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See this sectino on Wikipedia for Quantum error correction

  • Peter Shor's 9-qubit-code, a.k.a. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for arbitrary errors in a single qubit.
  • Andrew Steane found a code that does the same with 7 instead of 9 qubits, see Steane code.
  • Raymond Laflamme and collaborators found a class of 5-qubit codes that do the same, which also have the property of being fault-tolerant. A 5-qubit code is the smallest possible code that protects a single logical qubit against single-qubit errors.

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Emphasis mine. The Raymond Laflamme code is highlighted as being fault tolerant. What does fault tolerant mean? I thought all error correction schemes were meant to make a quantum computer fault tolerant. But the grammar in this sentence makes it sound like "fault tolerant" is being used in some technical way to apply to error correcting codes. If so, what is the definition of a fault tolerant quantum error correction code? And what are examples of non-fault-tolerant and fault-tolerant error correction codes?

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  • $\begingroup$ This deserves a longer and more canonical answer but it's my (probably incorrect or incomplete) understanding that Shor's code didn't take into account errors occurring in the gates used to correct the errors themselves. The threshold is achieved when you can correct errors faster than they accumulate because of your correction. With Shor's 9-qubit code you'd be limited to circuits that grow like $\log n$-depth but with a fully fault-tolerant code your circuits could grow as $\mathrm { poly} n$ depth. $\endgroup$ Jun 1, 2023 at 21:40
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    $\begingroup$ @MarkSpinelli that is an important consideration but it's a different theoretical problem. One problem is: You get an error, is there any way to detect and correct it? For that there's no notion of fault-tolerant or not I guess. Now the problem you're suggesting is: you get errors that are caused by gates. Is there now a scheme that involves correcting that error using gates that improves your logical error rate. If there is, that scheme would be "fault tolerant" Two different, but important theoretical problems. Have I got it right? $\endgroup$
    – Jagerber48
    Jun 1, 2023 at 21:51

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Fault-tolerance isn't really a property of the code, more of the operations that perform encoded logic and syndrome measurement. A good working definition of fault-tolerance for a circuit implementing either of those is

one for which a single operational error can only produce one error within a single encoded block

This is from an early paper by Daniel Gottesman where he shows fault-tolerant universal logic is possible for any stabilizer code. An earlier paper by DiVincenzo and Shor had shown that fault-tolerant error correction was also possible for such codes. So, to partially answer your question, all stabilizer codes can be considered fault-tolerant.

As to where there are any non-stabilizer codes that don't admit fault-tolerant techniques for logic and error correction, I don't know. There are almost certainly some for which there are no known techniques.

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  • $\begingroup$ "one for which a single operational error can only produce one error within a single encoded block" and I guess this should also include that the single error in the encoded block can be corrected, right? $\endgroup$
    – Jagerber48
    Jun 2, 2023 at 13:18
  • $\begingroup$ Yes that's an implicit assumption $\endgroup$
    – ChrisD
    Jun 2, 2023 at 17:51

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