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)


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.


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  1. 1.
    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM Journal on Computing 26(5), 1411–1473 (1997); Earlier version in STOC 1993MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Simon, D.: On the power of quantum computation. SIAM Journal on Computing 26(5), 1474–1483 (1997); Earlier version in FOCS 1994MathSciNetCrossRefMATHGoogle 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 1994MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Evans, W.S., Schulman, L.J.: Signal propagation and noisy circuits. IEEE Trans. Inform. Theory 45(7), 2367–2373 (1999)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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

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