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My question concerns fault-tolerant measurement of the syndrome of an quantum error-correcting code.

Somewhat recently, Chao and Riechardt, in https://arxiv.org/abs/1705.02329, proposed a method involving only two qubits, now known as "flag qubits". (Other improvements also exist, e.g., https://arxiv.org/abs/2108.02184.) In the introduction, they mention a few other methods, such as Shor's method, Steane's method, and Knill's method (http://arxiv.org/abs/quant-ph/9605011, http://arxiv.org/abs/quant-ph/0607047, http://arxiv.org/abs/quant-ph/9611027.

My question is, is there any place all these are explained? And is there any advantage to using older methods with more qubits, (for example, to the noise theshold)? Or are the new schemes just better by all metrics?

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I highly recommend chapter 12 of Daniel Gottesman's book Surviving as a Quantum Computer in a Classical World

  • Section 12.2 Shor Error Correction
  • Section 12.3 Steane Error Correction and Measurement
  • Section 12.4 Knill Error Correction and Measurement
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As far as I know, there is not a paper that details all these schemes and make comparisons(correct me if I'm wrong). The results are indeed scattered, especially assumptions on error models are sometimes different, some use circuit-level noise while others use phenomenological noise. However, in Chao and Riechardt Figure 6 they do make a comparison for Shor's Steane's and Flag methods.

I came across the same issue when I started to learn all these. At the end, I just did the simulation and comparison by myself. My conclusion is (based on circuit-level noise) overall Knill EC has the highest threshold, while others, for the [7,1,3] Steane code case, the thresholds are roughly the same, with Flag method slightly outperforms Steane's and Shor's.

The biggest advantage of flag qubit is the resource savings. However, I think it also involves more time-overhead compared to Steane's method. I didn't explicitly check this, but from their constructions I feel this is the case(since Flag requires more measurements and qubit resetting), which then requires the data qubits to have longer coherence time. Perhaps if we scale up this could be an issue.

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    $\begingroup$ I actually feel like quantum error correction misses a proper overarching review paper that goes into mathematical details what families of codes have been constructed and what their properties are, and what their advantages are (if any) compared to other constructions. The results are very scattered indeed and that makes getting into the topic a bit harder. I've actually been thinking about writing a review paper myself at this point simply because not having one when I started out frustrated me as well. The only good review paper I know of is about quantum LDPC codes, with history and all. $\endgroup$
    – JoJo P
    Commented Apr 25 at 18:44
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    $\begingroup$ @JoJoP Maybe let's wait for the second half of Daniel Gottesman's book $\endgroup$
    – AndyLiuin
    Commented Apr 25 at 18:50

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