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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?

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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:

Practical Challenges:

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:

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    $\begingroup$ Thank you! Are there comments in the literature about the predicted performance of such codes against loss\leakage\faulty entangling gates? "Non-local" connections are connections between qubits that are O(d) far away from each other? $\endgroup$ Jun 30, 2023 at 15:21
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    $\begingroup$ Christopher, Krishna and Preskill 2023 does not consider leakage but does consider faulty entangling gates. The same is true for Tremblay, Delfosse, Beverland 2021. $\endgroup$
    – Peter-Jan
    Jun 30, 2023 at 16:30
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    $\begingroup$ Yes with lon-local connections usually something like O(d) is meant, the exact definition can differ by paper. $\endgroup$
    – Peter-Jan
    Jun 30, 2023 at 16:33

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