Decodable quantum LDPC codes beyond the square root distance barrier using high dimensional expanders
| Autor: | Tali Kaufman, Gilles Zémor, Shai Evra |
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| Rok vydání: | 2020 |
| Předmět: |
Discrete mathematics
Dimension (graph theory) 020206 networking & telecommunications 02 engineering and technology 01 natural sciences Square (algebra) Ramanujan's sum symbols.namesake Square root Qubit 0103 physical sciences 0202 electrical engineering electronic engineering information engineering symbols Low-density parity-check code 010306 general physics Quantum Decoding methods Mathematics |
| Zdroj: | FOCS |
| DOI: | 10.1109/focs46700.2020.00029 |
| Popis: | Constructing quantum 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 generalises 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 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance $n^{1/2}\log n$ , and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$ . 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. |
| Databáze: | OpenAIRE |
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