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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?

Jagerber48
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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.

ChrisD
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