Upper Bounds on the Noise Threshold for Fault-Tolerant Quantum Computing

  • Julia Kempe
  • Oded Regev
  • Falk Unger
  • Ronald de Wolf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)

Abstract

We prove new upper bounds on the tolerable level of noise in a quantum circuit. Our circuits consist of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, and arbitrary one-qubit gates that are essentially noise-free. We assume the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for \(p>1-{\it \Theta}(1/\sqrt{k})\), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k = 2, our bound is p > 35.7%. Moreover, if the only gate on more than one qubit is the CNOT, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet incomparable because of the somewhat different circuit model that we are using. Our main technique is a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM Journal on Computing 26(5), 1411–1473 (1997); Earlier version in STOC 1993MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Simon, D.: On the power of quantum computation. SIAM Journal on Computing 26(5), 1474–1483 (1997); Earlier version in FOCS 1994MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26(5), 1484–1509 (1997); Earlier version in FOCS 1994MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of 28th ACM STOC, pp. 212–219 (1996)Google Scholar
  5. 5.
    Shor, P.W.: Scheme for reducing decoherence in quantum memory. Physical Review A 52, 2493 (1995)CrossRefGoogle Scholar
  6. 6.
    Shor, P.W.: Fault-tolerant quantum computation. In: 37th FOCS, pp. 56–65 (1996)Google Scholar
  7. 7.
    Steane, A.: Multiple particle interference and quantum error correction. Proceedings of the Royal Society of London 452, 2551–2577 (1996)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Knill, M., Laflamme, R., Zurek, W.: Accuracy threshold for quantum computation (October 15, 1996) quant-ph/9610011Google Scholar
  9. 9.
    Knill, E., Laflamme, R., Zurek, W.H.: Resilient quantum computation. Science 279(5349), 342–345 (1998)CrossRefGoogle Scholar
  10. 10.
    Aharonov, D., Ben-Or, M.: Fault tolerant quantum computation with constant error. In: Proceedings of 29th ACM STOC, pp. 176–188 (1997)Google Scholar
  11. 11.
    Kitaev, A.Y.: Quantum computations: Algorithms and error correction. Russian Mathematical Surveys 52(6), 1191–1249 (1997)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gottesman, D.: Stabilizer Codes and Quantum Error Correction. PhD thesis, Caltech (1997) quant-ph/9702052Google Scholar
  13. 13.
    Knill, M.: Quantum computing with realistically noisy devices. Nature 434, 39–44 (2005)CrossRefGoogle Scholar
  14. 14.
    Knill, M.: Fault-tolerant postselected quantum computation: Threshold analysis (April 19, 2004) quant-ph/0404104Google Scholar
  15. 15.
    Aliferis, P., Gottesman, D., Preskill, J.: Accuracy threshold for postselected quantum computation. Quantum Information and Computation 8(3), 181–244 (2008)MATHMathSciNetGoogle Scholar
  16. 16.
    Aliferis, P.: Threshold lower bounds for Knill’s Fibonacci scheme (September 22, 2007) quant-ph/0709.3603Google Scholar
  17. 17.
    Aliferis, P.: Level Reduction and the Quantum Threshold Theorem. PhD thesis, Caltech (2007) quant-ph/0703264Google Scholar
  18. 18.
    Reichardt, B.: Error-Detection-Based Quantum Fault Tolerance Against Discrete Pauli Noise. PhD thesis, UC Berkeley (2006) quant-ph/0612004Google Scholar
  19. 19.
    Buhrman, H., Cleve, R., Laurent, M., Linden, N., Schrijver, A., Unger, F.: New limits on fault-tolerant quantum computation. In: 47th FOCS, pp. 411–419 (2006)Google Scholar
  20. 20.
    Razborov, A.: An upper bound on the threshold quantum decoherence rate. Quantum Information and Computation 4(3), 222–228 (2004)MATHMathSciNetGoogle Scholar
  21. 21.
    Kempe, J., Regev, O., Unger, F., de Wolf, R.: Upper bounds on the noise threshold for fault-tolerant quantum computing (2008) quant-ph/0802.1464Google Scholar
  22. 22.
    Barenco, A., Bennett, C., Cleve, R., DiVincenzo, D., Margolus, N., Shor, P., Sleator, T., Smolin, J., Weinfurter, H.: Elementary gates for quantum computation. Physical Review A 52, 3457–3467 (1995)CrossRefGoogle Scholar
  23. 23.
    Bravyi, S., Kitaev, A.: Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A 71 (022316) (2005)Google Scholar
  24. 24.
    Reichardt, B.: Quantum universality from Magic States Distillation applied to CSS codes. Quantum Information Processing 4, 251–264 (2005)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Evans, W.S., Schulman, L.J.: Signal propagation and noisy circuits. IEEE Trans. Inform. Theory 45(7), 2367–2373 (1999)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Evans, W.S., Schulman, L.J.: On the maximum tolerable noise of k-input gates for reliable computation by formulas. IEEE Trans. Inform. Theory 49(11), 3094–3098 (2003)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Reichardt, B.: Quantum universality by distilling certain one- and two-qubit states with stabilizer operations (2006) quant-ph/0608085Google Scholar
  28. 28.
    Virmani, S., Huelga, S., Plenio, M.: Classical simulability, entanglement breaking, and quantum computation thresholds. Physical Review A 71 (042328) (2005)Google Scholar
  29. 29.
    Bruss, D., DiVincenzo, D., Ekert, A., Fuchs, C., Macchiavello, C., Smolin, J.: Optimal universal and state-dependent quantum cloning. Physical Review A 43, 2368–2378 (1998)CrossRefGoogle Scholar
  30. 30.
    Ruskai, M.B., Szarek, S., Werner, E.: An analysis of completely-positive trace-preserving maps on \({\cal M}_2\). Linear Algebra and its Applications 347, 159–187 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Julia Kempe
    • 1
  • Oded Regev
    • 1
  • Falk Unger
    • 2
  • Ronald de Wolf
    • 2
  1. 1.Department of Computer ScienceTel-Aviv UniversityTel-AvivIsrael
  2. 2.CWIAmsterdamThe Netherlands

Personalised recommendations