Skip to main content
Springer Nature Link
Log in

Algorithms for Quantum Computers

  • Reference work entry
Handbook of Natural Computing

Abstract

This chapter surveys the field of quantum computer algorithms. It gives a taste of both the breadth and the depth of the known algorithms for quantum computers, focusing on some of the more recent results. It begins with a brief review of quantum Fourier transform-based algorithms, followed by quantum searching and some of its early generalizations. It continues with a more in-depth description of two more recent developments: algorithms developed in the quantum walk paradigm, followed by tensor network evaluation algorithms (which include approximating the Tutte polynomial).

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
€34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 1,176.99
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
EUR 1,176.99
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

') var buybox = document.querySelector("[data-id=id_"+ timestamp +"]").parentNode var buyingOptions = buybox.querySelectorAll(".buying-option") ;[].slice.call(buyingOptions).forEach(initCollapsibles) var buyboxMaxSingleColumnWidth = 480 function initCollapsibles(subscription, index) { var toggle = subscription.querySelector(".buying-option-price") subscription.classList.remove("expanded") var form = subscription.querySelector(".buying-option-form") var priceInfo = subscription.querySelector(".price-info") var buyingOption = toggle.parentElement if (toggle && form && priceInfo) { toggle.setAttribute("role", "button") toggle.setAttribute("tabindex", "0") toggle.addEventListener("click", function (event) { var expandedBuyingOptions = buybox.querySelectorAll(".buying-option.expanded") var buyboxWidth = buybox.offsetWidth ;[].slice.call(expandedBuyingOptions).forEach(function(option) { if (buyboxWidth <= buyboxMaxSingleColumnWidth && option != buyingOption) { hideBuyingOption(option) } }) var expanded = toggle.getAttribute("aria-expanded") === "true" || false toggle.setAttribute("aria-expanded", !expanded) form.hidden = expanded if (!expanded) { buyingOption.classList.add("expanded") } else { buyingOption.classList.remove("expanded") } priceInfo.hidden = expanded }, false) } } function hideBuyingOption(buyingOption) { var toggle = buyingOption.querySelector(".buying-option-price") var form = buyingOption.querySelector(".buying-option-form") var priceInfo = buyingOption.querySelector(".price-info") toggle.setAttribute("aria-expanded", false) form.hidden = true buyingOption.classList.remove("expanded") priceInfo.hidden = true } function initKeyControls() { document.addEventListener("keydown", function (event) { if (document.activeElement.classList.contains("buying-option-price") && (event.code === "Space" || event.code === "Enter")) { if (document.activeElement) { event.preventDefault() document.activeElement.click() } } }, false) } function initialStateOpen() { var buyboxWidth = buybox.offsetWidth ;[].slice.call(buybox.querySelectorAll(".buying-option")).forEach(function (option, index) { var toggle = option.querySelector(".buying-option-price") var form = option.querySelector(".buying-option-form") var priceInfo = option.querySelector(".price-info") if (buyboxWidth > buyboxMaxSingleColumnWidth) { toggle.click() } else { if (index === 0) { toggle.click() } else { toggle.setAttribute("aria-expanded", "false") form.hidden = "hidden" priceInfo.hidden = "hidden" } } }) } initialStateOpen() if (window.buyboxInitialised) return window.buyboxInitialised = true initKeyControls() })()

Institutional subscriptions

Similar content being viewed by others

References

  • Aaronson S, Shi Y (2004) Quantum lower bounds for the collision and the element distinctness problems. J ACM 51(4):595–605. doi: http://doi.acm.org/10.1145/1008731.1008735

    Google Scholar 

  • Aharonov D, Ambainis A, Kempe J, Vazirani U (2001) Quantum walks on graphs. In: STOC’01: proceedings of the 33rd annual ACM symposium on theory of computing. ACM Press, New York, pp 50–59. doi: http://doi.acm.org/10.1145/380752.380758

  • Aharonov D, Arad I (2006) The BQP-hardness of approximating the Jones polynomial. http://arxiv.org/abs/quant-ph/0605181

  • Aharonov D, Arad I, Eban E, Landau Z (2007) Polynomial quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane. http://arxiv.org/abs/quant-ph/0702008

  • Aharonov D, Jones V, Landau Z (2008) A polynomial quantum algorithm for approximating the Jones polynomial. Algorithmica 55(3):395–421

    Article  MathSciNet  Google Scholar 

  • Ambainis A (2003) Quantum walks and their algorithmic applications. Int J Quantum Inform 1:507–518

    Article  MATH  Google Scholar 

  • Ambainis A (2004) Quantum walk algorithm for element distinctness. In: Proceedings of the 45th annual IEEE symposium on foundations of computer science, pp 22–31. doi: 10.1109/FOCS.2004.54

    Google Scholar 

  • Ambainis A, Bach E, Nayak A, Vishwanath A, Watrous J (2001) One-dimensional quantum walks. In: STOC’ 01: proceedings of the 33rd annual ACM symposium on theory of computing. ACM Press, New York, pp 37–49. doi: http://doi.acm.org/10.1145/380752.380757

  • Ambainis A, Childs A, Reichardt B, Spalek R, Zhang S (2007) Any and-or formula of size n can be evaluated in time n 1 ∕ 2+o(1) on a quantum computer. In: Proceedings of the 48th annual IEEE symposium on foundations of computer science, pp 363–372. doi: 10.1109/FOCS.2007.57

    Google Scholar 

  • Arad I, Landau Z (2008) Quantum computation and the evaluation of tensor networks. http://arxiv.org/abs/0805.0040

  • Beaudin L, Ellis-Monaghan J, Pangborn G, Shrock R (2008) A little statistical mechanics for the graph theorist. http://arxiv.org/abs/0804.2468

  • Bernstein BK, Vazirani U (1997) Quantum complexity theory. SIAM J Comput 26:1411–1473

    Article  MathSciNet  MATH  Google Scholar 

  • Berry DW, Ahokas G, Cleve R, Sanders BC (2007) Efficient quantum algorithms for simulating sparse Hamiltonians. Commun Math Phys 270:359

    Article  MathSciNet  MATH  Google Scholar 

  • Boixo S, Knill E, Somma R (2009) Quantum state preparation by phase randomization. http://arxiv.org/abs/0903.1652

  • Boneh D, Lipton R (1995) Quantum cryptanalysis of hidden linear functions (extended abstract). In: Proceedings of the 15th annual international cryptology conference on advances in cryptology. Lecture notes in computer science, vol. 963. Springer, London, UK, pp 424–437

    Google Scholar 

  • Boyer M, Brassard G, Høyer P, Tapp A (1998) Tight bounds on quantum searching. Fortschritte der Physik 56(5–5):493–505

    Article  Google Scholar 

  • Brassard G, Høyer P (1997) An exact quantum polynomial-time algorithm for Simon's problem. In: Proceedings of the fifth Israeli symposium on theory of computing and systems (ISTCS’97). IEEE Press, Piscataway, pp 12–23

    Google Scholar 

  • Brassard G, Høyer P, Mosca M, Tapp A (2002) Quantum amplitude amplification and estimation. Quantum Computation & Information, AMS Contemporary Math Series 305:53–74

    Google Scholar 

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

    Article  Google Scholar 

  • Brouwer AE (1989) Distance-regular graphs. Springer, New York

    Book  MATH  Google Scholar 

  • Childs A (2008) CO781 Topics in quantum information: quantum algorithms. Lecture notes on quantum algorithms. http://www.math.uwaterloo.ca/~amchilds/teaching/w08/co781.html

  • Childs AM, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman DA (2003) Exponential algorithmic speedup by a quantum walk. In: STOC ’03: proceedings of the 35th annual ACM symposium on theory of computing. ACM Press, New York, pp 59–68. doi: http://doi.acm.org/10.1145/780542.780552

  • Childs A, van Dam W (2010) Quantum algorithms for algebraic problems. Rev Mod Phys 82(1):1–52

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • D'Ariano GM, van Dam W, Ekert E, Macchiavello C, Mosca M (2007) General optimized schemes for phase estimation. Phys Rev Lett 98(9):090,501

    Google Scholar 

  • Das A, Chakrabarti BK (2008) Quantum annealing and analog quantum computation. Rev Mod Phys 80:1061

    Article  MathSciNet  MATH  Google Scholar 

  • De las Cuevas G, Dür W, Van den Nest M, Briegel HJ (2008) Completeness of classical spin models and universal quantum computation. http://arxiv.org/abs/0812.2368

  • Deutsch D (1985) Quantum theory, the Church-Turing principle and the universal quantum computer. Proc Roy Soc Lond A 400:97–117

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Farhi E, Goldstone J, Gutmann S (2007) A quantum algorithm for the Hamiltonian NAND tree. http://arxiv.org/abs/quant-ph/0702144

  • Feynman R (1982) Simulating physics with computers. Int J Theor Phys 21(6,7):467–488

    Article  MathSciNet  Google Scholar 

  • Freedman MH, Kitaev A, Larsen MJ, Wang Z (2001) Topological quantum computation. http://arxiv.org/abs/quant-ph/0101025

  • Freedman MH, Kitaev A, Wang Z (2000) Simulation of topological field theories by quantum computers. http://arxiv.org/abs/quant-ph/0001071

  • Geraci J (2008) A BQP-complete problem related to the Ising model partition function via a new connection between quantum circuits and graphs. http://arxiv.org/abs/0801.4833

  • Geraci J, Lidar DA (2008) On the exact evaluation of certain instances of the Potts partition function by quantum computers. Commun Math Phys 279(3):735–768

    Article  MathSciNet  MATH  Google Scholar 

  • Grigoriev D (1997) Testing shift-equivalence of polynomials by deterministic, probabilistic and quantum machines. Theor Comput Sci 180:217–228

    Article  MathSciNet  MATH  Google Scholar 

  • 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, 96). ACM Press, New York, pp 212–219

    Google Scholar 

  • Grover L (1998) A framework for fast quantum mechanical algorithms. In: Proceedings of the 13th annual ACM symposium on theory of computing (STOC' 98). ACM Press, New York, pp 53–62

    Chapter  Google Scholar 

  • Hirvensalo M (2001) Quantum computing. Series: Natural Computing Series. Springer

    Google Scholar 

  • Hübener R, Van den Nest M, Dür W, Briegel HJ (2008) Classical spin systems and the quantum stabilizer formalism: general mappings and applications. http://arxiv.org/abs/0812.2127

  • Jordan S (2008) Quantum computation beyond the circuit model. PhD thesis, MIT University, Cambridge

    Google Scholar 

  • Karchmer M, Wigderson A (1993) On span programs. In: Proceedings of the 8th IEEE structures in complexity conference. IEEE Press, Piscataway, pp 102–111

    Google Scholar 

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

    Google Scholar 

  • Kempe J (2003) Quantum random walks - an introductory overview. Contemp Phys 44(4):307–327

    Article  MathSciNet  Google Scholar 

  • Kitaev AY (1995) Quantum measurements and the Abelian stabilizer problem. http://arxiv.org/abs/quant-ph/9511026

  • Kitaev A, Shen A, Vyalvi M (2002) Classical and quantum computation. American Mathematical Society, Providence, RI

    MATH  Google Scholar 

  • Magniez F, Nayak A, Roland J, Santha M (2007) Search via quantum walk. In: STOC ’07: proceedings of the 39th annual ACM symposium on theory of computing. ACM, New York, pp 575–584. doi: http://doi.acm.org/10.1145/1250790.1250874

  • Menezes A, van Oorschot P, Vanstone S (1996) Handbook of applied cryptography. CRC Press, Boca Raton

    Book  Google Scholar 

  • Mermin ND (2007) Quantum computer science: an introduction. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Mosca M (2001) Counting by quantum eigenvalue estimation. Theor Comput Sci 264:139–153

    Article  MathSciNet  MATH  Google Scholar 

  • Mosca M (2008) Abelian hidden subgroup problem. In: Kao M-Y (ed) Encyclopedia of algorithms. Springer, Berlin

    Google Scholar 

  • Mosca M (2009) Quantum algorithms. In: Meyers R (ed) Encyclopedia of complexity and systems science. Springer

    Google Scholar 

  • Nayak A, Vishwanath A (2000) Quantum walk on the line. http://arxiv.org/abs/quant-ph/0010117

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

    MATH  Google Scholar 

  • Reichardt BW, Spalek R (2008) Span-program-based quantum algorithm for evaluating formulas. In: STOC ’08: proceedings of the 40th annual ACM symposium on theory of computing. ACM Press, New York, pp 103–112. doi: http://doi.acm.org/10.1145/1374376.1374394

  • Santha M (2008) Quantum walk based search algorithms. In: Agrawal M, Du D-Z, Duan Z, Li A (eds) Theory and applications of models of computation. Lecture notes in computer science, vol 4978. Springer, Berlin, Heidelberg, pp 31–46. doi: 10.1007/978-3-540-79228-4_3

    Google Scholar 

  • Shor P (1994) Algorithms for quantum computation: Discrete logarithms and factoring. In: Proceedings of the 35th annual symposium on foundations of computer science. IEEE Computer Society, Washington, DC, pp 124–134

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Simon D (1994) On the power of quantum computation. In: Proceedings of the 35th IEEE symposium on the foundations of computer science (FOCS). IEEE Computer Society, Washington, DC, pp 116–123

    Google Scholar 

  • Szegedy M (2004) Quantum speed-up of Markov chain based algorithms. In: Proceedings of the 45th annual IEEE symposium on foundations of computer science (FOCS). IEEE Computer Society, Washington, DC, pp 32–41. doi: http://dx.doi.org/10.1109/FOCS.2004.53

  • Tulsi T, Grover L, Patel A (2006) A new algorithm for fixed point quantum search. Quant Inform Comput 6(6):483–494

    MathSciNet  MATH  Google Scholar 

  • Welsh D (1993) Complexity: knots, colourings and countings. Cambridge University Press, Cambridge, UK

    Google Scholar 

  • Wiebe N, Berry DW, Høyer P, Sanders BC (2008) Higher order decompositions of ordered operator exponentials. http://arxiv.org/abs/0812.0562

  • Wocjan P, Yard J (2006) The Jones polynomial: quantum algorithms and applications in quantum complexity theory. http://arxiv.org/abs/quant-ph/0603069

Download references

Author information

Authors and Affiliations

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

    Jamie Smith

  2. Institute for Quantum Computing and Department of Combinatorics & Optimization, University of Waterloo and St. Jerome's University and Perimeter Institute for Theoretical Physics, Waterloo, Canada

    Michele Mosca

Authors
  1. Jamie Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Mosca
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LIACS, Leiden University, Leiden, The Netherlands

    Grzegorz Rozenberg

  2. Computer Science Department, University of Colorado, Boulder, USA

    Grzegorz Rozenberg

  3. LIACS, Leiden University, Leiden, The Netherlands

    Thomas Bäck

  4. LIACS, Leiden University, Leiden, The Netherlands

    Joost N. Kok

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

Smith, J., Mosca, M. (2012). Algorithms for Quantum Computers. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds) Handbook of Natural Computing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92910-9_43

Download citation

Publish with us

Policies and ethics