quantum computer

A quantum computer can solve computational problems considerably faster than classical computers. There are a variety of models of quantum computers, including the quantum circuit model and the quantum Turing machine. Central to the heart of quantum computers are the individual elements, called qubit’s, which equate to the classical computational bit. They are represented in many states, including a 1 or 0 quantum state, or they can be in a superposition of the 1 and 0. This multi-permutation holds the advantage over binary systems, and thus opens up the possibilities of far reaching computation power based on quantum theory.


How does a quantum computer work?

Quantum computing began when Benioff proposed a quantum model of the Turing machine, in the 1980’s. Feynman and Manin later suggested that a quantum computer had the potential to simulate things that couldn’t be achieved classically. An algorithm was later developed for factoring integers that had the potential to decrypt RSA-encrypted messages. Investment into quantum research has increased and an example in 2019, Google AI and NASA performed a quantum computation that is infeasible on any classical computer.

Any computational problem that can be solved by a classical computer can also be solved by a quantum computer. Quantum computers obey the Church–Turing method. They can quickly solve certain problems that no classical computer could solve in any feasible amount of time, often referred to as quantum supremacy.  Quantum complexity theory is the field of study.


Definition of a quantum computer

The model of quantum computation explains the computation in terms of a network of quantum logic gates. This is an abstract linear-algebraic sumamry of a classical circuit. Obeying quantum mechanics leads to a quantum computer capable of efficiently running these circuits and is theoretically believed to be physically realisable.


Scaling quantum computing

To maximise computational power we optimise along two dimensions, qubit count and low error rates. The greater the number of qubits, the greater number of states that can be manipulated. Low error rates are also needed to to be accurate and perform more precise sequential operations.

Quantum volume is a critical metric and forms the relationship between number and quality of qubits, circuit connectivity and error. Achieving higher quantum volume will lead to applications which can offer a real computational advantage for real world challenges.


Quantum algorthims

Quantum algorithms focus on the quantum circuit model. Quantum algorithms are classified by the uplift achieved. Algorithms that output a polynomial uplift over the best known classical algorithms e.g. Shor’s algorithm or Pell’s equation. These algorithms rely on the quantum Fourier transform. Oracle problems like Simon’s problem do give speedups, though this is a restricted model and doesn’t always translate for practical problems.

Further, simulations of quantum physical processes from chemistry, or the quantum algorithm for linear systems have quantum algorithms that give super-polynomial speedups. Some quantum algorithms, like Grover’s algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms. Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.


Quantum computer applications and advantages


A simple application would be the defence of cryptographic systems. Factorisation is agreed to be practically impossible for large integers if they are derived from several prime numbers. A quantum computer could solve this using Shor’s algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve algorithms could be solved. These are used to protect secure web pages and encrypted email. Lattice-based crypto systems are also a benchmark and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, is an interesting problem.


Machine learning

As quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.  For example, the quantum algorithm for linear systems of equations, or HHL Algorithm, is believed to provide speedup over classical counterpart.


Quantum supremacy

Quantum supremacy refers to the hypothetical speedup advantage that a quantum computer would have. Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year, whilst IBM said in 2018 that the best classical computers will be beaten within five years. In 2019, a Sycamore processor (Google AI Quantum) was reported to have achieved quantum supremacy, with calculations more than 3,000,000 times as fast as those of Summit, the world’s fastest computer.


Disadvantages of quantum computing

There are a number of technical challenges in building a quantum computer.

  • Scalability of qubits
  • Qubits that can be initialised to arbitrary values
  • Quantum gates that are faster than decoherence time
  • Universal gate set

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need Helium-3, a nuclear research byproduct, and special superconducting cables.


Quantum computer models

There are a number of models. The four main models of practical importance are:

  • Quantum gate array (computation decomposed into a sequence of few-qubit quantum gates)
  • One-way quantum computer (computation decomposed into a sequence of one-qubit measurements applied to a highly entangled cluster state)
  • Adiabatic quantum computer (computation decomposed into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution)
  • Topological quantum computer (computation decomposed into the braiding of anyons in a 2D lattice)


In 2020, Honeywell forged ahead with the method of trapped ions. They are taking on this method, against the grain as other global progress has not seen this as the preferred route. IonQ have since revealed they are looking at a trapped-ion machine, to compete with the technologies arising from IBM. Trapped-ion quantum computers were the actually studied before progress in superconducting loops.


Quantum computing in the finance world

In finance, the gains from huge computing performance could be game changing. For example, in valuation, the ability to identify an optimal risk-adjusted portfolio would offer a leading edge in the market. Across loans, fast estimates of more detailed credit exposures would mitigate risk. Additionally, multi-capital investment could be analysed and fine tuned in a far more eloquent manner. As highlighted previously, data encryption would move into a new field.

The advantages for equity would be enormous. The financial sector spends billions in investment into computational power.



Recent category posts


  1.  The National Academies of Sciences, Engineering, and Medicine (2019). Grumbling, Emily; Horowitz, Mark (eds.). Quantum Computing : Progress and Prospects (2018). Washington, DC: National Academies Press. p. I-5. doi:10.17226/25196ISBN 978-0-309-47969-1OCLC 1081001288.
  2. ^ Benioff, Paul (1980). “The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines”. Journal of Statistical Physics22 (5): 563–591. Bibcode:1980JSP….22..563Bdoi:10.1007/bf01011339S2CID 122949592.
  3. ^ Feynman, Richard (June 1982). “Simulating Physics with Computers” (PDF)International Journal of Theoretical Physics21 (6/7): 467–488. Bibcode:1982IJTP…21..467Fdoi:10.1007/BF02650179S2CID 124545445. Archived from the original (PDF) on 8 January 2019. Retrieved 28 February 2019.
  4. ^ Manin, Yu. I. (1980). Vychislimoe i nevychislimoe[Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 2013-05-10. Retrieved 2013-03-04.
  5. ^ Mermin, David (March 28, 2006). “Breaking RSA Encryption with a Quantum Computer: Shor’s Factoring Algorithm”(PDF)Physics 481-681 Lecture Notes. Cornell University. Archived from the original (PDF) on 2012-11-15.
  6. ^ John Preskill (2018). “Quantum Computing in the NISQ era and beyond”. Quantum2: 79. arXiv:1801.00862doi:10.22331/q-2018-08-06-79S2CID 44098998.
  7. ^ Gibney, Elizabeth (2 October 2019). “Quantum gold rush: the private funding pouring into quantum start-ups”Nature574 (7776): 22–24. Bibcode:2019Natur.574…22Gdoi:10.1038/d41586-019-02935-4PMID 31578480.
  8. ^ Rodrigo, Chris Mills (12 February 2020). “Trump budget proposal boosts funding for artificial intelligence, quantum computing”The Hill.
  9. ^ “On “Quantum SupremacyIBM Research Blog. 2019-10-22. Retrieved 2020-01-21.
  10. ^ Franklin, Diana; Chong, Frederic T. (2004). “Challenges in Reliable Quantum Computing”. Nano, Quantum and Molecular Computing. pp. 247–266. doi:10.1007/1-4020-8068-9_8ISBN 1-4020-8067-0.
  11. ^ Pakkin, Scott; Coles, Patrick (10 June 2019). “The Problem with Quantum Computers”. Scientific American.
  12. ^ Nielsen, p. 29
  13. ^ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511976667ISBN 9780511976667.
  14. Jump up to:a b c Quantum Algorithm Zoo Archived 2018-04-29 at the Wayback Machine – Stephen Jordan’s Homepage
  15. ^ Schiller, Jon (2009-06-19). Quantum ComputersISBN 9781439243497.[self-published source?]
  16. ^ Nielsen, p. 42
  17. ^ Nielsen, p. 7
  18. ^ Brassard, Gilles; Høyer, Peter; Tapp, Alain (2016), Kao, Ming-Yang (ed.), “Quantum Algorithm for the Collision Problem”Encyclopedia of Algorithms, New York, NY: Springer, pp. 1662–1664, doi:10.1007/978-1-4939-2864-4_304ISBN 978-1-4939-2864-4, retrieved 2020-12-06
  19. ^ Farhi, Edward; Goldstone, Jeffrey; Gutmann, Sam (2008-12-23). “A Quantum Algorithm for the Hamiltonian NAND Tree”Theory of Computing4 (1): 169–190. doi:10.4086/toc.2008.v004a008ISSN 1557-2862.
  20. ^ Lenstra, Arjen K. (2000). “Integer Factoring” (PDF)Designs, Codes and Cryptography19 (2/3): 101–128. doi:10.1023/A:1008397921377S2CID 9816153. Archived from the original (PDF) on 2015-04-10.
  21. Jump up to:a b Bernstein, Daniel J. (2009). “Introduction to post-quantum cryptography”. Post-Quantum CryptographyNature549. pp. 1–14. doi:10.1007/978-3-540-88702-7_1ISBN 978-3-540-88701-0PMID 28905891.
  22. ^ See also pqcrypto.org, a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing.
  23. ^ McEliece, R. J. (January 1978). “A Public-Key Cryptosystem Based On Algebraic Coding Theory” (PDF)DSNPR44: 114–116. Bibcode:1978DSNPR..44..114M.
  24. ^ Kobayashi, H.; Gall, F.L. (2006). “Dihedral Hidden Subgroup Problem: A Survey”Information and Media Technologies1(1): 178–185. doi:10.2197/ipsjdc.1.470.
  25. ^ Bennett, Charles H.; Bernstein, Ethan; Brassard, Gilles; Vazirani, Umesh (October 1997). “Strengths and Weaknesses of Quantum Computing”. SIAM Journal on Computing26 (5): 1510–1523. arXiv:quant-ph/9701001Bibcode:1997quant.ph..1001Bdoi:10.1137/s0097539796300933S2CID 13403194.
  26. ^ Katwala, Amit (5 March 2020). “Quantum computers will change the world (if they work)”Wired UK.
  27. ^ Ambainis, Ambainis (June 2004). “Quantum search algorithms”. ACM SIGACT News35 (2): 22–35. arXiv:quant-ph/0504012Bibcode:2005quant.ph..4012Adoi:10.1145/992287.992296S2CID 11326499.
  28. ^ Rich, Steven; Gellman, Barton (2014-02-01). “NSA seeks to build quantum computer that could crack most types of encryption”Washington Post.
  29. ^ Norton, Quinn (2007-02-15). “The Father of Quantum Computing”Wired.
  30. ^ Ambainis, Andris (Spring 2014). “What Can We Do with a Quantum Computer?”. Institute for Advanced Study.
  31. ^ https://www.youtube.com/watch?v=7susESgnDv8
  32. ^ Biamonte, Jacob; Wittek, Peter; Pancotti, Nicola; Rebentrost, Patrick; Wiebe, Nathan; Lloyd, Seth (September 2017). “Quantum machine learning”Nature549 (7671): 195–202. doi:10.1038/nature23474ISSN 0028-0836.
  33. Jump up to:a b Preskill, John (2018-08-06). “Quantum Computing in the NISQ era and beyond”Quantum2: 79. doi:10.22331/q-2018-08-06-79.
  34. ^ Harrow, Aram; Hassidim, Avinatan; Lloyd, Seth (2009). “Quantum algorithm for solving linear systems of equations”. Physical Review Letters103 (15): 150502. arXiv:0811.3171Bibcode:2009PhRvL.103o0502Hdoi:10.1103/PhysRevLett.103.150502PMID 19905613S2CID 5187993.
  35. ^ Boixo, Sergio; Isakov, Sergei V.; Smelyanskiy, Vadim N.; Babbush, Ryan; Ding, Nan; Jiang, Zhang; Bremner, Michael J.; Martinis, John M.; Neven, Hartmut (2018). “Characterizing Quantum Supremacy in Near-Term Devices”. Nature Physics14 (6): 595–600. arXiv:1608.00263Bibcode:2018NatPh..14..595Bdoi:10.1038/s41567-018-0124-xS2CID 4167494.
  36. ^ Savage, Neil. “Quantum Computers Compete for “Supremacy.
  37. ^ “Quantum Supremacy and Complexity”. 23 April 2016.
  38. ^ Kalai, Gil. “The Quantum Computer Puzzle” (PDF). AMS.
  39. ^ Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C.; Barends, Rami; Biswas, Rupak; Boixo, Sergio; Brandao, Fernando G. S. L.; Buell, David A.; Burkett, Brian; Chen, Yu; Chen, Zijun; Chiaro, Ben; Collins, Roberto; Courtney, William; Dunsworsth, Andrew; Farhi, Edward; Foxen, Brooks; Fowler, Austin; Gidney, Craig; Giustina, Marissa; Graff, Rob; Guerin, Keith; Habegger, Steve; Harrigan, Matthew P.; Hartmann, Michael J.; Ho, Alan; Hoffman, Markus; Huang, Trent; Humble, Travis S.; Isakov, Sergei V.; Jeffery, Evan; Jiang, Zhang; Kafri, Dvir; Kechedzhi, Kostyantyn; Kelly, Julian; Klimov, Paul V.; Knysh, Sergey; Korotov, Alexander; Kostritsa, Fedor; Landhuis, David; Lindmark, Mike; Lucero, Erik; Lyakh, Dmitry; Mandrà, Salvatore; McClean, Jarrod R.; McEwen, Matthew; Megrant, Anthony; Mi, Xiao; Michielsen, Kristel; Mohseni, Masoud; Mutus, Josh; Naaman, Ofer; Neeley, Matthew; Neill, Charles; Niu, Murphy Yuezhen; Ostby, Eric; Petukhov, Andre; Platt, John C.; Quintana, Chris; Rieffel, Eleanor G.; Roushan, Pedram; Rubin, Nicholas C.; Sank, Daniel; Satzinger, Kevin J.; Smelyanskiy, Vadim; Sung, Kevin J.; Trevithick, Matthew D.; Vainsencher, Amit; Villalonga, Benjamin; White, Theodore; Yao, Z. Jamie; Yeh, Ping; Zalcman, Adam; Neven, Hartmut; Martinis, John M. (23 October 2019). “Quantum supremacy using a programmable superconducting processor”. Nature574 (7779): 505–510. arXiv:1910.11333Bibcode:2019Natur.574..505Adoi:10.1038/s41586-019-1666-5PMID 31645734S2CID 204836822.
  40. ^ “Google researchers have reportedly achieved “quantum supremacyMIT Technology Review.
  41. ^ Unruh, Bill (1995). “Maintaining coherence in Quantum Computers”. Physical Review A51 (2): 992–997. arXiv:hep-th/9406058Bibcode:1995PhRvA..51..992Udoi:10.1103/PhysRevA.51.992PMID 9911677S2CID 13980886.
  42. ^ Davies, Paul. “The implications of a holographic universe for quantum information science and the nature of physical law” (PDF). Macquarie University.
  43. ^ Dyakonov, Mikhail (2018-11-15). “The Case Against Quantum Computing”IEEE Spectrum.
  44. ^ DiVincenzo, David P. (2000-04-13). “The Physical Implementation of Quantum Computation”. Fortschritte der Physik48 (9–11): 771–783. arXiv:quant-ph/0002077Bibcode:2000ForPh..48..771Ddoi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E.
  45. ^ Giles, Martin (17 January 2019). “We’d have more quantum computers if it weren’t so hard to find the damn cables”. MIT Technology Review.
  46. ^ DiVincenzo, David P. (1995). “Quantum Computation”. Science270 (5234): 255–261. Bibcode:1995Sci…270..255DCiteSeerX 220110562.(subscription required)
  47. ^ Jones, Nicola (19 June 2013). “Computing: The quantum company”Nature498 (7454): 286–288. Bibcode:2013Natur.498..286Jdoi:10.1038/498286aPMID 23783610.
  48. ^ Vepsäläinen, Antti P.; Karamlou, Amir H.; Orrell, John L.; Dogra, Akshunna S.; Loer, Ben; et al. (August 2020). “Impact of ionizing radiation on superconducting qubit coherence”Nature584 (7822): 551–556. arXiv:2001.09190Bibcode:2020Natur.584..551Vdoi:10.1038/s41586-020-2619-8ISSN 1476-4687PMID 32848227S2CID 210920566.
  49. ^ Amy, Matthew; Matteo, Olivia; Gheorghiu, Vlad; Mosca, Michele; Parent, Alex; Schanck, John (November 30, 2016). “Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3”. arXiv:1603.09383 [quant-ph].
  50. ^ Dyakonov, M. I. (2006-10-14). S. Luryi; J. Xu; A. Zaslavsky (eds.). “Is Fault-Tolerant Quantum Computation Really Possible?”. Future Trends in Microelectronics. Up the Nano Creek: 4–18. arXiv:quant-ph/0610117Bibcode:2006quant.ph.10117D.
  51. ^ Freedman, Michael H.Kitaev, AlexeiLarsen, Michael J.; Wang, Zhenghan (2003). “Topological quantum computation”. Bulletin of the American Mathematical Society40 (1): 31–38. arXiv:quant-ph/0101025doi:10.1090/S0273-0979-02-00964-3MR 1943131.
  52. ^ Monroe, Don (2008-10-01). “Anyons: The breakthrough quantum computing needs?”New Scientist.
  53. ^ Dyakonov, Mikhail. “The Case Against Quantum Computing”IEEE Spectrum. Retrieved 3 December 2019.
  54. ^ Dyakonov, Mikhail (24 March 2020). Will We Ever Have a Quantum Computer?. Springer. ISBN 9783030420185. Retrieved 22 May 2020.[page needed]
  55. ^ Das, A.; Chakrabarti, B. K. (2008). “Quantum Annealing and Analog Quantum Computation”. Rev. Mod. Phys. 80 (3): 1061–1081. arXiv:0801.2193Bibcode:2008RvMP…80.1061DCiteSeerX 14255125.
  56. ^ Nayak, Chetan; Simon, Steven; Stern, Ady; Das Sarma, Sankar (2008). “Nonabelian Anyons and Quantum Computation”. Reviews of Modern Physics80 (3): 1083–1159. arXiv:0707.1889Bibcode:2008RvMP…80.1083Ndoi:10.1103/RevModPhys.80.1083S2CID 119628297.
  57. ^ Clarke, John; Wilhelm, Frank K. (18 June 2008). “Superconducting quantum bits”Nature453 (7198): 1031–1042. Bibcode:2008Natur.453.1031Cdoi:10.1038/nature07128PMID 18563154S2CID 125213662.
  58. ^ Kaminsky, William M.; Lloyd, Seth; Orlando, Terry P. (12 March 2004). “Scalable Superconducting Architecture for Adiabatic Quantum Computation”. arXiv:quant-ph/0403090Bibcode:2004quant.ph..3090K.
  59. ^ Khazali, Mohammadsadegh; Mølmer, Klaus (2020-06-11). “Fast Multiqubit Gates by Adiabatic Evolution in Interacting Excited-State Manifolds of Rydberg Atoms and Superconducting Circuits”Physical Review X10 (2): 021054. Bibcode:2020PhRvX..10b1054Kdoi:10.1103/PhysRevX.10.021054.
  60. ^ Henriet, Loic; Beguin, Lucas; Signoles, Adrien; Lahaye, Thierry; Browaeys, Antoine; Reymond, Georges-Olivier; Jurczak, Christophe (2020-06-22). “Quantum computing with neutral atoms”. Quantum4: 327. arXiv:2006.12326doi:10.22331/q-2020-09-21-327S2CID 219966169.
  61. ^ Imamog¯lu, A.; Awschalom, D. D.; Burkard, G.; DiVincenzo, D. P.; Loss, D.; Sherwin, M.; Small, A. (15 November 1999). “Quantum Information Processing Using Quantum Dot Spins and Cavity QED”. Physical Review Letters83 (20): 4204–4207. arXiv:quant-ph/9904096Bibcode:1999PhRvL..83.4204Idoi:10.1103/PhysRevLett.83.4204S2CID 18324734.
  62. ^ Fedichkin, L.; Yanchenko, M.; Valiev, K. A. (June 2000). “Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot”. Quantum Computers and Computing1: 58. arXiv:quant-ph/0006097Bibcode:2000quant.ph..6097F.
  63. ^ Ivády, Viktor; Davidsson, Joel; Delegan, Nazar; Falk, Abram L.; Klimov, Paul V.; Whiteley, Samuel J.; Hruszkewycz, Stephan O.; Holt, Martin V.; Heremans, F. Joseph; Son, Nguyen Tien; Awschalom, David D.; Abrikosov, Igor A.; Gali, Adam (6 December 2019). “Stabilization of point-defect spin qubits by quantum wells”Nature Communications10 (1): 5607. arXiv:1905.11801Bibcode:2019NatCo..10.5607Idoi:10.1038/s41467-019-13495-6PMC 6898666PMID 31811137.
  64. ^ “Scientists Discover New Way to Get Quantum Computing to Work at Room Temperature”interestingengineering.com. 24 April 2020.
  65. ^ Bertoni, A.; Bordone, P.; Brunetti, R.; Jacoboni, C.; Reggiani, S. (19 June 2000). “Quantum Logic Gates based on Coherent Electron Transport in Quantum Wires”. Physical Review Letters84 (25): 5912–5915. Bibcode:2000PhRvL..84.5912Bdoi:10.1103/PhysRevLett.84.5912hdl:11380/303796PMID 10991086.
  66. ^ Ionicioiu, Radu; Amaratunga, Gehan; Udrea, Florin (20 January 2001). “Quantum Computation with Ballistic Electrons”. International Journal of Modern Physics B15 (2): 125–133. arXiv:quant-ph/0011051Bibcode:2001IJMPB..15..125ICiteSeerX 119389613.
  67. ^ Ramamoorthy, A; Bird, J P; Reno, J L (11 July 2007). “Using split-gate structures to explore the implementation of a coupled-electron-waveguide qubit scheme”. Journal of Physics: Condensed Matter19 (27): 276205. Bibcode:2007JPCM…19A6205Rdoi:10.1088/0953-8984/19/27/276205.
  68. ^ Leuenberger, Michael N.; Loss, Daniel (April 2001). “Quantum computing in molecular magnets”. Nature410(6830): 789–793. arXiv:cond-mat/0011415Bibcode:2001Natur.410..789Ldoi:10.1038/35071024PMID 11298441S2CID 4373008.
  69. ^ Harneit, Wolfgang (27 February 2002). “Fullerene-based electron-spin quantum computer”. Physical Review A65 (3): 032322. Bibcode:2002PhRvA..65c2322Hdoi:10.1103/PhysRevA.65.032322.https://www.researchgate.net/publication/257976907_Fullerene-based_electron-spin_quantum_computer
  70. ^ K. Igeta and Y. Yamamoto. “Quantum mechanical computers with single atom and photon fields.” International Quantum Electronics Conference (1988) https://www.osapublishing.org/abstract.cfm?uri=IQEC-1988-TuI4
  71. ^ I.L. Chuang and Y. Yamamoto. “Simple quantum computer.” Physical Review A 52, 5, 3489 (1995) https://doi.org/10.1103/PhysRevA.52.3489
  72. ^ Knill, G. J.; Laflamme, R.; Milburn, G. J. (2001). “A scheme for efficient quantum computation with linear optics”Nature409 (6816): 46–52. Bibcode:2001Natur.409…46Kdoi:10.1038/35051009PMID 11343107S2CID 4362012.
  73. ^ Nizovtsev, A. P. (August 2005). “A quantum computer based on NV centers in diamond: Optically detected nutations of single electron and nuclear spins”Optics and Spectroscopy99 (2): 248–260. Bibcode:2005OptSp..99..233Ndoi:10.1134/1.2034610S2CID 122596827.
  74. ^ Dutt, M. V. G.; Childress, L.; Jiang, L.; Togan, E.; Maze, J.; Jelezko, F.; Zibrov, A. S.; Hemmer, P. R.; Lukin, M. D. (1 June 2007). “Quantum Register Based on Individual Electronic and Nuclear Spin Qubits in Diamond”. Science316 (5829): 1312–1316. Bibcode:2007Sci…316…..Ddoi:10.1126/science.1139831PMID 17540898S2CID 20697722Lay summary.
  75. ^ Neumann, P.; et al. (June 6, 2008). “Multipartite Entanglement Among Single Spins in Diamond”. Science320(5881): 1326–1329. Bibcode:2008Sci…320.1326Ndoi:10.1126/science.1157233PMID 18535240S2CID 8892596.
  76. ^ Anderlini, Marco; Lee, Patricia J.; Brown, Benjamin L.; Sebby-Strabley, Jennifer; Phillips, William D.; Porto, J. V. (July 2007). “Controlled exchange interaction between pairs of neutral atoms in an optical lattice”. Nature448 (7152): 452–456. arXiv:0708.2073Bibcode:2007Natur.448..452Adoi:10.1038/nature06011PMID 17653187S2CID 4410355Lay summary.
  77. ^ Ohlsson, N.; Mohan, R. K.; Kröll, S. (January 1, 2002). “Quantum computer hardware based on rare-earth-ion-doped inorganic crystals”. Opt. Commun201 (1–3): 71–77. Bibcode:2002OptCo.201…71Odoi:10.1016/S0030-4018(01)01666-2.
  78. ^ Longdell, J. J.; Sellars, M. J.; Manson, N. B. (September 23, 2004). “Demonstration of conditional quantum phase shift between ions in a solid”. Phys. Rev. Lett93 (13): 130503. arXiv:quant-ph/0404083Bibcode:2004PhRvL..93m0503Ldoi:10.1103/PhysRevLett.93.130503PMID 15524694S2CID 41374015.
  79. ^ Náfrádi, Bálint; Choucair, Mohammad; Dinse, Klaus-Peter; Forró, László (18 July 2016). “Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres”Nature Communications7 (1): 12232. arXiv:1611.07690Bibcode:2016NatCo…712232Ndoi:10.1038/ncomms12232PMC 4960311PMID 27426851.
  80. ^ Nielsen, p. 29
  81. ^ Nielsen, p. 126
  82. ^ Ozhigov, Yuri (1999). “Quantum Computers Speed Up Classical with Probability Zero”. Chaos, Solitons & Fractals10(10): 1707–1714. arXiv:quant-ph/9803064Bibcode:1998quant.ph..3064Odoi:10.1016/S0960-0779(98)00226-4.
  83. ^ Nielsen, p. 41
  84. ^ Nielsen, p. 201
  85. ^ Nielsen, p. 42
  86. Jump up to:a b Bernstein, Ethan; Vazirani, Umesh (1997). “Quantum Complexity Theory”SIAM Journal on Computing26 (5): 1411–1473. CiteSeerX
  87. ^ Aaronson, Scott. “Quantum Computing and Hidden Variables” (PDF).
  88. ^ Scott Aaronson (2005). “NP-complete Problems and Physical Reality”. ACM SIGACT News2005arXiv:quant-ph/0502072Bibcode:2005quant.ph..2072A. See section 7 “Quantum Gravity”: “[…] to anyone who wants a test or benchmark for a favorite quantum gravity theory,[author’s footnote: That is, one without all the bother of making numerical predictions and comparing them to observation] let me humbly propose the following: can you define Quantum Gravity Polynomial-Time? […] until we can say what it means for a ‘user’ to specify an ‘input’ and ‘later’ receive an ‘output’—there is no such thing as computation, not even theoretically.” (emphasis in original)
  89. ^ “D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation”. D-Wave. 2011-05-25. Retrieved 2011-05-30.
  90. https://www.ibm.com/quantum-computing/learn/what-is-quantum-computing/
  91. https://www.nature.com/articles/d41586-020-03237-w
  92. https://www.mckinsey.com/industries/financial-services/our-insights/how-quantum-computing-could-change-financial-services#