finding and associativity testing, Algorithmica, 2017, 77: 459–486.MathSciNetzbMATHCrossRefGoogle Scholar
  77. [77]
    Bernstein D J, Jeffery S, Lange T, et al., Quantum algorithms for the subset-sum problem, Post-Quantum Cryptography, Ed. by Gaborit P, LNCS 7932, Springer, Berlin, Heidelberg, 2013, 16–33.CrossRefzbMATHGoogle Scholar
  78. [78]
    Ambainis A, Quantum walk algorithm for element distinctness, SIAM J. Comput., 2007, 37: 210–239.MathSciNetzbMATHCrossRefGoogle Scholar
  79. [79]
    Magniez F and Nayak A, Quantum complexity of testing group commutativity, Algorithmica, 2007, 48(3): 221–232.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [80]
    Buhrman H and Spalek R, Quantum verification of matrix products, Proc. SODA 2006, SIAM, Philadelphia, 2006, 880–889.Google Scholar
  81. [81]
    Le Gall F, Improved output-sensitive quantum algorithms for Boolean matrix multiplication, Proc. SODA 2012, SIAM, Philadelphia, 2012, 1464–1476.Google Scholar
  82. [82]
    Dorn S and Thierauf T, The quantum query complexity of algebraic properties, Fundamentals of Computation Theory, Eds. by Csuhaj-Varjú E, et al, LNCS 4639, Springer, Berlin, Heidelberg, 2007, 250–260.CrossRefzbMATHGoogle Scholar
  83. [83]
    Feynman R, Quantum mechanical computer, Optics News, 1985, 11: 11–20.CrossRefGoogle Scholar
  84. [84]
    Chase B A and Landhal A J, Universal quantum walks and adiabatic algorithms by 1d Hamiltonians, arXiv: 0802.1207 [quant-ph].Google Scholar
  85. [85]
    Farhi E and Gutmann S, Quantum computation and decision trees, Phys. Rev. A, 1998, 58: 915–928.MathSciNetCrossRefGoogle Scholar
  86. [86]
    Meyer D, From quantum cellular automata to quantum lattice gases, J. Stat. Phys., 1996, 85: 551–574.MathSciNetzbMATHCrossRefGoogle Scholar
  87. [87]
    Nayak A and Vishvanath A, Quantum walk on the line, arXiv: quant-ph/0010117.Google Scholar
  88. [88]
    Strauch F W, Connecting the discrete and continuous-time quantum walks, Phys. Rev. A, 2006, 74: 030301.MathSciNetCrossRefGoogle Scholar
  89. [89]
    Childs A M, On the relationship between continuous- and discrete-time quantum walk, Communications in Mathematical Physics, 2010, 294(2): 581–603.MathSciNetzbMATHCrossRefGoogle Scholar
  90. [90]
    Ambainis A, Bach E, Nayak A, et al., One-dimensional quantum walks, Proc. STOC 2001, ACM Press, New York, 2001, 37–49.Google Scholar
  91. [91]
    Kempe J, Discrete quantum walks hit exponentially faster, Probability Theory and Related Fields, 2005, 133(2): 215–235.MathSciNetzbMATHCrossRefGoogle Scholar
  92. [92]
    Shenvi N, Kempe J, and Whaley K B, A quantum random-walk search algorithm, Phys. Rev. A, 2003, 67: 052307.CrossRefGoogle Scholar
  93. [93]
    Childs A M, Cleve R, Deotto E, et al., Exponential algorithmic speedup by quantum walk, Proc. STOC 2003, ACM Press, New York, 2003, 59–68.Google Scholar
  94. [94]
    Childs A M, Farhi E, and Gutmann S, An example of the difference between quantum and classical random walks, Quantum Inf. Process, 2002, 1(1 & 2): 35–43.MathSciNetzbMATHCrossRefGoogle Scholar
  95. [95]
    Szegedy M, Quantum speed-up of markov chain based algorithms, Proc. FOCS 2004, IEEE, Rome, 2004, 32–41.Google Scholar
  96. [96]
    Magniez F, Nayak A, Roland J, et al., Search via quantum walk, SIAM J. Comput., 2011, 40(1): 142–164.MathSciNetzbMATHCrossRefGoogle Scholar
  97. [97]
    Childs A M, Jeffery S, Kothari R, et al., A time-efficient quantum walk for 3-distinctness using nested updates, arXiv: 1302.7316 [quant-ph].Google Scholar
  98. [98]
    Farhi E, Goldstone J, and Gutmann S, A quantum algorithm for the Hamiltonian NAND tree, Theory of Computing, 2008, 4: 169–190.MathSciNetzbMATHCrossRefGoogle Scholar
  99. [99]
    Harrow A W, Hassidim A, and Lloyd S, Quantum algorithm for solving linear systems of equations, Phys. Rev. Lett., 2009, 103(15): 150502.MathSciNetCrossRefGoogle Scholar
  100. [100]
    Lloyd S, Universal quantum simulators, Science, 1996, 273(5278): 1073–1078.MathSciNetzbMATHCrossRefGoogle Scholar
  101. [101]
    Suzuki M, General theory of fractal path integrals with applications to many-body theories and statistical physics, Journal of Mathematical Physics, 1991, 32(2): 400–407.MathSciNetzbMATHCrossRefGoogle Scholar
  102. [102]
    Aharonov D and Ta-Shma A, Adiabatic quantum state generation and statistical zero knowledge, Proc. STOC 2003, ACM Press, New York, 2003, 20–29.Google Scholar
  103. [103]
    Berry D W, Ahokas G, Cleve R, et al., Efficient quantum algorithms for simulating sparse Hamiltonians, Comm. Math. Phys., 2007, 270(2): 359–371.MathSciNetzbMATHCrossRefGoogle Scholar
  104. [104]
    Childs A M and Kothari R, Simulating sparse Hamiltonians with star decompositions, Theory of Quantum Computation, Communication, and Cryptography, Eds. by van Dam W, et al., LNCS 6519, Springer-Verlag Berlin Heidelberg, 2011, 94–103.CrossRefGoogle Scholar
  105. [105]
    Berry D W and Childs A M, Black-box Hamiltonian simulation and unitary implementation, Quantum Information and Computation, 2012, 12: 29–62.MathSciNetzbMATHGoogle Scholar
  106. [106]
    Berry D W, Childs A M, Cleve R, et al., Simulating Hamiltonian dynamics with a truncated Taylor series, Phys. Rev. Lett., 2015, 114: 090502.CrossRefGoogle Scholar
  107. [107]
    Berry D W, Childs A M, and Kothari R, Hamiltonian simulation with nearly optimal dependence on all parameters, Proc. FOCS 2015, IEEE, Berkeley, 2015, 792–809.Google Scholar
  108. [108]
    Low G H and Chuang I L, Hamiltonian simulation by qubitization, arXiv: 1610.06546v2 [quantph].Google Scholar
  109. [109]
    Chakraborty S, Gilyén A, and Jeffery S, The power of block-encoded matrix powers: Improved regression techniques via faster Hamiltonian simulation, arXiv: 1804.01973v1 [quant-ph].Google Scholar
  110. [110]
    Gilyén A, Su Y, Low G H, et al., Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics, arXiv: 1806.01838 [quant-ph].Google Scholar
  111. [111]
    Childs A M and Kothari R, Limitations on the simulation of non-sparse Hamiltonians, Quantum Information and Computation, 2010, 10: 669–684.MathSciNetzbMATHGoogle Scholar
  112. [112]
    Rebentrost P, Steffens A, and Lloyd S, Quantum singular value decomposition of non-sparse low-rank matrices, Phys. Rev. A, 2018, 97: 012327.MathSciNetCrossRefGoogle Scholar
  113. [113]
    Kerenidis I and Prakash A, Quantum recommendation system, Proc. ITCSC 2017, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl Publishing, 2017, 49: 1–21.Google Scholar
  114. [114]
    Wang C H and Wossnig L, A quantum algorithm for simulating non-sparse Hamiltonians, arXiv: 1803.08273v1 [quant-ph].Google Scholar
  115. [115]
    Childs A M, Kothari R, and Somma R D, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision, SIAM J. Comput., 2017, 46: 1920–1950.MathSciNetzbMATHCrossRefGoogle Scholar
  116. [116]
    Ambainis A, Variable time amplitude amplification and quantum algorithms for linear algebra problems, Proc. STACS 2012, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl Publishing, 2012, 636–647.Google Scholar
  117. [117]
    Saad Y, Iterative Methods for Sparse Linear Systems, 2nd edition, Society for Industrial and Applied Mathematics, 2003.zbMATHCrossRefGoogle Scholar
  118. [118]
    Clader B D, Jacobs B C, and Sprouse C R, Preconditioned quantum linear system algorithm, Phys. Rev. Lett., 2013, 110: 250504.CrossRefGoogle Scholar
  119. [119]
    Wossnig L, Zhao Z K, and Prakash A, Quantum linear system algorithm for dense matrices, Phys. Rev. Lett., 2018, 120: 050502.MathSciNetCrossRefGoogle Scholar
  120. [120]
    Chen Y A and Gao X S, Quantum algorithms for Boolean equation solving and quantum algebraic attack on cryptosystems, arXiv: 1712.06239v3 [quant-ph].Google Scholar
  121. [121]
    Chen Y A, Gao X S, and Yuan C M, Quantum algorithms for optimization and polynomial systems solving over finite fields, arXiv: 1802.03856v2 [quant-ph].Google Scholar
  122. [122]
    Schuld M and Petruccione F, Supervised Learning with Quantum Computers, Springer, 2018.zbMATHCrossRefGoogle Scholar
  123. [123]
    Wittek P, Quantum Machine Learning: What Quantum Computing Mean to Data Mining, Academic Press, 2014.zbMATHGoogle Scholar
  124. [124]
    Wiebe N, Braun D, and Lloyd S, Quantum algorithm for data fitting, Phys. Rev. Lett., 2012, 109(5): 050505.CrossRefGoogle Scholar
  125. [125]
    Schuld M, Sinayskiy I, and Petruccione F, Prediction by linear regression on a quantum computer, Phys. Rev. A, 2016, 94: 022342.CrossRefGoogle Scholar
  126. [126]
    Wang G M, Quantum algorithm for linear regression, Phy. Rev. A, 2017, 96: 012335.MathSciNetCrossRefGoogle Scholar
  127. [127]
    Lloyd S, Mohseni M, and Rebentrost P, Quantum algorithms for supervised and unsupervised machine learning, arXiv: 1307.0411v2 [quant-ph].Google Scholar
  128. [128]
    Lloyd S, Rebentrost P, and Mohseni M, Quantum principal component analysis, Nature Physics, 2014, 10: 631–633.CrossRefGoogle Scholar
  129. [129]
    Rebentrost P, Mohseni M, and Lloyd S, Quantum support vector machine for big data classification, Phys. Rev. Lett., 2014, 113(13): 130503.CrossRefGoogle Scholar
  130. [130]
    Rebentrost P, Bromley T R, Weedbrook C, et al., Quantum recurrent neural network, Phys. Rev. A, 2018, 98: 042308.CrossRefGoogle Scholar
  131. [131]
    Altaisky M V, Quantum neural network, arXiv: quant-ph/0107012v2.Google Scholar
  132. [132]
    Schuld M, Sinayskiy I, and Petruccione F, The quest for a quantum neural network, Quantum Inf. Process, 2014, 13: 2567–2586.MathSciNetzbMATHCrossRefGoogle Scholar
  133. [133]
    Schuld M, Sinayskiy I, and Petruccione F, Simulating a perceptron on a quantum computer, Phys. Lett. A, 2015, 379: 660–663.zbMATHCrossRefGoogle Scholar
  134. [134]
    Wan K H, Dahlsten O, Kristjánsson H, et al., Quantum generalisation of feedforward neural networks, NPJ Quantum Information, 2017, 3(1): 36–43.CrossRefGoogle Scholar
  135. [135]
    Wiebe N, Kapoor A, and Svore K, Quantum perceptron models, Advances in Neural Information Processing Systems, 2016, 29: 3999–4007.Google Scholar
  136. [136]
    Yamamoto A Y, Sundqvist K M, Li P, et al., Simulation of a multidimensional input quantum perceptron, Quantum Inf. Process, 2018, 17(6): 128–139.MathSciNetzbMATHCrossRefGoogle Scholar
  137. [137]
    Schützhold R, Pattern recognition on a quantum computer, Phys. Rev. A, 2003, 67: 062311.CrossRefGoogle Scholar
  138. [138]
    Trugenberger C A, Quantum pattern recognition, Quantum Inf. Process, 2002, 1(6): 471–493.MathSciNetCrossRefGoogle Scholar
  139. [139]
    Ventura D and Martinez T, Quantum associative memory, Information Sciences, 2000, 124(1): 273–296.MathSciNetCrossRefGoogle Scholar
  140. [140]
    Low G H, Yoder T J, and Chuang I L, Quantum inference on Bayesian networks, Phys. Rev. A, 2014, 89: 062315.CrossRefGoogle Scholar
  141. [141]
    Sentís G, Calsamiglia J, Muñoz-Tapia R, et al., Quantum learning without quantum memory, Scientific Reports, 2012, 2(708): 1–8.Google Scholar
  142. [142]
    Barry J, Barry D T, and Aaronson S, Quantum POMDPs, Phys. Rev. A, 2014, 90: 032311.CrossRefGoogle Scholar
  143. [143]
    Clark L A, Huang W, Barlow T M, et al., Hidden quantum Markov models and open quantum systems with instantaneous feedback, Interdisciplinary Symposium on Complex Systems, Eds. by Sanayei A, et al., ECC 14, Springer, Cham, 2015, 143–151.Google Scholar
  144. [144]
    Adachi S H and Henderson M P, Application of quantum annealing to training of deep neural networks, arXiv: 1510.06356 [quant-ph].Google Scholar
  145. [145]
    Wiebe N, Kapoor A, and Svore K, Quantum deep learning, arXiv: 1412.3489 [quant-ph].Google Scholar
  146. [146]
    Amin M H, Andriyash E, Rolfe J, et al., Quantum Boltzmann machine, Phys. Rev. X, 2018, 8: 021050.Google Scholar
  147. [147]
    Crawford D, Levit A, Ghadermarzy N, et al., Reinforcement learning using quantum Boltzmann machines, arXiv: 1612.05695 [quant-ph].Google Scholar
  148. [148]
    Kieferova M and Wiebe N, Tomography and generative data modeling via quantum Boltzmann training, Phys. Rev. A, 2017, 96: 062327.CrossRefGoogle Scholar
  149. [149]
    Messiah A, Quantum Mechanics, Vol. II. Wiley, New Jersey, 1976.zbMATHGoogle Scholar
  150. [150]
    Farhi E, Goldstone J, Gutmann S, et al., Quantum computation by adiabatic evolution, arXiv: quant-ph/0001106v1.Google Scholar
  151. [151]
    Childs A M, Farhi E, and Preskill J, Robustness of adiabatic quantum computation, Phys. Rev. A, 2001, 65: 012322.CrossRefGoogle Scholar
  152. [152]
    Roland J and Cerf N, Quantum search by local adiabatic evolution, Phys. Rev. A, 2002, 65: 042308.CrossRefGoogle Scholar
  153. [153]
    Aharonov D, van Dam W, Kempe J, et al., Adiabatic quantum computation is equivalent to standard quantum computation, SIAM J. Comput., 2007, 37(1): 166–194.MathSciNetzbMATHCrossRefGoogle Scholar
  154. [154]
    Childs A M, Farhi E, Goldstone J, et al., Finding cliques by quantum adiabatic evolution, Quantum Information and Computation, 2002, 2(181): 181–191.MathSciNetzbMATHGoogle Scholar
  155. [155]
    Farhi E, Goldstone F, Gutmann S, et al., A quantum adiabatic evolution algorithm applied to instances of an NP-complete problem, Science, 2001, 292(5516): 472–475.MathSciNetzbMATHCrossRefGoogle Scholar
  156. [156]
    Hogg T, Adiabatic quantum computing for random satisfiability problems, Phys. Rev. A, 2003, 67: 022314.CrossRefGoogle Scholar
  157. [157]
    Reichardt B W, The quantum adiabatic optimization algorithm and local minima, Proc. STOC 2004, ACM Press, New York, 2004, 502–510.Google Scholar
  158. [158]
    van Dam W and Vazirani U V, Limits on quantum adiabatic optimization, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.394.2905&rep=rep1&type=pdf.Google Scholar
  159. [159]
    Neven H, Denchev V S, Rose E, et al., Training a binary classifier with the quantum adiabatic algorithm, arXiv: 0811.0416 [quant-ph].Google Scholar
  160. [160]
    Pudenz K L and Lidar D A, Quantum adiabatic machine learning, Quantum Inf. Process, 2013, 12(5): 2027–2070.MathSciNetzbMATHCrossRefGoogle Scholar
  161. [161]
    Gaitan F and Clark L, Ramsey numbers and adiabatic quantum computing, Phys. Rev. Lett., 2012, 108: 010501.CrossRefGoogle Scholar
  162. [162]
    Gaitan F and Clark L, Graph isomorphism and adiabatic quantum computing, Phys. Rev. A, 2014, 89(2): 022342.CrossRefGoogle Scholar
  163. [163]
    Kitaev A Y, Fault-tolerant quantum computation by anyons, Annals of Physics, 2003, 303(1): 2–30.MathSciNetzbMATHCrossRefGoogle Scholar
  164. [164]
    Freedman M H, Kitaev A, Larsen M J, et al., Topological quantum computation, Bull. Amer. Math. Soc., 2003, 40: 31–38.MathSciNetzbMATHCrossRefGoogle Scholar
  165. [165]
    Aharonov D, Jones V, and Landau Z, A polynomial quantum algorithm for approximating the jones polynomial, Proc. STOC 2006, ACM Press, New York, 2006, 427–436.Google Scholar
  166. [166]
    Aharonov D, Arad I, Eban E, et al., Polynomial quantum algorithms for additive approximations of the potts model and other points of the tutte plane, arXiv: quant-ph/0702008.Google Scholar
  167. [167]
    Wocjan P and Yard J, The Jones polynomial: Quantum algorithms and applications in quantum complexity theory, Quantum Information and Computation, 2008, 8(1): 147–180.MathSciNetzbMATHGoogle Scholar
  168. [168]
    Arad I and Landau Z, Quantum computation and the evaluation of tensor networks, SIAM J. Comput., 2010, 39(7): 3089–3121.MathSciNetzbMATHCrossRefGoogle Scholar
  169. [169]
    Aaronson S, The limits of quantum computers, Scientific American, 2008, 298(3): 62–69.CrossRefGoogle Scholar
  170. [170]
    Aaronson S, BQP and the polynomial hierarchy, Proc. STOC 2010, ACM Press, New York, 2010, 141–150.Google Scholar
  171. [171]
    Mahadev U, Classical Verification of Quantum Computations, arXiv: 1804.01082v2 [quant-ph].Google Scholar
  172. [172]
    Cirac J and Zoller P, Quantum computation with cold trapped ions, Phys. Rev. Lett., 1995, 74: 4091–4094.CrossRefGoogle Scholar
  173. [173]
    Gershenfeld N and Chuang I L, Bulk spin resonance quantum computing, Science, 1997, 275: 350–356.MathSciNetzbMATHCrossRefGoogle Scholar
  174. [174]
    Loss D and Di Vincenzo D, Quantum computation with quantum dots, Phys. Rev. A, 1998, 57: 120–126.CrossRefGoogle Scholar
  175. [175]
    Vandersypen L M K, Steffen M, Breyta G, et al., Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance, Nature, 2001, 414(6866): 883–887.CrossRefGoogle Scholar
  176. [176]
    Zhao Z, Chen Y A, Zhang A N, et al., Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature, 2004, 430(6995): 54–58.CrossRefGoogle Scholar
  177. [177]
    12-qubits reached in quantum information quest, https://www.sciencedaily.com/releases/2006/05/060508164700.htm.Google Scholar
  178. [178]
    World’s First 28 qubit Quantum Computer Demonstrated Online at Supercomputing 2007 Conference, https://www.nanowerk.com/news/newsid=3274.php.Google Scholar
  179. [179]
    Dwave System’s 128 qubit chip has been made, https://www.nextbigfuture.com/2008/12/dwavesystems-128-qubit-chip-has-been.html.Google Scholar
  180. [180]
    Monz T, Schindler P, Barreiro J T, et al., 14-Qubit Entanglement: Creation and Coherence, Phys. Rev. Lett., 2011, 106(13): 130506.CrossRefGoogle Scholar
  181. [181]
    D-wave systems breaks the 1000 qubit quantum computing barrier, https://www.dwavesys.com/press-releases/d-wave-systems-breaks-1000-qubit-quantum-computing-barrier.Google Scholar
  182. [182]
    O’Malley P J J, Babbush R, Kivlichan I D, et al., Scalable quantum simulation of molecular energies, Phys. Rev. X, 2016, 6: 031007.Google Scholar
  183. [183]
    IBM just made a 17 qubit quantum processor, its most powerful one yet, https://motherboard.vice.com/enus/article/wnwk5w/ibm-17-qubit-quantum-processor-computer-google.Google Scholar
  184. [184]
    Quantum inside: Intel manufactures an exotic new chip, https://www.technologyreview.com/s/609094/quantum-inside-intel-manufactures-an-exotic-new-chip/.Google Scholar
  185. [185]
    IBM Q Experience, https://quantumexperience.ng.bluemix.net/qx/experience.Google Scholar
  186. [186]
    Coles P J, Eidenbenz S, Pakin S, et al., Quantum Algorithm Implementations for Beginners, arXiv: 1804.03719v1 [cs.ET].Google Scholar
  187. [187]
    China builds five qubit quantum computer sampling and will scale to 20 qubits by end of this year and could any beat regular computer next year, https://www.nextbigfuture.com/2017/05/chinabuilds-ten-qubit-quantum-computer-and-will-scale-to-20-qubits-by-end-of-this-year-and-couldany-beat-regular-computer-next-year.html.Google Scholar
  188. [188]
    IBM raises the bar with a 50-qubit quantum computer, https://www.technologyreview.com/s/609451/ibm-raises-the-bar-with-a-50-qubit-quantum-computer/.Google Scholar
  189. [189]
    D-wave announces d-wave 2000q quantum computer and first system order, https://www.dwavesys.com/press-releases/d-wave-announces-d-wave-2000q-quantum-computer-and-first-system-order.Google Scholar
  190. [190]
    CES 2018: Intel’s 49-qubit chip shoots for quantum supremacy, https://spectrum.ieee.org/techtalk/computing/hardware/intels-49qubit-chip-aims-for-quantum-supremacy.Google Scholar
  191. [191]
    Google moves toward quantum supremacy with 72-qubit computer, https://www.sciencenews.org/article/google-moves-toward-quantum-supremacy-72-qubit-computer.Google Scholar
  192. [192]
    Lu C Y, Browne D E, and Yang T, Demonstration of a compiled version of Shor’s quantum factoring algorithm Using Photonic Qubits, Phys. Rev. Lett., 2007, 99: 250504.CrossRefGoogle Scholar
  193. [193]
    Lanyon B P, Weinhold T J, Langford N K, et al., Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement, Phys. Rev. Lett., 2007, 99(25): 250505.CrossRefGoogle Scholar
  194. [194]
    Martin-Lopez E, Laing A, Lawson T, et al., Experimental realisation of Shor’s quantum factoring algorithm using qubit recycling, Nat. Photon., 2012, 6: 773–776.CrossRefGoogle Scholar
  195. [195]
    Chuang I L, Gershenfeld N, and Kubinec M, Experimental implementation of fast quantum searching, Phys. Rev. Lett., 1998, 80: 3408–3411.CrossRefGoogle Scholar
  196. [196]
    Vandersypen L M K, Steffen M, Sherwood M H, et al., Implementation of a three-quantum-bit search algorithm, Appl. Phys. Lett., 2000, 76: 646–648.CrossRefGoogle Scholar
  197. [197]
    Barz S, Kassal I, and Ringbauer M, Solving systems of linear equations on a quantum computer, Sci. Rep., 2014, 4: 115.Google Scholar
  198. [198]
    Cai X D, Weedbrook C, Su Z E, et al., Experimental quantum computing to solve systems of linear equations, Phys. Rev. Lett., 2013, 110(23): 230501.CrossRefGoogle Scholar
  199. [199]
    Pan J W, Cao Y D, Yao X W, et al., Experimental realization of quantum algorithm for solving linear systems of equations, Phys. Rev. A, 2014, 89: 022313.CrossRefGoogle Scholar
  200. [200]
    Ambainis A, New Developments in Quantum Algorithms, Mathematical Foundations of Computer Science, Eds. by Hlineny P, et al., LNCS 6281, Springer, Berlin, Heidelberg, 2010, 1–11.Google Scholar
  201. [201]
    Cleve R, Ekert A, Macchiavello C, et al., Quantum algorithms revisited, Proc. R. Soc. A, 1997, 454(1969): 339–354.MathSciNetzbMATHCrossRefGoogle Scholar
  202. [202]
    Montanaro A, Quantum algorithms: An overview, npj Quantum Information, 2016, 2: 15023.CrossRefGoogle Scholar
  203. [203]
    Shor P W, Progress in quantum algorithms, Quantum Inf. Process, 2004, 3: 5–13.MathSciNetzbMATHCrossRefGoogle Scholar
  204. [204]
    Sun X M, A survey on quantum computing, Sci. China Inf. Sci., 2016, 8: 982–1002 (in Chinese).Google Scholar
  205. [205]
    Dervovic D, Herbster M, Mountney P, et al., Quantum linear systems algorithms: A primer, arXiv: 1802.08227v1 [quant-ph].Google Scholar
  206. [206]
    Kaye P, Laflamme R, and Mosca M, An Introduction to Quantum Computing, Oxford University Press, New York, 2007.zbMATHGoogle Scholar
  207. [207]
    Rieffel E and Polak W, Quantum Computing — A Gentle Introduction, The MIT Press, Cambridge, Massachusetts London, England, 2011.zbMATHGoogle Scholar
  208. [208]
    Ambainis A, Quantum walks and their algorithmic applications, International Journal of Quantum Information, 2003, 1: 507–518.zbMATHCrossRefGoogle Scholar
  209. [209]
    Elías S, Venegas-Andraca S E, Quantum walks: A comprehensive review, Quantum Inf. Process, 2012, 11(5): 1015–1106.MathSciNetzbMATHCrossRefGoogle Scholar
  210. [210]
    Nayak A, Richter P C, and Szegedy M, Quantum analogs of markov chains, Encyclopedia of Algorithms, Ed. by Kao M Y, Springer, Berlin, Heidelberg, 2014, 1–10.Google Scholar
  211. [211]
    Santha M, Quantum walk based search algorithms, Proc. TAMC 2008, Xi’an, 2008, 31–46.Google Scholar
  212. [212]
    Kempe J, Quantum random walks - an introductory overview, Contemporary Physics, 2003, 44(4): 307–327.CrossRefGoogle Scholar
  213. [213]
    Childs A M, Maslov D, Nam Y, et al., Toward the first quantum simulation with quantum speedup, Proceedings of the National Academy of Sciences, 2018, 115: 9456–9461.MathSciNetCrossRefGoogle Scholar
  214. [214]
    Adcock J C, Allen E, Day M, et al., Advances in quantum machine learning, arXiv: 1512.02900v1 [quant-ph].Google Scholar
  215. [215]
    Arunachalam S and de Wolf R, A Survey of Quantum Learning Theory, arXiv:1701.06806v3[quant-ph].Google Scholar
  216. [216]
    Biamonte J, Wittek P, Pancotti N, et al., Quantum machine learning, Nature, 2017, 549: 195–202.CrossRefGoogle Scholar
  217. [217]
    Ciliberto C, Herbster M, Ialongo A D, et al., Quantum machine learning: A classical perspective, Proc. R. Soc. A, 2018, 474(2209): 20170551.MathSciNetzbMATHCrossRefGoogle Scholar
  218. [218]
    Dunjko V and Briegel H J, Machine learning & artificial intelligence in the quantum domain, arXiv: 1709.02779v1 [quant-ph].Google Scholar
  219. [219]
    Schuld M, Sinayskiy I, and Petruccione F, An introduction to quantum machine learning, Contemporary Physics, 2015, 56(2): 172–185.zbMATHCrossRefGoogle Scholar
  220. [220]
    Albash T and Lidar D A, Adiabatic quantum computing, Rev. Mod. Phys., 2018, 90: 015002.CrossRefGoogle Scholar
  221. [221]
    Quantum algorithm zoo, https://math.nist.gov/quantum/zoo/.Google Scholar
  222. [222]
    Childs A M, Lecture notes on quantum algorithms, http://www.cs.umd.edu/amchilds/qa/.Google Scholar
  223. [223]
    Christopher M D and Nielsen A, The Solovay-Kitaev algorithm, Quantum Information and Computation, 2006, 6(1): 81–95.MathSciNetzbMATHGoogle Scholar
  224. [224]
    Abrams D S and Lloyd S, Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors, Phys. Rev. Lett., 1999, 83(24): 5162–5165.CrossRefGoogle Scholar
  225. [225]
    Buhrman H, Cleve R, Watrous J, et al., Quantum fingerprinting, Phys. Rev. Lett., 2001, 87(16): 167902.CrossRefGoogle Scholar
  226. [226]
    Long G L, General quantum interference principle and duality computer, Common. Theor. Phys., 2006, 45: 825–844.MathSciNetCrossRefGoogle Scholar
  227. [227]
    Childs A M and Wiebe N, Hamiltonian simulation using linear combinations of unitary operations, Quantum Information and Computation, 2012, 12: 901–924.MathSciNetzbMATHGoogle Scholar
  228. [228]
    Kerenidis I and Prakash A, Quantum gradient descent for linear systems and least squares, arXiv: 1704.04992v3 [quant-ph].Google Scholar
  229. [229]
    Rebentrost P, Schuld M, Wossnig L, et al., Quantum gradient descent and Newton’s method for constrained polynomial optimization, arXiv: 1612.01789v2 [quant-ph].Google Scholar
  230. [230]
    Grover L K and Rudolph T, Creating superpositions that correspond to efficiently integrable probability distributions, arXiv: quant-ph/0208112.Google Scholar
  231. [231]
    Soklakov A N and Schack R, Efficient state preparation for a register of quantum bits, Phys. Rev. A, 2006, 73: 012307.CrossRefGoogle Scholar
  232. [232]
    Alpaydin E, Introduction to Machine Learning, 3rd Edition, The MIT Press, Massachusetts, 2015.zbMATHGoogle Scholar
  233. [233]
    Mackay D, Information Theory, Inference and Learning Algorithms, Cambridge University Press, Cambridge, 2003.zbMATHGoogle Scholar
  234. [234]
    Sra S, Nowozin S, and Wright S J, Optimization for Machine Learning, The MIT Press, Massachusetts, 2011.CrossRefGoogle Scholar
  235. [235]
    Golub G H and Van Loan C F, Matrix Computations, 4th Edition, The John Hopkins University Press, Baltimore, MD, 2013.zbMATHGoogle Scholar
  236. [236]
    Hestenes M R and Stiefel E, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 1952, 49: 409–436.MathSciNetzbMATHCrossRefGoogle Scholar
  237. [237]
    Aaronson S, Quantum Machine Learning Algorithms: Read the Fine Print, Nature Physics, 2015, 11(4): 291–293.CrossRefGoogle Scholar
  238. [238]
    Childs A M, Quantum algorithms: Equation solving by simulation, Nature Physics, 2009, 5(5): 861.CrossRefGoogle Scholar
  239. [239]
    Harrow A W, Quantum algorithms for systems of linear equations, Encyclopedia of Algorithms, Ed. by Kao M Y, Springer, Berlin, Heidelberg, 2016, 1680–1683.CrossRefGoogle Scholar
  240. [240]
    Giovannetti V, Lloyd S, and Maccone L, Architectures for a quantum random access memory, Phys. Rev. A, 2008, 78: 052310.zbMATHCrossRefGoogle Scholar
  241. [241]
    Giovannetti V, Lloyd S, and Maccone L, Quantum random access memory, Phys. Rev. Lett., 2008, 100: 160501.MathSciNetzbMATHCrossRefGoogle Scholar
  242. [242]
    Kerenidis I and Prakash A, A quantum interior point method for LPs and SDPs, arXiv: 1808.09266v1 [quant-ph].Google Scholar
  243. [243]
    Bardet M, Faugère J C, Salvy B, et al., On the complexity of solving quadratic boolean systems, Journal of Complexity, 2013, 29(1): 53–75.MathSciNetzbMATHCrossRefGoogle Scholar
  244. [244]
    Hastie T, Tibshirani R, and Friedman J, The Elements of Statistical Learning, Springer, New York, 2001.zbMATHCrossRefGoogle Scholar
  245. [245]
    Haykin S, Neural Networks and Learning Machines, 3rd Edition, Pearson, 2009.Google Scholar
  246. [246]
    Cleveland W S, Robust locally weighted regression and smoothing scatterplots, Journal of the American Statistical Association, 1979, 74(368): 829–836.MathSciNetzbMATHCrossRefGoogle Scholar
  247. [247]
    Cleveland W S and Devlin S J, Locally-weighted regression: An approach to regression analysis by local fitting, Journal of the American Statistical Association, 1988, 83(403): 596–610.zbMATHCrossRefGoogle Scholar
  248. [248]
    Ng A, Supervised learning, discriminative algorithms, http://cs229.stanford.edu/notes/cs229-notes1.pdf.Google Scholar
  249. [249]
    Suykens J A K and Vandewalle J, Least squares support vector machine classifiers, Neural Processing Letters, 1999, 9(3): 293–300.CrossRefGoogle Scholar
  250. [250]
    Levin D A, Peres Y, and Wilmer E L, Markov Chains and Mixing Times, American Mathematical Society, Providence, Rhode Island, 2009.Google Scholar
  251. [251]
    Lovász L, Random walks on graphs — A survey, Combinatorics, 1993, 1–46.zbMATHGoogle Scholar
  252. [252]
    Gerschgorin S, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, 1931, 7: 749–754.zbMATHGoogle Scholar
  253. [253]
    Høyer P and Komeili M, Efficient quantum walk on the grid with multiple marked elements, STACS 2017, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl Publishing, 2017, 42:1–42:14.Google Scholar
  254. [254]
    Watrous J, Quantum simulations of classical random walks and undirected graph connectivity, Journal of Computer and System Sciences, 2001, 62(2): 376–391.MathSciNetzbMATHCrossRefGoogle Scholar
  255. [255]
    Aharonov D, Ambainis A, Kempe J, et al., Quantum walks on graphs, Proc. STOC 2001, ACM Press, New York, 2001, 50–59.Google Scholar
  256. [256]
    Kendon V, Quantum walks on general graphs, Int. J. Quantum Inf., 2006, 4(5): 791–805.zbMATHCrossRefGoogle Scholar
  257. [257]
    Potocek V, Gabris A, Kiss T, et al., Optimized quantum random-walk search algorithms on the hypercube, Phys. Rev. A, 2009, 79: 012325.CrossRefGoogle Scholar
  258. [258]
    Tonchev H, Alternative coins for quantum random walk search optimized for a hypercube, Journal of Quantum Information Science, 2015, 5: 6–15.CrossRefGoogle Scholar
  259. [259]
    Ambainis A, Quantum search with variable times, Theory of Computing Systems, 2010, 47(3): 786–807.MathSciNetzbMATHCrossRefGoogle Scholar
  260. [260]
    Boyer M, Brassard G, Høyer P, et al., Tight bounds on quantum searching, Fortsch. Phys. Prog. Phys., 1998, 46(4–5): 493–505.CrossRefGoogle Scholar
  261. [261]
    Grover L K, Fixed-point quantum search, Phys. Rev. Lett., 2005, 95: 150501.zbMATHCrossRefGoogle Scholar
  262. [262]
    Yoder T J, Low G H, Chuang I L, Optimal fixed-point quantum amplitude amplification using Chebyshev polynomials, Phys. Rev. Lett., 2014, 113: 210501.CrossRefGoogle Scholar
  263. [263]
    Brassard G, Høyer P, and Tapp A, Quantum algorithm for the collision problem, ACM SIGACT News, 1997, 28: 14–19.CrossRefGoogle Scholar
  264. [264]
    Falk M, Quantum search on the spatial grid, arXiv: 1303.4127 [quant-ph].Google Scholar
  265. [265]
    Buhrman H, Dürr C, Heiligman M, et al., Quantum algorithms for element distinctness, SIAM J. Comput., 2005, 34(6): 1324–1330.MathSciNetzbMATHCrossRefGoogle Scholar
  266. [266]
    Belovs A, Learning-graph-based quantum algorithm for k-distinctness, Proc. FOCS 2012, IEEE, New Brunswick, 2012, 207–216.Google Scholar
  267. [267]
    Jeffery S, Frameworks for quantum algorithms, PhD thesis, University of Waterloo, 2014.Google Scholar
  268. [268]
    Krovi H, Magniez F, Ozols M, et al., Quantum walks can find a marked element on any graph, Algorithmica, 2016, 74(2): 851–907.MathSciNetzbMATHCrossRefGoogle Scholar
  269. [269]
    Dohotaru C and Høyer P, Controlled quantum amplification, Proc. ICALP 2017, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl Publishing, 2017, 18: 1–13.Google Scholar
  270. [270]
    Ambainis A and Kokainis M, Analysis of the extended hitting time and its properties, Proc. QIP 2015, Philadelphia, 2012.Google Scholar
  271. [271]
    Portugal R, Santos R A M, Fernandes T D, et al., The staggered quantum walk model, Quantum Information Processing, 2016, 15: 85–101.MathSciNetzbMATHCrossRefGoogle Scholar
  272. [272]
    Aggarwal D, Dadush D, and Regev O, et al., Solving the shortest vector problem in 2n time via discrete Gaussian sampling, Proc. STOC 2014, ACM Press, New York, 2014, 733–742.Google Scholar
  273. [273]
    Chen Y, Chung K, and Lai C, Space-efficient classical and quantum algorithms for the shortest vector problem, Quantum Information and Computation, 2018, 18(3 & 4): 285–307.MathSciNetGoogle Scholar
  274. [274]
    Becker A, Ducas L, and Gama N, et al., New directions in nearest neighbor searching with applications to lattice sieving, Proc. SODA 2016, SIAM, Philadelphia, 2016, 10–24.Google Scholar
  275. [275]
    Laarhoven T, Search problems in cryptography, PhD dissertation, Eindhoven University of Technology, Eindhoven, 2015.zbMATHGoogle Scholar
  276. [276]
    Ettinger M and Høyer P, On quantum algorithms for noncommutative hidden subgroups, Advances in Applied Mathematics, 2000, 25(3): 239–251.MathSciNetzbMATHCrossRefGoogle Scholar
  277. [277]
    Bacon D, Childs A M, and van Dam W, Optimal measurements for the dihedral hidden subgroup problem, Chicago Journal of Theoretical Computer Science, 2006, article 2.Google Scholar
  278. [278]
    Regev O, A subexponential time algorithm for the dihedral hidden subgroup problem with polynomial space, arXiv: quant-ph/0406151.Google Scholar
  279. [279]
    Babai L, On the complexity of canonical labeling of strongly regular graphs, SIAM J. Comput., 1980, 9: 212–216.MathSciNetzbMATHCrossRefGoogle Scholar
  280. [280]
    Spielman D A, Faster isomorphism testing of strongly regular graphs, Proc. STOC 1996, ACM Press, New York, 576–584.Google Scholar
  281. [281]
    Wang F, The Hidden Subgroup Problem, Master’s Project, Aarhus Universitet, 2010.Google Scholar
  282. [282]
    Ettinger M and Høyer P, A quantum observable for the graph isomorphism problem, arXiv: quant-ph/9901029.Google Scholar
  283. [283]
    Beals R, Quantum computation of Fourier transforms over symmetric groups, Proc. STOC 1997, ACM Press, New York, 1997, 48–53.Google Scholar
  284. [284]
    Boneh D and Lipton R J, Quantum cryptanalysis of hidden linear functions, Advances in Cryptology CRYPTO’95, Ed. by Coppersmith D, LNCS 963, Springer-Verlag Berlin, Heidelberg, 1995, 424–437.Google Scholar
  285. [285]
    Hallgren S, Moore C, Roetteler M, et al., Limitations of quantum coset states for graph isomorphism, J. ACM, 2010, 57(6): 1–33.MathSciNetCrossRefGoogle Scholar
  286. [286]
    Moore C, Russell A, and Schulman L J, The symmetric group defies strong fourier sampling, SIAM J. Comput., 2008, 37(6): 1842–1864.MathSciNetzbMATHCrossRefGoogle Scholar
  287. [287]
    Kawano Y and Sekigawa H, Quantum fourier transform over symmetric groups, Proc. ISSAC 2013, ACM Press, New York, 2013, 227–234.Google Scholar
  288. [288]
    Kawano Y and Sekigawa H, Quantum Fourier transform over symmetric groups - improved result, J. Symb. Comp., 2016, 75: 219–243.MathSciNetzbMATHCrossRefGoogle Scholar
  289. [289]
    Williams V V and Williams R, Subcubic equivalences between path, matrix and triangle problems, Proc. FOCS 2010, IEEE, Las Vegas, 2010, 645–654.Google Scholar
  290. [290]
    Williams V V and Williams R, Finding, minimizing, and counting weighted subgraphs, SIAM J. Comput., 2013, 42(3): 831–854.MathSciNetzbMATHCrossRefGoogle Scholar
  291. [291]
    Williams R, A new algorithm for optimal 2-constraint satisfaction and its implications, Theor. Comput. Sci., 348(2–3): 357–365.Google Scholar
  292. [292]
    Belovs A, Span programs for functions with constant-sized 1-certificates, Proc. STOC 2012, ACM Press, New York, 2012, 77–84.Google Scholar
  293. [293]
    Grötschel M, Lovász L, and Schrijver A, Geometric Algorithms and Combinatorial Optimization, Springer, New York, 1988.zbMATHCrossRefGoogle Scholar
  294. [294]
    Lee Y T, Sidford A, and Wong S C, A faster cutting plane method and its implications for combinatorial and convex optimization, Proc. FOCS 2015, IEEE, Berkeley, 2015, 1049–1065.Google Scholar
  295. [295]
    Brandão F G S L and Svore K M, Quantum speed-ups for solving semidefinite programs, Proc. FOCS 2017, IEEE, Berkeley, 2017, 415–426.Google Scholar
  296. [296]
    Apeldoorn J van, Gilyén A, Gribling S, et al., Quantum SDP-solvers: Better upper and lower bounds, Proc. FOCS 2017, IEEE, Berkeley, 2017, 403–414.Google Scholar
  297. [297]
    Apeldoorn J van and Gilyén A, Improvements in quantum SDP-solving with applications, arXiv: 1804.05058 [quant-ph].Google Scholar
  298. [298]
    Brandão F G S L, Kalev A, Li T Y, et al., Quantum SDP solvers: Large speed-ups, optimality, and applications to quantum learning, arXiv: 1710.02581v2 [quant-ph].Google Scholar
  299. [299]
    Lee Y T, Sidford A, and Vempala S S, Efficient convex optimization with membership oracles, arXiv: 1706.07357 [cs.DS].Google Scholar
  300. [300]
    Chakraborty S, Childs A M, Li T Y, et al., Quantum algorithms and lower bounds for convex optimization, arXiv: 1809.01731 [quant-ph].Google Scholar
  301. [301]
    Apeldoorn J van, Gilyén A, Gribling S, et al., Convex optimization using quantum oracles, arXiv: 1809.00643v1 [quant-ph].Google Scholar
  302. [302]
    Grassl M, Langenberg B, and Roetteler M, et al., Applying Grover’s Algorithm to AES: Quantum Resource Estimates, PQCrypto 2016, Ed. by Takagi T, LNCS 9606, Springer, Cham 2016, 29–43.Google Scholar
  303. [303]
    Amy M, Di Matteo O, and Gheorghiu V, et al., Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3, Selected Areas in Cryptography, Eds. by Avanzi R and Heys H, LNCS 10532, Springer, Cham, 2017, 317–337.Google Scholar
  304. [304]
    Dunjko V, Ge Y M, and Cirac I, Computational speedups using small quantum devices, arXiv: 1807.08970 [quant-ph].Google Scholar
  305. [305]
    Schuld M, Bocharov A, Svore K, et al., Circuit-centric quantum classifiers, arXiv: 1804.00633 [quant-ph].Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems Science, University of Chinese Academy of SciencesChinese Academy of SciencesBeijingChina

Personalised recommendations

Cite article

Buy options