Journal of Electronic Testing

, Volume 26, Issue 5, pp 499–511 | Cite as

Fault Models for Quantum Mechanical Switching Networks

  • Jacob D. BiamonteEmail author
  • Jeff S. Allen
  • Marek A. Perkowski


In classical test and verification one develops a test set separating a correct circuit from a circuit containing any considered fault. Classical faults are modelled at the logical level by fault models that act on classical states. The stuck fault model, thought of as a lead connected to a power rail or to a ground, is most typically considered. A classical test set complete for the stuck fault model propagates both binary basis states, 0 and 1, through all nodes in a network and is known to detect many physical faults. A classical test set complete for the stuck fault model allows all circuit nodes to be completely tested and verifies the function of many gates. It is natural to ask if one may adapt any of the known classical methods to test quantum circuits. Of course, classical fault models do not capture all the logical failures found in quantum circuits. The first obstacle faced when using methods from classical test is developing a set of realistic quantum-logical fault models (a question which we address, but will likely remain largely open until the advent of the first quantum computer). Developing fault models to abstract the test problem away from the device level motivated our study. Several results are established. First, we describe typical modes of failure present in the physical design of quantum circuits. From this we develop fault models for quantum binary quantum circuits that enable testing at the logical level. The application of these fault models is shown by adapting the classical test set generation technique known as constructing a fault table to generate quantum test sets. A test set developed using this method will detect each of the considered faults.


Testing Fault modeling Diagnostics Error-checking Reversible computers Quantum computers Formal test and verification 



The Quantum \ in LATEX using Q-circuit [16]. The simulation tool QuIDDPro [55] was used during this study. We would like to thank G.F. Viamontes, D. Maslov, J.A. Jones and P.J. Love.


  1. 1.
    Aharonov D, Ben-Or M (1998) Fault-tolerant quantum computation with constant error rate. In: Proc 29th ann ACM symp on theory of computing, New York, p 176. quant-ph/9611025; quant-ph/9906129
  2. 2.
    Allen JS, Biamonte JD, Perkowski MA (2005) ATPG for reversible circuits using technology-related fault models. In: Proc 7th international symposium on representations and methodology of future computing technologies, RM2005, Tokyo, Japan, 5–6 September 2005, 8 ppGoogle Scholar
  3. 3.
    Amin MHS, Grajcar M, II’ichev’ E, Maassen van den Bringk AM, Rose G, Smirnov AY, Zagoskin AM (2004) Superconducting quantum storage and processing. In: IEEE interational solid-state circuits conference, ISSCC, Session 16, 10 ppGoogle Scholar
  4. 4.
    Anwar MS, Bazina D, Carteret H, Duckett SB, Halstead TK, Jones JA, Kozak CM, Taylor RJK (2004) Preparing high purity initial states for nuclear magnetic resonance quantum computing. Phys Rev Lett 93:040501. quant-ph/0312014, 3 ppGoogle Scholar
  5. 5.
    Barenco A, Bennett CH, Cleve R, DiVincenzo DP, Margolus N, Shor PW, Sleator T, Smolin J, Weinfurter H (1995) Elementary gates of quantum computation. Phys Rev A 52(5):3457–3467. quant-ph/9503016, 31 ppGoogle Scholar
  6. 6.
    Barenco A, Brun TA, Schack R, Spiller TP (1998) Effects of noise on quantum error correction algorithms. Mod Phys Lett A 13:2503–2512. quant-ph/9612047 Google Scholar
  7. 7.
    Bettelli S (2004) Quantitative model for the effective decoherence of a quantum computer with imperfect unitary operations. Phys Rev A 69:042310. quant-ph/0310152, 14 pp
  8. 8.
    Biamonte JD, Perkowski MA (2004) Testing a quantum computer. In: Proceedings of KIAS-KAIST 5th workshop on quantum information science, Seoul Korea, 29–31 August 2004, p 16Google Scholar
  9. 9.
    Bowdrey MD, Jones JA (2001) A simple and convenient measure of NMR rotor fidelity. JAJ-QP-01-01. quant-ph/0103060
  10. 10.
    Childs AM, Preskill J, Renes J (2000) Quantum information and precision measurement. J Mod Opt 47:155–176. quant-ph/9904021 Google Scholar
  11. 11.
    Childs AM, Chuang IL, Leung DW (2001) Realization of quantum process tomography in NMR. Phys Rev A 64:012314. quant-ph/0012032, 8 pp
  12. 12.
    Chuang IL, Nielsen MA (1997) Prescrition for experimental determination of the dynamics of a quantum black box. J Mod Opt 44:2455. quant-ph/9610001, 6 ppGoogle Scholar
  13. 13.
    Cummins HK, Jones JA (2000) Use of composite rotations to correct systematic errors in NMR quantum computation. New J Phys 2:6.1–6.12. quant-ph/9911072, 11 pp
  14. 14.
    Cummins HK, Llewellyn G, Jones JA (2003) Tackling systematic errors in quantum logic gates with composite rotation. Phys Rev A 67:042308. quant-ph/0208092, 7 ppGoogle Scholar
  15. 15.
    Dodd JL, Nielsen MA (2002) A simple operational interpretation of the fidelity. Phys Rev A 66:044301. quant-ph/0111053, 1 pGoogle Scholar
  16. 16.
    Eastin B, Flammia ST (2004) Q-circuit tutorial. Free online, quant-ph/0406003, 7 pp
  17. 17.
    Gilchrist A, Langford NK, Nielsen MA (2005) Distance measures to compare real and ideal quantum processes. Phys Rev A 71:062310. quant-ph/0408063, 14 ppGoogle Scholar
  18. 18.
    Grover LK (2005) A different kind of quantum search. quant-ph/0503205, 13 pp
  19. 19.
    Hayes JP, Polian I, Becker B (2004) Testing for missing-gate faults in reversible circuits. In: Proc Asian test symposium, Taiwan, pp 100–105Google Scholar
  20. 20.
    Howard P (2004) Nuclear magnetic resonance quantum computation. In: Esteve D, Raimond J-M, Dalibard J (eds) Quantum entanglement and information processing. Elsevier.
  21. 21.
    James DFV, Kwiat PG, Munro WJ, White AG (2001) On the measurement of qubits. Phys Rev A 64:052312. quant-ph/0103121, 21 pp
  22. 22.
    Jones JA (2001) NMR quantum computing. In: Quantum computation and quantum information theory. World Scientific, Singapore.
  23. 23.
    Jones JA, Knill E (1999) Efficient refocussing of one spin and two spin interactions for NMR quantum computation. J Magn Reson 141:322–325. quant-ph/9905008 Google Scholar
  24. 24.
    Jones JA, Mosca M (1998) Implementation of a quantum algorithm to solve Deutsch’s problem on a nuclear magnetic resonance quantum computer. J Chem Phys 109:1648–1653. quant-ph/9801027 Google Scholar
  25. 25.
    Jones J, Mosca M (1999) Approximate quantum counting on an NMR ensemble quantum computer. Phys Rev Lett 83:1050. quant-ph/9808056, 4 ppGoogle Scholar
  26. 26.
    Jones JA, Hansen RH, Mosca M (1998) Quantum logic gates and nuclear magnetic resonance pulse sequences. J Magn Reson 135:353–360. quant-ph/9805070 Google Scholar
  27. 27.
    Jones J, Mosca M, Hansen R (1998) Implementation of a quantum search algorithm on a nuclear magnetic resonance quantum computer. Nature 393:344–346. quant-ph/9805069 Google Scholar
  28. 28.
    Kak S (1999) The initialization problem in quantum computing. Found Phys 29:267–279. quant-ph/9805002 Google Scholar
  29. 29.
    Kalay U, Perkowski MA, Hall DV (2000) A minimal universal test set for self-test of EXOR-sum-of-product circuits. IEEE Trans Comput 49(3):267–276CrossRefGoogle Scholar
  30. 30.
    Kautz W (1961) Automatic fault detection in combinatoinal switching networks. In: Proc AIEE 2nd switching circuit theory and logical design symp, pp 195–214Google Scholar
  31. 31.
    Kautz WH (1971) Testing faults in combinational cellular logic arrays. In: Proceedings of 8th annu symp switching and automata theory, pp 161–174Google Scholar
  32. 32.
    Kim K, Song M, Lee S, Lee J-S (2005) Quantum process tomography with arbitary number of ancillary qubits in nuclear magnetic resonance. J Korean Phys Soc 47:736–739Google Scholar
  33. 33.
    Knill E, Laflamme R, Zurek WH (1997) Resilient quantum computation: error models and thresholds. In: Proc mathematical, physical engineering sciences, vol 454, pp 365–384. quant-ph/9702058
  34. 34.
    Knill E, Laflamme R, Ashikhmin A, Barnum H, Viola L, Zurek WH (2002) Introduction to quantum error correction. LA Science 27:188–225. quant-ph/0207170, 22 ppGoogle Scholar
  35. 35.
    Lee S, Lee JS, Kim T, Biamonte JD, Perkowski MA (2005) The cost of quantum gate primitives. J Mult-Valued Log Soft ComputGoogle Scholar
  36. 36.
    Leung DW (2000) Towards robust quantum computation. PhD Dissertation, Stanford University, July 2000. cs/0012017, 243 pp
  37. 37.
    Maslov D, Young C, Miller DM, Dueck GW (2005) Quantum circuit simplification using templates. In: Proc DATE conference, Munich, Germany, pp 1208–1213Google Scholar
  38. 38.
    McCluskey EJ, Tseng CW (2000) Stuck-fault tests vs actual defects. In: Proc 2000 int test conf, Atlantic City, pp 336–343Google Scholar
  39. 39.
    Nielsen MA (1998) Quantum information theory. PhD thesis, University of New Mexico, Report UNM-98-08. quant-ph/0011036, 259 pp
  40. 40.
    Nielsen MA (2002) A simple formula for the average gate fidelity of a quantum dynamical operation. Phys Lett A 303(4):249–252. quant-ph/0205035 Google Scholar
  41. 41.
    Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University PressGoogle Scholar
  42. 42.
    Obenland KM, Despain AM (1996) Impact of errors on a quantum computer architecture. Technical report, Information Sciences Institute, University of Southern California, 1 October 1996.
  43. 43.
    O’Brien JL, Pryde GJ, White AG, Ralph TC, Branning D (2003) Demonstration of an all-optical quantum controlled-NOT gate. Nature 426:264. quant-ph/0403062, 5 ppGoogle Scholar
  44. 44.
    O’Brien JL, Pryde GJ, Gilchrist A, James DFV, Langford NK, Ralph TC, White AG (2004) Quantum process tomography of a controlled-NOT gate. Phys Rev Lett 93:080502. quant-ph/0402166, 4 ppGoogle Scholar
  45. 45.
    Oskin M (2004) Quantum computing lecture notes. Class notes, University of Washington.
  46. 46.
    Patel KN, Hayes JP, Markov IL (2004) Fault testing for reversible circuits. IEEE Trans CAD 23(8):1220–1230. quant-ph/0404003 Google Scholar
  47. 47.
    Perkowski MA et al (2005) Test generation and fault localization for quantum circuits. In: Proc 35th ISMVL, pp 62–68. doi:10.1109/ISMVL.2005.46
  48. 48.
    Reichardt BW, Grover LK (2005) Quantum error correction of systematic errors using a quantum search framework. Available online at quant-ph/0506242, 6 pp
  49. 49.
    Shenvi N, Brown KR, Whaley KB (2003). Effects of random noisy oracle on search algorithm complexity. Phys Rev A 68:052313. quant-ph/0304138, 11 ppGoogle Scholar
  50. 50.
    Shor PW (1996) Fault-tolerant quantum computation. In: 37th symposium on foundations of computing, vol 37. IEEE Computer Society Press, pp 56–65. quant-ph/9605011
  51. 51.
    Shukla SK, Kam R, Goldstein SC, Brewer F, Banejee K, Basu S (2003) Nano, quantum, and molecular computing: are we ready for the validation and test challenges? In: IEEE pannel disscussion, 0-7803-8236-6, pp 3–7.
  52. 52.
    Steane AM, Lucas DM (2000) Quantum computing with trapped ions, atoms and light. Fortschr Phys (special issue). quant-ph/0004053, 17 pp
  53. 53.
    Steffen M, Vandersypen LMK, Chuang IL (2001) Toward quantum computation: a five-qubit quantum processor. IEEE MICRO 21(2):24–34. doi:10.1109/40.918000 CrossRefGoogle Scholar
  54. 54.
    Vandersypen LMK, Yannoni CS, Chuang IL (2002) Liquid state NMR quantum computing. In: Encyclopedia of nuclear magnetic resonance, vol 9. Advances in NMR, pp 687–697.
  55. 55.
    Viamontes GF, Markov IL, Hayes JP (2005) Graph-based simulation of quantum computation in the density matrix representation. Quantum Information and Computation 5(2):113–130zbMATHMathSciNetGoogle Scholar
  56. 56.
    White AG, Gilchrist A, Pryde GJ, O’Brien JL, Bremner MJ, Langford NK (2003) Measuring controlled-NOT and two-qubit gate operation. quant-ph/0308115, 10 pp
  57. 57.
    Williams CP, Clearwater SH (1998) Explorations in quantum computing. SpringerGoogle Scholar
  58. 58.
    Xiao L, Jones JA (2005) Error tolerance in an NMR implementation of Grover’s fixed-point quantum search algorithm. Phys Rev A 72:032326. doi: 10.1103/PhysRevA.72.032326 zbMATHCrossRefGoogle Scholar
  59. 59.
    Zurek WH (1984) Reversibility and stability of information processing systems. Phys Rev Lett 53:391–394. doi:10.1103/PhysRevLett.53.391 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Jacob D. Biamonte
    • 1
    Email author
  • Jeff S. Allen
    • 1
  • Marek A. Perkowski
    • 1
  1. 1.Portland State UniversityPortlandUSA

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