What are reviews on the search for good quantum LDPC codes?

Question

Can anyone provide a link for an up-to-date review of the current situation in researching good quantum LDPC codes and prospects?

I am interested in questions like: Does the quantum computing community believe this research will provide practical code examples soon (2-4 years)? Is it believed that such codes will improve only the overhead or also the error correction power (considering the probably more complicated circuits required to implement such codes)? What are the "fundamental" papers of this field? What are the predicted requirements from the hardware to implement such codes?

Answer 1

I'll try to answer your questions by giving the main motivations for implementing ldpc codes and the main practical challenges, along with fundamental papers. I only focus on quantum memory and ignore performing logical operations. Most of this answer is inspired by a talk given by Christopher A. Pattison.

Motivation:

• LDPC codes exist with parameters $[n,Θ(n),Θ(n)]$. ('Good' LDPC codes exist). Panteleev, Kalachev 2022
• Good LPDC codes can be decoded efficiently. Shouzhen Gu, Christopher A. Pattison, Eugene Tang 2022 Anthony Leverrier, Jean-Pierre Tillich, Gilles Zémor 2015
• Good LDPC codes are single shot decodable: Shouzhen Gu et al. 2023

Practical Challenges:

• For an $[n, k, d]$ code with geometrically local checks in 2D Euclidean space $k$ and $d$ are bounded by $kd^2 =O(n)$ Bravyi, Poulin, Terhal 2010
• Circumventing BPT requires many long-range connections. Baspin and Krishna 2022
• If you want to implement an LDPC code using swap gates in a nearest neighbor 2D architecture, you get deep syndrome extraction circuits Delfosse, Beverland, and Tremblay 2021

Can we overcome these challenges?

Can we design a practical architecture that encodes information in LDPC codes that is more efficient than an architecture using topological codes? This is an important open question.

Here are some promising recent results:

• If one has 2 vertically connected layers consisting of square grids of qubits, a large ldpc code concatenated with a surface code is more efficient than just a surface code.Christopher A. Pattison, Anirudh Krishna, John Preskill 2023
• One can measure generators acting on spatially distant qubits less frequently than those which do not. Noah Berthusen, Daniel Gottesman 2023
• If one has a device with a small number of planar layers each containing long-range connections which do not cross, ldpc codes outperform the surface code. Maxime A. Tremblay, Nicolas Delfosse, Michael E. Beverland 2021