Abstract
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.
Original language | English |
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Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |
Publisher | IEEE Computer Society |
Pages | 218-227 |
Number of pages | 10 |
ISBN (Electronic) | 9781728196213 |
DOIs | |
State | Published - Nov 2020 |
Externally published | Yes |
Event | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States Duration: 16 Nov 2020 → 19 Nov 2020 |
Publication series
Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2020-November |
ISSN (Print) | 0272-5428 |
Conference
Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
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Country/Territory | United States |
City | Virtual, Durham |
Period | 16/11/20 → 19/11/20 |
Bibliographical note
Publisher Copyright:© 2020 IEEE.
Keywords
- Quantum code
- Ramanujan Complex
- expander