Quantum LDPC Codes of Almost Linear Distance via Homological Products
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arXiv
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| Format: | Preprint |
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2024
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| author | Golowich, Louis Guruswami, Venkatesan |
| author_facet | Golowich, Louis Guruswami, Venkatesan |
| contents | We present new constructions of quantum codes of linear or close-to-linear distance and dimension with low-weight stabilizers. Only a few constructions of such codes were previously known, and were primarily based on a specific operation from homological algebra, namely the balanced product. In contrast, our constructions are based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Our results help address the natural question: When do homological products preserve good code distance?
Our first main result constructs asymptotically good $[[N,Θ(N),Θ(N)]]$ quantum codes with small polynomial stabilizer weight from homological products of codes with a property called product-expansion. This notion was recently introduced and used to bound the distance of balanced product quantum codes; we apply it instead to homological products.
For every $ε>0$, our second main result constructs close-to-linear distance $[[N,N^{1-ε},N^{1-ε}]]$ (subsystem) quantum LDPC codes with constant stabilizer weight from iterated homological products of a constant-sized quantum locally testable code. The key insight here is that by using subsystem codes (but still with constant-weight stabilizers), we can circumvent a particular obstruction that limited the distance of many prior product code constructions to at most $\tilde{O}(\sqrt{N})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_03646 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum LDPC Codes of Almost Linear Distance via Homological Products Golowich, Louis Guruswami, Venkatesan Quantum Physics Information Theory We present new constructions of quantum codes of linear or close-to-linear distance and dimension with low-weight stabilizers. Only a few constructions of such codes were previously known, and were primarily based on a specific operation from homological algebra, namely the balanced product. In contrast, our constructions are based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Our results help address the natural question: When do homological products preserve good code distance? Our first main result constructs asymptotically good $[[N,Θ(N),Θ(N)]]$ quantum codes with small polynomial stabilizer weight from homological products of codes with a property called product-expansion. This notion was recently introduced and used to bound the distance of balanced product quantum codes; we apply it instead to homological products. For every $ε>0$, our second main result constructs close-to-linear distance $[[N,N^{1-ε},N^{1-ε}]]$ (subsystem) quantum LDPC codes with constant stabilizer weight from iterated homological products of a constant-sized quantum locally testable code. The key insight here is that by using subsystem codes (but still with constant-weight stabilizers), we can circumvent a particular obstruction that limited the distance of many prior product code constructions to at most $\tilde{O}(\sqrt{N})$. |
| title | Quantum LDPC Codes of Almost Linear Distance via Homological Products |
| topic | Quantum Physics Information Theory |
| url | https://arxiv.org/abs/2411.03646 |