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Quantum Error Correction

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 397 Accesses

Years and Authors of Summarized Original Work

  • 1995; Shor

Problem Definition

A quantum system can never be seen as being completely isolated from its environment, thereby permanently causing disturbance to the state of the system. The resulting noise problem threatens quantum computers and their great promise, namely, to provide a computational advantage over classical computers for certain problems (see also the cross-references in the section “Cross-References”). Quantum noise is usually modeled by the notion of a quantum channel which generalizes the classical case and, in particular, includes scenarios for communication (space) and storage (time) of quantum information. For more information about quantum channels and quantum information in general, see [19]. A basic channel is the quantum mechanical analog of the classical binary symmetric channel [17]. This quantum channel is called the depolarizing channel and depends on a real parameter p∈ [0, 1]. Its effect is to randomly...

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Recommended Reading

  1. Aharonov D, Kitaev A, Preskill J (2006) Fault-tolerant quantum computation with long-range correlated noise. Phys Rev Lett 96:050504

    Article  Google Scholar 

  2. Ashikhmin A, Litsyn S, Tsfasman MA (2001) Asymptotically good quantum codes. Phys Rev A 63:032311

    Article  Google Scholar 

  3. Bennett CH, DiVincenzo DP, Smolin JA, Wootters WK (1996) Mixed-state entanglement and quantum error correction. Phys Rev A 54:3824–3851

    Article  MathSciNet  Google Scholar 

  4. Calderbank AR, Shor PW (1996) Good quantum error-correcting codes exist. Phys Rev A 54:1098–1105

    Article  Google Scholar 

  5. Calderbank AR, Rains EM, Shor PW, Sloane NJA (1998) Quantum error correction via codes over GF(4). IEEE Trans Inf Theory 44:1369–1387

    Article  MathSciNet  MATH  Google Scholar 

  6. Chiaverini J, Leibfried D, Schaetz T, Barrett MD, Blakestad RB, Britton J, Itano WM, Jost JD, Knill E, Langer C, Ozeri R, Wineland DJ (2004) Realization of quantum error correction. Nature 432:602–605

    Article  MATH  Google Scholar 

  7. Fowler AG, Mariantoni M, Martinis JM, Cleland AN (2012) Surface codes: towards practical large-scale quantum computation. Phys Rev A 86:032324

    Article  Google Scholar 

  8. Gottesman D (1996) Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys Rev A 54:1862–1868

    Article  MathSciNet  Google Scholar 

  9. Gottesman D (1997) Stabilizer codes and quantum error correction, Ph.D. thesis, Caltech. See also: arXiv preprint quant-ph/9705052

    Google Scholar 

  10. Gottesman D (2013) Fault-tolerant quantum computation with constant overhead. arXiv.org preprint, arXiv:1310.2984

    Google Scholar 

  11. Hastings MB (2014) Decoding in hyperbolic spaces: quantum LDPC codes with linear rate and efficient error correction. Quantum Inf Comput 14:1187–1202

    MathSciNet  Google Scholar 

  12. Kelly J, Barends R, Fowler AG, Megrant A, Jeffrey E, White TC, Sank D, Mutus JY, Campbell B, Chen Y, Chen Z, Chiaro B, Dunsworth A, Hoi I-C, Neill C, O’Malley PJJ, Quintana C, Roushan P, Vainsencher A, Wenner J, Cleland AN, Martinis JM (2014) State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519:66–69

    Article  Google Scholar 

  13. Knill E, Laflamme R (1997) Theory of quantum error-correcting codes. Phys Rev A 55:900–911

    Article  MathSciNet  Google Scholar 

  14. Knill E, Laflamme R, Martinez R, Negrevergne C (2001) Benchmarking quantum computers: the five-qubit error correcting code. Phys Rev Lett 86:5811–5814

    Article  Google Scholar 

  15. Kovalev AA, Pryadko LP (2013) Fault-tolerance of “bad” quantum low-density parity check codes. Phys Rev A 87:020304(R)

    Google Scholar 

  16. Kribs D, Laflamme R, Poulin D (2005) Unified and generalized approach to quantum error correction. Phys Rev Lett 94(4):180501

    Article  Google Scholar 

  17. MacWilliams FJ, Sloane NJA (1977) The theory of error-correcting codes. North-Holland, Amsterdam

    MATH  Google Scholar 

  18. Monroe C, Raussendorf R, Ruthven A, Brown KR, Maunz P, Duan L-M, Kim J (2014) Large scale modular quantum computer architecture with atomic memory and photonic interconnects. Phys Rev A 89:022317

    Article  Google Scholar 

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

    MATH  Google Scholar 

  20. Shor PW (1995) Scheme for reducing decoherence in quantum computer memory. Phys Rev A 52:R2493–R2496

    Article  Google Scholar 

  21. Steane A (1996) Error correcting codes in quantum theory. Phys Rev Lett 77:793–797

    Article  MathSciNet  MATH  Google Scholar 

  22. Steane A (1996) Multiple-particle interference and quantum error correction. Proc R Soc Lond A 452:2551–2577

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Martin Roetteler

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  1. Martin Roetteler
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Correspondence to Martin Roetteler .

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Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Roetteler, M. (2016). Quantum Error Correction. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_315

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