Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Quantum Algorithms and Complexity for Continuous Problems

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_424-3

Definition of the Subject

Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we are dealing with a numerical approximation to the solution.

There are two major motivations for studying quantum algorithms and complexity for continuous problems:
  1. 1.

    Are quantum computers more powerful than classical computers for important scientific problems? How much more powerful? This would answer the question posed by Nielsen and Chuang (2000, p. 47).

     
  2. 2.

    Many important scientific and engineering problems have continuous formulations. These problems occur in fields such as physics, chemistry, engineering, and finance. The continuous formulations include path integration, partial differential equations (in particular, the Schrödinger equation), and continuous optimization.

     

To answer the first question, we must know the classical computational complexity (for brevity, complexity) of the...

Keywords

Boolean Function Quantum Computer Path Integration Quantum Algorithm Query Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Notes

Acknowledgments

We are grateful to Erich Novak, University of Jena, and Henryk Woźniakowski, Columbia University and University of Warsaw, for their very helpful comments. We thank Jason Petras, Columbia University, for checking the complexity estimates appearing in the tables.

Bibliography

  1. Abrams DS, Lloyd S (1997) Simulation of many-body fermi systems on a universal quantum computer. Phys Rev Lett 79(13):2586–2589, http://arXiv.org/quant-ph/9703054 ADSCrossRefGoogle Scholar
  2. Abrams DS, Lloyd S (1999) Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys Rev Lett 83:5162–5165ADSCrossRefGoogle Scholar
  3. Abrams DS, Williams CP (1999) Fast quantum algorithms for numerical integrals and stochastic processes. http://arXiv.org/quant-ph/9908083
  4. Babuska I, Osborn J (1991) Eigenvalue problems. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol II. North-Holland, Amsterdam, pp 641–787Google Scholar
  5. Bakhvalov NS (1977) Numerical methods. Mir Publishers, MoscowGoogle Scholar
  6. Beals R, Buhrman H, Cleve R, Mosca M, de Wolf R (2001) Quantum lower bounds by polynomials. J ACM 48(4):778–797, http://arXiv.org/quant-ph/9802049
  7. Bennett CH, Bernstein E, Brassard G, Vazirani U (1997) Strengths and weaknesses of quantum computing. SIAM J Comput 26(5):1510–1523CrossRefMATHMathSciNetGoogle Scholar
  8. Bernstein E, Vazirani U (1997) Quantum complexity theory. SIAM J Comput 26(5):1411–1473CrossRefMATHMathSciNetGoogle Scholar
  9. Berry DW, Ahokas G, Cleve R, Sanders BC (2007) Efficient quantum algorithms for simulating sparse Hamiltonians. Commun Math Phys 270(2):359–371, http://arXiv.org/quant-ph/0508139 ADSCrossRefMATHMathSciNetGoogle Scholar
  10. Bessen AJ (2007) On the complexity of classical and quantum algorithms for numerical problems in quantum mechanics. PhD thesis. Department of Computer Science, Columbia UniversityGoogle Scholar
  11. Boghosian BM, Taylor W (1998) Simulating quantum mechanics on a quantum computer. Physica D 120:30–42, http://arXiv.org/quant-ph/9701019 ADSCrossRefMATHMathSciNetGoogle Scholar
  12. Brassard G, Hoyer P, Mosca M, Tapp A (2002) Quantum amplitude amplification and estimation. Contemporary mathematics. Am Math Soc Providence 305:53–74, http://arXiv.org/quant-ph/0005055 MathSciNetGoogle Scholar
  13. Brown KR, Clark RJ, Chuang IL (2006) Limitations of quantum simulation examined by simulating a pairing Hamiltonian using magnetic resonance. Phys Rev Lett 97(5):050504, http://arXiv.org/quant-ph/0601021 ADSCrossRefGoogle Scholar
  14. Cameron RH (1951) A Simpson’s rule for the numerical evaluation of Wiener’s integrals in function space. Duke Math J 8:111–130CrossRefGoogle Scholar
  15. Chen Z, Yepez J, Cory DG (2006) Simulation of the burgers equation by NMR quantum information processing. Phys Rev A 74:042321, http://arXiv.org/quant-ph/0410198 ADSCrossRefGoogle Scholar
  16. Chorin AJ (1973) Accurate evaluation of Wiener integrals. Math Comput 27:1–15CrossRefMATHMathSciNetGoogle Scholar
  17. Cleve R, Ekert A, Macchiavello C, Mosca M (1969) Quantum algorithms revisited. Proc R Soc Lond A 454:339–354ADSCrossRefMathSciNetGoogle Scholar
  18. Collatz L (1960) The numerical treatment of differential equations. Springer, BerlinCrossRefMATHGoogle Scholar
  19. Courant C, Hilbert D (1989) Methods of mathematical physics, vol 1. Wiley, New YorkCrossRefGoogle Scholar
  20. Curbera F (2000) Delayed curse of dimension for Gaussian integration. J Complex 16(2):474–506CrossRefMATHMathSciNetGoogle Scholar
  21. Dawson CM, Eisert J, Osborne TJ (2007) Unifying variational methods for simulating quantum many-body systems. http://arxiv.org/abs/0705.3456v1
  22. Demmel JW (1997) Applied numerical linear algebra. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  23. Egorov AD, Sobolevsky PI, Yanovich LA (1993) Functional integrals: approximate evaluation and applications. Kluwer, DordrechtCrossRefMATHGoogle Scholar
  24. Feynman RP (1982) Simulating physics with computers. Int J Theor Phys 21:476CrossRefMathSciNetGoogle Scholar
  25. Forsythe GE, Wasow WR (2004) Finite-difference methods for partial differential equations. Dover, New YorkMATHGoogle Scholar
  26. Grover L (1997) Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett 79(2):325–328, http://arXiv.org/quant-ph/9706033 ADSCrossRefGoogle Scholar
  27. Heinrich S (2002) Quantum summation with an application to integration. J Complex 18(1):1–50, http://arXiv.org/quant-ph/0105116 CrossRefMATHMathSciNetGoogle Scholar
  28. Heinrich S (2003a) From Monte Carlo to quantum computation. In: Entacher K, Schmid WC, Uhl A (eds) Proceedings of the 3rd IMACS Seminar on Monte Carlo Methods MCM2001, Salzburg. Special Issue of Math Comput Simul 62:219–230Google Scholar
  29. Heinrich S (2003b) Quantum integration in Sobolev spaces. J Complex 19:19–42CrossRefMATHMathSciNetGoogle Scholar
  30. Heinrich S (2004) Quantum approximation II. Sobolev embeddings. J Complex 20:27–45, http://arXiv.org/quant-ph/0305031 CrossRefMATHMathSciNetGoogle Scholar
  31. Heinrich S (2006a) The randomized complexity of elliptic PDE. J Complex 22(2):220–249CrossRefMATHMathSciNetGoogle Scholar
  32. Heinrich S (2006b) The quantum query complexity of elliptic PDE. J Complex 22(5):691–725CrossRefMATHMathSciNetGoogle Scholar
  33. Heinrich S, Milla B (2007) The randomized complexity of initial value problems. Talk presented at First Joint International Meeting between the American Mathematical Society and the Polish Mathematical Society, WarsawGoogle Scholar
  34. Heinrich S, Novak E (2002) Optimal summation by deterministic, randomized and quantum algorithms. In: Fang KT, Hickernell FJ, Niederreiter H (eds) Monte Carlo and Quasi-Monte Carlo methods 2000. Springer, BerlinGoogle Scholar
  35. Heinrich S, Kwas M, Woźniakowski H (2004) Quantum boolean summation with repetitions in the worst-average setting. In: Niederreiter H (ed) Monte Carlo and Quasi-Monte Carlo methods, 2002. Springer, New York, pp 27–49CrossRefGoogle Scholar
  36. Jaksch P, Papageorgiou A (2003) Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation. Phys Rev Lett 91:257902, http://arXiv.org/quant-ph/0308016 ADSCrossRefGoogle Scholar
  37. Jordan SP (2005) Fast quantum algorithm for numerical gradient estimation. Phys Rev Lett 95:050501, http://arXiv.org/quant-ph/0405146 ADSCrossRefGoogle Scholar
  38. Kacewicz BZ (1984) How to increase the order to get minimal-error algorithms for systems of ODEs. Numer Math 45:93–104CrossRefMATHMathSciNetGoogle Scholar
  39. Kacewicz BZ (2006) Almost optimal solution of initial-value problems by randomized and quantum algorithms. J Complex 22(5):676–690CrossRefMATHMathSciNetGoogle Scholar
  40. Keller HB (1968) Numerical methods for two-point boundary-value problems. Blaisdell, WalthamMATHGoogle Scholar
  41. Knuth DE (1997) The art of computer programming, vol 2, 3rd edn, Seminumerical algorithms. Addison-Wesley Professional, CambridgeGoogle Scholar
  42. Kwas M (2005) Quantum algorithms and complexity for certain continuous and related discrete problems. PhD thesis. Department of Computer Science, Columbia UniversityGoogle Scholar
  43. Kwas M, Li Y (2003) Worst case complexity of multivariate Feynman-Kac path integration. J Complex 19:730–743CrossRefMATHMathSciNetGoogle Scholar
  44. Manin Y (1980) Computable and uncomputable. Sovetskoye Radio, Moscow (in Russian)Google Scholar
  45. Manin YI (1999) Classical computing, quantum computing, and Shor’s factoring algorithm. http://arXiv.org/quant-ph/9903008
  46. Morita S, Nishimori H (2007) Convergence of quantum annealing with real-time Schrödinger dynamics. J Phys Soc Jpn 76(6):064002, http://arXiv.org/quant-ph/0702252 ADSCrossRefMathSciNetGoogle Scholar
  47. Nayak A, Wu F (1999) The quantum query complexity of approximating the median and related statistics. In: Proc STOC 1999, Association for Computing Machinery, New York, pp 384–393. http://arXiv.org/quant-ph/9804066
  48. Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press, CambridgeMATHGoogle Scholar
  49. Novak E (1988) Deterministic and stochastic error bounds in numerical analysis, vol 1349, Lecture notes in mathematics. Springer, BerlinMATHGoogle Scholar
  50. Novak E (2001) Quantum complexity of integration. J Complex 17:2–16, http://arXiv.org/quant-ph/0008124 CrossRefMATHGoogle Scholar
  51. Ortiz G, Gubernatis JE, Knill E, Laflamme R (2001) Quantum algorithms for fermionic simulations. Phys Rev A 64(2):022319, http://arXiv.org/cond-mat/0012334 ADSCrossRefGoogle Scholar
  52. Papageorgiou A (2004) Average case quantum lower bounds for computing the boolean mean. J Complex 20(5):713–731CrossRefMATHGoogle Scholar
  53. Papageorgiou A (2007) On the complexity of the multivariate Sturm-Liouville eigenvalue problem. J Complex 23(4–6):802–827CrossRefMATHGoogle Scholar
  54. Papageorgiou A, Traub JF (2005) Qubit complexity of continuous problems. http://arXiv.org/quant-ph/0512082
  55. Papageorgiou A, Woźniakowski H (2005) Classical and quantum complexity of the sturm-liouville eigenvalue problem. Quantum Inf Process 4(2):87–127, http://arXiv.org/quant-ph/0502054 CrossRefMathSciNetGoogle Scholar
  56. Paredes B, Verstraete F, Cirac JI (2005) Exploiting quantum parallelism to simulate quantum random many-body systems. Phys Rev Lett 95:140501, http://arXiv.org/cond-mat/0505288 ADSCrossRefGoogle Scholar
  57. Plaskota L (1996) Noisy information and computational complexity. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  58. Plaskota L, Wasilkowski GW, Woźniakowski H (2000) A new algorithm and worst case complexity for Feynman-Kac path integration. J Comp Phys 164(2):335–353ADSCrossRefMATHGoogle Scholar
  59. Ritter K (2000) Average-case analysis of numerical problems, vol 1733, Lecture notes in mathematics. Springer, BerlinCrossRefMATHGoogle Scholar
  60. Shor PW (1997) Polynomial-time algorithms for prime factorization and discrete logarithm on a quantum computer. SIAM J Comput 26(5):1484–1509CrossRefMATHMathSciNetGoogle Scholar
  61. Somma R, Ortiz G, Knill E, Gubernatis (2003) Quantum simulations of physics problems. In: Pirich AR, Brant HE (eds) Quantum information and computation. Proc SPIE 2003, vol 5105. The International Society for Optical Engineering, Bellingham, pp 96–103. http://arXiv.org/quant-ph/0304063Google Scholar
  62. Sornborger AT, Stewart ED (1999) Higher order methods for simulations on quantum computers. Phys Rev A 60(3):1956–1965, http://arXiv.org/quant-ph/9903055 ADSCrossRefGoogle Scholar
  63. Strang G, Fix GJ (1973) An analysis of the finite element method. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  64. Szkopek T, Roychowdhury V, Yablonovitch E, Abrams DS (2005) Egenvalue estimation of differential operators with a quantum algorithm. Phys Rev A 72:062318ADSCrossRefGoogle Scholar
  65. Titschmarsh EC (1958) Eigenfunction expansions associated with second-order differential equations, part B. Oxford University Press, OxfordGoogle Scholar
  66. Traub JF (1999) A continuous model of computation. Phys Today May:39–43Google Scholar
  67. Traub JF, Werschulz AG (1998) Complexity and information. Cambridge University Press, CambridgeMATHGoogle Scholar
  68. Traub JF, Woźniakowski H (1980) A general theory of optimal algorithms, ACM monograph series. Academic, New YorkMATHGoogle Scholar
  69. Traub JF, Woźniakowski H (1992) The Monte Carlo algorithm with a pseudorandom generator. Math Comp 58(197):323–339ADSCrossRefMATHMathSciNetGoogle Scholar
  70. Traub JF, Woźniakowski H (2002) Path integration on a quantum computer. Quantum Inf Process 1(5):365–388, http://arXiv.org/quant-ph/0109113 CrossRefMathSciNetGoogle Scholar
  71. Traub JF, Wasilkowski GW, Woźniakowski H (1988) Information-based complexity. Academic, New YorkMATHGoogle Scholar
  72. Wasilkowski GW, Woźniakowski H (1996) On tractability of path integration. J Math Phys 37(4):2071–2088ADSCrossRefMATHMathSciNetGoogle Scholar
  73. Weinberger HF (1956) Upper and lower bounds for eigenvalues by finite difference methods. Commun Pure Appl Math IX:613–623CrossRefMathSciNetGoogle Scholar
  74. Weinberger HF (1958) Lower bounds for higher eigenvalues by finite difference methods. Pac J Math 8(2):339–368CrossRefMATHMathSciNetGoogle Scholar
  75. Werschulz AG (1991) The computational complexity of differential and integral equations. Oxford University Press, OxfordMATHGoogle Scholar
  76. Wisner S (1996) Simulations of many-body quantum systems by a quantum computer. http://arXiv.org/quant-ph/96
  77. Woźniakowski H (2006) The quantum setting with randomized queries for continuous problems. Quantum Inf Process 5(2):83–130CrossRefMATHMathSciNetGoogle Scholar
  78. Yepez J (2002) An efficient and accurate quantum algorithm for the Dirac equation. http://arXiv.org/quant-ph/0210093
  79. Zalka C (1998) Simulating quantum systems on a quantum computer. Proc Royal Soc Lond A 454(1969):313–322, http://arXiv.org/quant-ph/9603026 ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Computer Science DepartmentColumbia UniversityNew YorkUSA