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I'm learning quantum error correction by myself and I want to know the relation between block codes and surface codes.

I find that many researchers in the field of coding theory are exploring quantum block codes with better parameters, like constructing quantum codes from various classical codes.

In other hand, I find that surface codes are popular in recent papers.


My questions are

  1. Which one are more promising in quantum error correction? and further in constructing fault-tolerant quantum computers?
  2. How many ancilla qubits are needed now approximataly for a computable logical qubit?
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I will attempt to provide some insight regarding your first question. For starters, both quantum surface codes and quantum block codes are stabilizer codes, which means that although they are significantly different in terms of their construction and utility, they still share some common ground. With regard to which code family is more promising, I believe both types of code are of equal value to the field of QEC, each one having a specific scenario in which it can outperform the other.

A quantum error correcting code will be good depending on the specific requirements of the problem at hand. Surface codes are particularly useful when implementing error correction in situations where the qubits that make up the code are placed in a lattice and only interact with neighbours that are nearby. In such a situation, it is desirable for syndrome extraction to be local (to facilitate fault tolerance). Surface codes (being the most basic example of topological codes) are extremely well suited to solve problems with locality constraints given that they have stabilizer generators with local support. This is essential, because it means that they are compatible with the locality constraints of realistic devices such as superconducting qubits, which, as you mention and has been shown in recent research, has caused a substantial rise in the popularity of these codes.

On the other hand, block codes are very useful because of how they coalesce the classical and quantum error correction paradigms. Most quantum block codes are built using the CSS construction, which enables you to essentially select classical codes and apply them to the quantum scenario almost seamlessly. The only issue is that a good classical code may not be applicable or as good when used to build a quantum CSS block code, and so most of the research is focused around finding good classical codes that can be used to make up good quantum codes. Due to the vast amounts of research and knowledge that we have regarding classical error correction, a big part of the value and promise of quantum block codes stems from the fact that they provide a gateway to apply our classical know-how to the quantum realm.

The subject of quantum surface codes and quantum block codes is a whole lot more nuanced than what I have mentioned in my answer, but it should serve as a succint summary of some of the differences of these code families.

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