By "faulty", I mean that you can have errors on the ancilla qubits, you can have faulty syndrome extraction, etc.

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    $\begingroup$ The key phrase you’re looking for is “fault tolerant threshold”. $\endgroup$ – DaftWullie Jul 20 '18 at 18:35
  • $\begingroup$ @DaftWullie, Yes, except that a lot of references (especially early references) ignore that the error correcting steps themselves can be faulty. I'm looking for some verification that a threshold still exists when error correcting steps are themselves faulty. There are fault-tolerant syndrome extraction approaches, but they require additional resources/time, potentially increasing chances for error to be introduced. I want to see a clear analysis of when a threshold exists under realistic conditions ... $\endgroup$ – Sherif F. Jul 21 '18 at 19:41

This is the paper I found really convincing once I’d worked through it: https://arxiv.org/abs/quant-ph/0504218 This was really the start of rigorous threshold proofs, and is the starting point for a number of subsequent improvements.


This paper by one of my group mates includes faulty prep on ancilla qubits and other errors in an analysis of Bacon Shor. I'm not sure what you mean by faulty error extraction (measurement errors?) but it has an explanation of the error model.

  • $\begingroup$ Yes, I'm referring to any faults in the syndrome extraction step, such as measurement error or any faulty ancilla states. $\endgroup$ – Sherif F. Jul 23 '18 at 15:20
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    $\begingroup$ Measurement errors are also included in the paper I linked. $\endgroup$ – Dripto Debroy Jul 23 '18 at 17:41

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