Advertisement

Quantum Computing

  • Viv KendonEmail author
Reference work entry
Part of the Encyclopedia of Complexity and Systems Science Series book series (ECSSS)

Glossary

Analog computing

Encoding into a continuous variable and processing in a continuous-time evolution.

Anyons

A type of quantum particle that can only exist in two dimensions and that has exotic statistics when two identical particles are exchanged.

Bit

A two-state classical system used to represent a binary digit, zero or one.

Bus

A communications channel in computer architecture to provide a high-speed link between different elements of memory or processing registers.

Bose-Einstein particles

Integer spin quantum particles like to be together: any number can occupy the same quantum state.

Classical computing

How we compute using classical logic and conventional computational devices.

Computability

What we can in principle compute with given physical resources (some things are uncomputable).

Computational complexity

How fast we can compute with given physical resources (some things are harder to compute than others).

Digital computing

Encoding into bits or qubitsand processing...

Bibliography

Books and Reviews

  1. For those still struggling with the concepts (which probably means most people without a physics degree or other formal study of quantum theory), there are plenty of popular science books and articles. Please dive in: it’s the way the world we all live in works, and there is no reason to not dig in deep enough to marvel at the way it fits together and puzzle with the best of us about the bits we can’t yet fathom.Google Scholar
  2. For those who want to learn the quantitative details and machinery of quantum computing, this is still the best textbook: Quantum Computation and Quantum Information: (10th Edition). Michael A. Nielsen, Isaac L. Chuang. ISBN 10: 1107002176 ISBN 13: 9781107002173. Publisher: CUP, Cambs., UKGoogle Scholar
  3. I have cited a number of accessible review articles and books in the primary literature. Especially useful among these are Venegas-Andraca (2012) on quantum versions of random walks; Lidar and Brun (2013), Devitt et al. (2009), and Paler and Devitt (2015) for quantum error correction; Pachos (2012) and Brennen and Pachos (2007) for topological quantum computing; and Brown et al. (2010) for quantum simulation.Google Scholar
  4. For the latest experimental details, the websites of the major academic and commercial players are the best up-to-date source of information. I have highlighted a few already in the main text, notably IBM Q http://research.ibm.com/ibm-q/ where you can use their demonstrator 5 and 16 qubit transmon quantum computers (current as of July 2017) and D-Wave Inc., https://www.dwavesys.com/ who build quantum annealers with thousands of superconducting qubits.
  5. Key academic research to watch includes Bristol Centre for Quantum Photonics. http://www.bristol.ac.uk/physics/research/quantum/ for photonic quantum processors and another online demonstrator; QuTech in Delft https://qutech.nl/; Google Santa Barbara John Martinis group http://web.physics.ucsb.edu/~mart JILA in Colorado https://jila.colorado.edu/research/quantum-information JQI in Maryland http://jqi.umd.edu/ for ion trap quantum simulators (and much else); and NQIT Oxford http://nqit.ox.ac.uk/ for modular ion trap quantum computers.
  6. Many of these websites include overviews and tutorials suitable for beginners.Google Scholar
  7. This is a fast-moving area, with major funding in the form of a European Union Quantum Technology Flagship, large national funding programs, and new companies starting up. Exciting developments are promised in the near future.Google Scholar

Primary Literature

  1. Aaronson S (2012) The complexity zoo. https://complexityzoo.uwaterloo.ca/Complexity_Zoo, a comprehensive cross-referenced list of computational complexity classes
  2. Abrams DS, Lloyd S (1999) Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys Rev Lett 83(24):5162. http://prola.aps.org/abstract/PRL/v83/i24/p5162_1CrossRefGoogle Scholar
  3. Aharonov D, Arad I (2006) The bqp-hardness of approximating the jones polynomial.  https://doi.org/10.1088/1367–2630/13/3/035019, arXiv:quant-ph/0605181
  4. Aharonov D, Ben-Or M (1996) Fault tolerant quantum computation with constant error. In: Proceedings of the 29th ACM STOC, ACM, NY, pp 176–188, arXiv:quantph/9611025
  5. Aharonov D, Ambainis A, Kempe J, Vazirani U (2001) Quantum walks on graphs. In: Proceedings of the 33rd annual ACM STOC, ACM, NY, pp 50–59, quant-ph/0012090
  6. Aharonov D, Jones V, Landau Z (2006) A polynomial quantum algorithm for approximating the Jones polynomial. In: STOC’06: Proceedings of the 38th annual ACM symposium on theory of computing, ACM, New York, pp 427–436.  https://doi.org/10.1145/1132516.1132579
  7. Aharonov D, van Dam W, Kempe J, Landau Z, Lloyd S, Regev O (2007) Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J Comput 37:166, arXiv:quant-ph/0405098
  8. Aharonov Y, Bohm D (1959) Significance of electromagnetic potentials in quantum theory. Phys Rev 115:485–491MathSciNetCrossRefGoogle Scholar
  9. Ambainis A (2003) Quantum walks and their algorithmic applications. Intl J Quantum Inf 1(4):507–518, ArXiv:quant-ph/0403120CrossRefGoogle Scholar
  10. Ambainis A (2004) Quantum walk algorithms for element distinctness. In: 45th annual IEEE symposium on foundations of computer science, Oct 17-19, 2004, IEEE computer society press, Los Alamitos, CA, pp 22–31, quant-ph/0311001
  11. Andersen UL, Neergaard-Nielsen JS, van Loock P, Furusawa A (2014) Hybrid quantum information processing. Nature Physics 11, 713–719 (2015).  https://doi.org/10.1038/nphys3410, arXiv:1409.3719
  12. Barenco A, Bennett CH, Cleve R, DiVincenzo DP, Margolus N, Shor P, Sleator T, Smolin JA, Weinfurter H (1995) Elementary gates for quantum computation. Phys Rev A 52(5):3457–3467.  https://doi.org/10.1103/PhysRevA.52.3457CrossRefGoogle Scholar
  13. Bartlett S, Sanders B, Braunstein SL, Moto KN (2002) Efficient classical simulation of continuous variable quantum information processes. Phys Rev Lett 88:097904, arXiv:quant-ph/0109047
  14. Bartlett SD, Rudolph T, Spekkens RW (2006) Reference frames, superselection rules, and quantum information. Rev Mod Phys 79:555, arXiv:quant-ph/0610030
  15. Bennett CH, Brassard G (1984) Quantum cryptography: public-key distribution and coin tossing. In: IEEE international conference on computers, systems and signal processing, IEEE Computer Society Press, Los Alamitos, CA, pp 175–179Google Scholar
  16. Bennett CH, Wiesner SJ (1992) Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys Rev Lett 69(20):2881–2884MathSciNetCrossRefGoogle Scholar
  17. Bennett CH, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters WK (1993) Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys Rev Lett 70:1895–1899MathSciNetCrossRefGoogle Scholar
  18. Bennett CH, Bernstein E, Brassard G, Vazirani U (1997) Strengths and weaknesses of quantum computing. SIAM J Comput 26(5):151–152MathSciNetCrossRefGoogle Scholar
  19. Bernien H, Schwartz S, Keesling A, Levine H, Omran A, Pichler H, Choi S, Zibrov AS, Endres M, Greiner M, Vuleti V, Lukin MD (2017) Probing many-body dynamics on a 51-atom quantum simulator. arXiv:1707.04344
  20. Berry DW, Ahokas G, Cleve R, Sanders BC (2007) Efficient quantum algorithms for simulating sparse hamiltonians. Commun Math Phys 270:359–371.  https://doi.org/10.1007/s00220-006-0150-x, http://springerlink.com/content/hk7484445j37r228/MathSciNetCrossRefzbMATHGoogle Scholar
  21. Blume-Kohout R, Caves CM, Deutsch IH (2002) Climbing mount scalable: physical resource requirements for a scalable quantum computer. Found Phys 32(11):1641–1670, ArXiv:quant-ph/0204157
  22. Bookatz AD, Farhi E, Zhou L (2014) Error suppression in hamiltonian based quantum computation using energy penalties. Phys. Rev. A 92, 022317.  https://doi.org/10.1103/PhysRevA.92.022317, arXiv:1407.1485
  23. Braunstein SL, van Loock P (2005) Quantum information with continuous variables. Rev Mod Phys 77:513–578, ArXiv:quant-ph/0410100v1MathSciNetCrossRefGoogle Scholar
  24. Bravyi S, DiVincenzo DP, Oliveira RI, Terhal BM (2006) The complexity of stoquastic local hamiltonian problems arXiv:quant-ph/0606140
  25. Brennen GK, Pachos JK (2007) Why should anyone care about computing with anyons? Proc Roy Soc Lond A 464(2089):1–24, ArXiv:0704.2241v2MathSciNetCrossRefGoogle Scholar
  26. Brown KR, Clark RJ, Chuang IL (2006) Limitations of quantum simulation examined by a pairing Hamiltonian using nuclear magnetic resonance. Phys Rev Lett 97(5):050504, http://link.aps.org/abstract/PRL/v97/e050504CrossRefGoogle Scholar
  27. Brown KL, Munro WJ, Kendon VM (2010) Using quantum computers for quantum simulation. Entropy 12(11):2268–2307.  https://doi.org/10.3390/e12112268MathSciNetCrossRefzbMATHGoogle Scholar
  28. Brown KL, Horsman C, Kendon VM, Munro WJ (2012) Layer by layer generation of cluster states. Phys Rev A 85:052305, http://arxiv.org/abs/1111.1774v1CrossRefGoogle Scholar
  29. Callison A, Chancellor NC, Kendon VM (2017) Continuous-time quantum walk algorithm for random spin-glass problems. In preparationGoogle Scholar
  30. Chancellor N (2016a) Modernizing quantum annealing ii: Genetic algorithms and inference. arXiv:1609.05875
  31. Chancellor N (2016b) Modernizing quantum annealing using local searches.  https://doi.org/10.1088/1367-2630/aa59c4, arXiv:1606.06833
  32. Childs A, Eisenberg JM (2005) Quantum algorithms for subset finding. Quantum Inf Comput 5:593–604, ArXiv:quant-ph/0311038zbMATHGoogle Scholar
  33. Childs A, Goldstone J (2004) Spatial search by quantum walk. Phys Rev A 70:022314, quant-ph/0306054CrossRefGoogle Scholar
  34. Childs AM (2009) Universal computation by quantum walk. Phys Rev Lett 102:180501MathSciNetCrossRefGoogle Scholar
  35. Childs AM, Farhi E, Preskill J (2002) Robustness of adiabatic quantum computation. Phys Rev A 65:012322, ArXiv:quant-ph/0108048CrossRefGoogle Scholar
  36. Childs AM, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman DA (2003) Exponential algorithmic speedup by a quantum walk. In: Proceedings of the 35th annual ACM STOC, ACM, NY, pp 59–68. arXiv:quant-ph/0209131
  37. Chun H, Choi I, Faulkner G, Clarke L, Barber B, George G, Capon C, Niskanen A, Wabnig J, OBrien D, Bitauld D (2016) Motion-compensated handheld quantum key distribution system. arXiv:1608.07465
  38. Cirac JI, Zoller P (1995) Quantum computations with cold trapped ions. Phys Rev Lett 74(20):4091.  https://doi.org/10.1103/PhysRevLett.74.4091, http://link.aps.org/abstract/PRL/v74/p4091CrossRefGoogle Scholar
  39. Coecke B, Edwards B, Spekkens RW (2010) Phase groups and the origin of non-locality for qubits.  https://doi.org/10.1016/j.entcs.2011.01.021, arXiv:1003.5005
  40. Collins RJ, Amiri R, Fujiwara M, Honjo T, Shimizu K, Tamaki K, Takeoka M, Andersson E, Buller GS, Sasaki M (2016) Experimental transmission of quantum digital signatures over 90-km of installed optical fiber using a differential phase shift quantum key distribution system. Opt Lett 41:4883.  https://doi.org/10.1364/OL.41.004883, arXiv:1608.04220CrossRefGoogle Scholar
  41. Courtland R (2016) Chinas 2,000-km quantum link is almost complete. IEEE Spectr. http://spectrum.ieee.org/telecom/security/chinas-2000km-quantum-link-is-almost-complete. iD Quantique, MagicQ
  42. Cubitt T, Montanaro A, Piddock S (2017) Universal quantum Hamiltonians. arXiv:1701.05182
  43. De Raedt K, Michielsen K, De Raedt H, Trieu B, Arnold G, Richter M, Lippert T, Watanabe H, Ito N (2007) Massive parallel quantum computer simulator. Comput Phys Commun 176:127–136, arXiv:quant-ph/0608239v1zbMATHGoogle Scholar
  44. Deutsch D (1985) Quantum-theory, the church-Turing principle and the universal quantum computer. Proc R Soc Lond A 400(1818):97–117MathSciNetCrossRefGoogle Scholar
  45. Deutsch D, Jozsa R (1992) Rapid solutions of problems by quantum computation. Proc Roy Soc Lon A 439:553MathSciNetCrossRefGoogle Scholar
  46. Devitt SJ, Nemoto K, Munro WJ (2009) Quantum error correction for beginners.  https://doi.org/10.1088/0034-4885/76/7/076001, arXiv:0905.2794
  47. Farhi E, Goldstone J, Gutmann S, Sipser M (2000) Quantum computation by adiabatic evolution. ArXiv:quant-ph/0001106
  48. Feynman RP (1982) Simulating physics with computers. Int J Theor Phys 21:467MathSciNetCrossRefGoogle Scholar
  49. Flitney AP, Abott D (2002) An introduction to quantum game theory. Fluctuation Noise Lett 02(04):R175–R187.  https://doi.org/10.1142/S0219477502000981MathSciNetCrossRefGoogle Scholar
  50. Gottesman D, Chuang IL (2001) Quantum digital signatures. http://arxiv.org/abs/quant-ph/0105032v2
  51. Greentree AD, Schirmer SG, Green F, Hollenberg LCL, Hamilton AR, Clark RG (2004) Maximizing the hilbert space for a finite number of distinguishable quantum states. Phys Rev Lett 92:097901, ArXiv:quant-ph/0304050CrossRefGoogle Scholar
  52. Grover LK (1996) A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th annual ACM STOC, ACM, NY, p 212, ArXiv:quant-ph/9605043
  53. Grover LK (1997) Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett 79:325, ArXiv:quant-ph/9706033CrossRefGoogle Scholar
  54. Hameroff S, Penrose R (1996) Conscious events as orchestrated spacetime selections. J Conscious Stud 3(1):36–53Google Scholar
  55. Hardy L (2001) Quantum theory from five reasonable axioms. arXiv:quant-ph/0101012
  56. Horsman C, Brown KL, Munro WJ, Kendon VM (2011) Reduce, reuse, recycle for robust cluster-state generation. Phys Rev A 83(4):042327. ArXiv:1005.1621[quant-ph]CrossRefGoogle Scholar
  57. Horsman C, Stepney S, Wagner RC, Kendon V (2014) When does a physical system compute? Proc Roy Soc A 470(2169):20140182, arXiv:1309.7979CrossRefGoogle Scholar
  58. Jozsa R (1998) Entanglement and quantum computation. In: Huggett SA, Mason LJ, Tod KP, Tsou S, Woodhouse NMJ (eds) The geometric universe, geometry, and the work of Roger Penrose. Oxford University Press, Oxford, pp 369–379zbMATHGoogle Scholar
  59. Jozsa R (2005) An introduction to measurement based quantum computation. ArXiv:quant-ph/0508124
  60. Kempe, Kitaev, Regev (2004) The complexity of the local hamiltonian problem. In: Proceedings of the 24th FSTTCS, pp 372–383. ArXiv:quant-ph/0406180
  61. Kempe, Kitaev, Regev (2006) The complexity of the local hamiltonian problem. SIAM J Comput 35(5):1070–1097MathSciNetCrossRefGoogle Scholar
  62. Kendon V, Tregenna B (2003) Decoherence can be useful in quantum walks. Phys Rev A 67:042315, ArXiv:quant-ph/0209005CrossRefGoogle Scholar
  63. Khrennikov A (2006) Brain as quantum-like computer. Biosystems 84:225–241, ArXiv:quant-ph/0205092v8CrossRefGoogle Scholar
  64. Kieu TD (2006) Quantum adiabatic computation and the travelling salesman problem, ArXiv:quant-ph/0601151v2
  65. Kitaev AY (2003) Fault-tolerant quantum computation by anyons. Ann Phys 303:2–30, ArXiv:quant-ph/9707021v1
  66. Kleinjung T, Aoki K, Franke J, Lenstra A, Thom E, Bos J, Gaudry P, Kruppa A, Montgomery P, Osvik DA, te Riele H, Timofeev A, Zimmermann P (2010) Factorization of a 768-bit rsa modulus. Cryptology ePrint Archive, Report 2010/006, http://eprint.iacr.org/2010/006
  67. Knill E, Laflamme R, Zurek W (1996) Threshold accuracy for quantum computation. ArXiv:quant-ph/9610011
  68. Kuhr S (2016) Quantum-gas microscopes – a new tool for cold-atom quantum simulators. Natl Sci Rev.  https://doi.org/10.1093/nsr/nww023, arXiv:1606.06990
  69. Ladd TD, van Loock P, Nemoto K, Munro WJ, Yamamoto Y (2006) Hybrid quantum repeater based on dispersive cqed interactions between matter qubits and bright coherent light. New J Phys 8:164.  https://doi.org/10.1088/1367-2630/8/9/184, ArXiv:quant-ph/0610154v1CrossRefGoogle Scholar
  70. Lidar DA, Brun TA (eds) (2013) Quantum error correction. Cambridge University Press, Cambridge, UKGoogle Scholar
  71. Lloyd S (1996) Universal quantum simulators. Science 273(5278):1073–1078MathSciNetCrossRefGoogle Scholar
  72. Lloyd S (2000) Ultimate physical limits to computation. Nature 406:1047–1054, ArXiv:quant-ph/9908043CrossRefGoogle Scholar
  73. Lloyd S, Braunstein SL (1999) Quantum computation over continuous variables. Phys Rev Lett 82:1784, ArXiv:quant-ph/9810082v1MathSciNetCrossRefGoogle Scholar
  74. Lomont C (2004) The hidden subgroup problem – review and open problems. arXiv:quant-ph/0411037
  75. Magniez F, Santha M, Szegedy M (2003) An o(n1.3) quantum algorithm for the triangle problem. ArXiv:quant-ph/0310134
  76. Magniez F, Santha M, Szegedy M (2005) Quantum algorithms for the triangle problem. In: Proceedings of 16th ACM-SIAM symposium on discrete algorithms, society for industrial and applied mathematics, Philadelphia, pp 1109–1117Google Scholar
  77. Margolus N, Levitin LB (1996) The maximum speed of dynamical evolution. In: Toffoli T, Biafore M, Liao J (eds) Physcomp96. NECSI, BostonGoogle Scholar
  78. Margolus N, Levitin LB (1998) The maximum speed of dynamical evolution. Physica D 120:188–195, ArXiv:quant-ph/9710043v2CrossRefGoogle Scholar
  79. Metodi TS, Thaker DD, Cross AW, deric T Chong F, Chuang IL (2005) A quantum logic array microarchitecture: scalable quantum data movement and computation. In: 38th annual IEEE/ACM international symposium on microarchitecture (MICRO’05), IEEE Computer Society Press, Los Alamitos, CA, pp 305–318. ArXiv:quant-ph/0509051v1
  80. Montanaro A (2015) Quantum algorithms: an overview.  https://doi.org/10.1038/npjqi.2015.23. arXiv:1511.04206
  81. Morley JG, Chancellor NC, Kendon VM, Bose S (2017a) Quantum search with hybrid adiabatic quantum-walk algorithms and realistic noise. https://arxiv.org/abs/1709.00371
  82. Morley JG, Chancellor NC, Kendon VM, Bose S (2017b) Quench vs adiabaticity: which is best for quantum search on realistic machines? In preparationGoogle Scholar
  83. Moses A, Covey P, Miecnikowski T, Jin DS, Ye J (2017) New frontiers for quantum gases of polar molecules. Nat Phys 13:13–20. http://www.nature.com/doifinder/10.1038/nphys3985CrossRefGoogle Scholar
  84. Neyenhuis B, Smith J, Lee AC, Zhang J, Richerme P, Hess PW, Gong ZX, Gorshkov AV, Monroe C (2016) Observation of prethermalization in long-range interacting spin chains. arXiv:1608.00681
  85. Nickerson NH, Fitzsimons JF, Benjamin SC (2014) Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links. Phys Rev X 4.  https://doi.org/10.1103/PhysRevX.4.041041, arXiv:1406.0880
  86. Nielsen M, Chuang I (1996) Talk at KITP workshop: quantum coherence and decoherence D. P. DiVencenzo, W. Zurek. http://www.kitp.ucsb.edu/activities/conferences/past/
  87. Nielsen MA (2004) Optical quantum computation using cluster states. Phys Rev Lett 93:040503CrossRefGoogle Scholar
  88. Pachos JK (2012) Introduction to topological quantum computation. Cambridge University Press, Cambs., UK. ISBN 9781107005044 1107005043CrossRefGoogle Scholar
  89. Paler A, Devitt SJ (2015) An introduction to fault-tolerant quantum computing. In: DAC’15 Proceedings of the 52nd annual design automation conference, p 60. arXiv:1508.03695
  90. Parekh O, Wendt J, Shulenburger L, Landahl A, Moussa J, Aidun J (2016) Benchmarking adiabatic quantum optimization for complex network analysis. Report number SAND2015–3025. arXiv:1604.00319
  91. Preskill J (1997) Fault-tolerant quantum computation. Check and update reference. arXiv:quant-ph/9712048
  92. Raussendorf R, Briegel HJ (2001) A one-way quantum computer. Phys Rev Lett 86(22):5188–5191.  https://doi.org/10.1103/PhysRevLett.86.5188CrossRefGoogle Scholar
  93. Raussendorf R, Browne DE, Briegel HJ (2003) Measurement-based quantum computation on cluster states. Phys Rev A 68(2):022312.  https://doi.org/10.1103/PhysRevA.68.022312CrossRefGoogle Scholar
  94. Richter P (2007a) Almost uniform sampling in quantum walks. New J Phys 9:72, ArXiv:quant-ph/0606202CrossRefGoogle Scholar
  95. Richter P (2007b) Quantum speedup of classical mixing processes. Phys Rev A 76:042306, ArXiv:quant-ph/0609204CrossRefGoogle Scholar
  96. Shenvi N, Kempe J, Birgitta Whaley K (2003) A quantum random walk search algorithm. Phys Rev A 67:052307, ArXiv:quant-ph/0210064CrossRefGoogle Scholar
  97. Shor P (1994) Algorithms for quantum computation: discrete logarithms and factoring. In: foundations of computer science, 1994 proceedings., 35th annual symposium on, IEEE Computer Society Press, Los Alamitos, pp 124–134.  https://doi.org/10.1109/SFCS.1994.365700
  98. Shor PW (1995) Scheme for reducing decoherence in quantum computer memory. Phys Rev A 52:R2493CrossRefGoogle Scholar
  99. Shor PW (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Sci Statist Comput 26:1484, quant-ph/9508027MathSciNetCrossRefGoogle Scholar
  100. Spekkens RW (2004) In defense of the epistemic view of quantum states: a toy theory.  https://doi.org/10.1103/PhysRevA.75.032110. arXiv:quant-ph/0401052
  101. Spiller TP, Nemoto K, Braunstein SL, Munro WJ, van Loock P, Milburn GJ (2006) Quantum computation by communication. New J Phys 8:30, ArXiv:quant-ph/0509202v3CrossRefGoogle Scholar
  102. Steane A (1996) Multiple particle interference and quantum error correction. Proc Roy Soc Lond A 452:2551, ArXiv:quant-ph/9601029
  103. Steffen M, van Dam W, Hogg T, Breyta G, Chuang I (2003) Experimental implementation of an adiabatic quantum optimization algorithm. Phys Rev Lett 90(6):067903, ArXiv:quant-ph/0302057CrossRefGoogle Scholar
  104. Venegas-Andraca SE (2012) Quantum walks: a comprehensive review. Quantum Inf Process 11(5):1015–1106.  https://doi.org/10.1007/s11128-012-0432-5, arXiv:1201.4780MathSciNetCrossRefzbMATHGoogle Scholar
  105. Verstraete F, Porras D, Cirac JI (2004) Density matrix renormalization group and periodic boundary conditions: a quantum information perspective. Phys Rev Lett 93:227205, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.93.227205
  106. Wang L, Piorn I, Verstraete F (2011) Monte carlo simulation with tensor network states. Phys Rev B 83:134421CrossRefGoogle Scholar
  107. Yoran N, Reznik B (2003) Deterministic linear optics quantum computation with single photon qubits. Phys Rev Lett 91:037903CrossRefGoogle Scholar
  108. Young T (1804) Experimental demonstration of the general law of the interference of light. Phil Trans Royal Soc Lon 94Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.JQC and Atmol, Department of PhysicsDurham UniversityDurhamUK

Personalised recommendations