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In the common stabilizer-based error correction procedure, one uses one syndrome qubit to measure the eigenvalue of a weight "w" syndrome by "w" times entangling the syndrome qubits to the relevant "w" stabilizer qubits.

However, there are methods, such as Shor's scheme, where one uses a "w" weight cat-state instead of the syndrome qubit, and entangle each of the cat-state qubits once to each of the stabilizer qubits (see here for example https://arxiv.org/abs/2208.05601). I know that other error correction schemes exist, such as the Steane protocol.

What are the differences between these schemes? What are the pros and cons of each scheme? For which error model or hardware each of these methods will be better? Are the other scheme besides the usual single syndrome qubit really useful, or they are only an academic research topic?

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Are the other scheme besides the usual single syndrome qubit really useful, or they are only an academic research topic?

For computation on realistic hardware, we need to convert a logical circuit $C$ to a fault-tolerant circuit $C_{FT}$, where $C_{FT}$ has the same logical outcomes as the that of $C$ with high probability.

The high probability is ensured by demanding that $C_{FT}$ be designed in a way that errors on physical qubits don't propagate "too much".

Fault-tolerance for generic stabilizer codes

For generic stabilizer codes, if you use one syndrome qubit for each stabilizer measurement, the errors will propagate too much. Therefore it is quite necessary to use something like Shor or Steane's schemes to do the stabilizer measurements. These scheme ensure that errors don't propagate too much, and hence your overall circuit is fault-tolerant.

Fault-tolerance for topological codes

Topological codes have "local" stabilizer measurements. This means that you can get away with using one syndrome qubit per stabilizer measurement, if you order the $CX$ gates correctly. See the zig-zag pattern of abcd for surface codes:

Fowler et al. 2013

One proof of this is that we do see reasonable fault-tolerance thresholds for topological codes under realistic noise models using the one-syndrome-qubit-per-stabilizer scheme.

Pros and cons

What are the differences between these schemes? What are the pros and cons of each scheme?

Suppose, we want to use some generic stabilizer code family and we decide to use one of Shor/Steane/Knill/flag-qubit scheme.

What are the metrics we should be looking at to compare these schemes? Given a $C$, what are the (a) depth and (b) width of the $C_{FT}$? And what is (c) the fault-tolerance threshold under realistic noise models?

I think the answers to (a) and (b) are quite easy to compute. See this paper by Gottesman that does a comparison

The answers to (c) are incomplete, and spread out across literature.

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  • $\begingroup$ if Shor's cat-state scheme is the best, why do most papers not use it, but the more straightforward single syndrome qubit method? Can you also please tell me where I can find a comparison of the Shor/Steane/Knill/flag-qubit schemes? $\endgroup$
    – Yaron Jarach
    Commented Jul 16, 2023 at 5:42
  • $\begingroup$ It depends on what you are interested in finding out from your simulation. In many papers, people are interested in discovering "code capacity thresholds" or "phenomenological noise thresholds". The first doesn't depend on your measurement scheme at all, the second not so much. So people just use one-qubit scheme. Its only when you want a "circuit noise threshold" that the measurement scheme becomes important - and as mentioned in the answer, only for non-local codes. $\endgroup$
    – Abdullah Khalid
    Commented Jul 16, 2023 at 6:50
  • $\begingroup$ As for your second question, I can't point to papers. There are papers out there that only do Shor's scheme, and papers out there that do flag-qubit. I haven't personally seen any simulations with Steane or Knill's method (probably because of complicated they are). So there is no apples to apples comparison out there, as far as I know. Hopefully that answers your questions. $\endgroup$
    – Abdullah Khalid
    Commented Jul 16, 2023 at 7:00
  • $\begingroup$ I have edited my answer to point to Gottesman's review arxiv.org/abs/0904.2557 which describes the Shor/Knill/Steane schemes. $\endgroup$
    – Abdullah Khalid
    Commented Jul 16, 2023 at 7:07

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