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The language of detectors and detecting regions in the context of quantum error correction is introduced in https://arxiv.org/pdf/2302.02192.pdf.

Detecting regions seem to be capable of specifying most relevant properties of a syndrome extraction circuit, e.g. their beginning and termination sets where resets and measurements occur, their expansion and contraction is associated with entangling and disentangling, their overlap determines the connectivity of the error graph to be decoded, etc.

My questions are:

  • Is it possible to make any statements about the fault-tolerance properties of the circuit implementation of a specific code by looking at its detecting regions?

  • Is it clear what are the requirements on detecting regions to guarantee fault tolerance?

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Detecting regions are the same thing as stabilizers, just spanning over spacetime instead of only over space. You can basically translate any statement about stabilizers in a stabilizer code into statements about detecting regions in a circuit, and it will be correct.

For example, detecting regions must commute. They must cover the locations where the observable is. The observable can't be equal to a product of detecting regions. The distance is the number of errors you need to flip the observable without flipping any detecting regions. Etc.

The reason they are so analogous is because you can take any circuit, and replace all cases of "then the qubit waits until the next operation" with "and as that's happening the qubit is teleported over to the next operation". This unfolds the circuit into a spacelike structure that will correspond to a stabilizer code (or maybe a gauge code? something simple like that).

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  • $\begingroup$ To what is extent is the relationship between stabilizers and detecting regions an analogy or a precise mathematical relationship? At the moment, I am struggling to understand what kind of mathematical objects are detecting regions. $\endgroup$ Mar 1 at 5:17
  • $\begingroup$ Importantly, detecting regions seem like a very powerful concept beyond just a tool to simulate QEC. The stabilizer formalism tells me that if I come up with a set of (cleverly chosen) Pauli operators I will be to store some information (logical qubits) and protect it against a certain amount of errors (code distance, etc.). How does one go about performing the same constructions using solely detecting regions? While this might be strictly tied up to the context of quantum circuits, I would expect that all properties mentioned in your answer would organically emerge from this construction. $\endgroup$ Mar 1 at 5:18

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