Citation: Bravyi, Sergey, et al. "High-threshold and low-overhead fault-tolerant quantum memory." Nature 627.8005 (2024): 778-782.
Web: https://arxiv.org/abs/2308.07915
Tags: Error-correcting-codes, Non-local-codes
This paper gives a lovely look at "what's out there" for LDPC codes.
The idea is that fault-tolerance is very important, and we want to use good error correcting codes to make it as practical as possible. The surface code is fine but it has problems - its encoding rate is bad, and it cannot fault tolerantly implement universal gates. Both of these issues are directly related to the fact that it has a 2D local geometry. Relative to its 2D nature, it is optimal both in its rate and it is ability to perform fault tolerant computations.
Going forward, the best approach might be to give up locality. There is a lot to gain: much higher encoding rates and the possibility of universal fault tolerant quantum computation. This paper gives a nice approach to this problem. First, the authors outline what they want out of a code. They want a high encoding rate. They want a large code distance. They want syndromes that can be measured with low-depth circuits. They want lots of fault-tolerant logical operators. They want all this, while not sacrificing locality that much. Namely, they want the graph to be locally two-dimensional with a few non-local connections.
The way they go about constructing such LDPC codes is by computer search. They get a look for "what's out there" before doing something more theoretically taxing. The results are very successful. They find codes which satisfy all of their criteria, while still being mostly local. This is a huge success for non-local codes, and is a good indicator for the fact that non-local LDPC codes might be the future of quantum error correction.