Abstract
Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large X-distance but a much smaller Z-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance n1/2 log1/2 n. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance n1/2 log n, which sets a new record for quantum LDPC codes.
Original language | English |
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Pages (from-to) | FOCS20-276 and FOCS20-316 |
Journal | SIAM Journal on Computing |
Volume | 53 |
Issue number | 6 |
DOIs | |
State | Published - 2024 |
Bibliographical note
Publisher Copyright:© 2022 Society for Industrial and Applied Mathematics.
Keywords
- chain complex
- expander
- quantum code