On asymmetric quantum error-correcting codes
This thesis focuses on the construction and analysis of the parameters of asymmetric quantum error-correcting codes (AQECCs). We use the so-called functional approach to connect AQECCs to classical codes via orthogonal arrays. Based on the functional approach, we derive pure AQECCs from CSS-like con...
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| Format: | Thesis |
| Language: | English |
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2011
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| Online Access: | https://hdl.handle.net/10356/46253 |
| _version_ | 1826114672672112640 |
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| author | Martianus Frederic Ezerman |
| author2 | Ling San |
| author_facet | Ling San Martianus Frederic Ezerman |
| author_sort | Martianus Frederic Ezerman |
| collection | NTU |
| description | This thesis focuses on the construction and analysis of the parameters of asymmetric quantum error-correcting codes (AQECCs). We use the so-called functional approach to connect AQECCs to classical codes via orthogonal arrays. Based on the functional approach, we derive pure AQECCs from CSS-like constructions that include the standard CSS construction known prior to this work. The CSS-like constructions remove both the specific choice of Euclidean inner product and the $\F_{q}$-linearity condition imposed by the standard variant. Hence, pure AQECCs can now be constructed by using pairs of nested $\F_{r}$-linear codes over $\F_{q}$ where $\F_{r}$ is any subfield of $\F_{q}$ under the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. Relationships between the various CSS-like constructions are also exhibited. A formal definition of asymmetric stabilizer codes is given and a connection between pure asymmetric stabilizer codes and AQECCs derived from the CSS-like constructions is established. It is shown that the class of pure CSS-like AQECCs forms a subset of asymmetric stabilizer codes. We show how classical linear MDS codes can be used to construct AQECCs that satisfy the quantum Singleton bound with $d_{z} \geq d_{x}\geq 2$ for all possible values of length $n$ for which linear MDS codes over $\F_{q}$ are known to exist. Beyond the lengths specified by the classical MDS conjecture, various explicit constructions of nested pairs of classical codes can be combined with linear programming to establish the optimality of pure CSS-like codes with parameters $[[n,k,d_{z}/d_{x}]]_{q}$. For $q=\{2,3,4,5,7,8,9\}$, lists of optimal asymmetric CSS-like codes for reasonable lengths are presented. |
| first_indexed | 2024-10-01T03:43:16Z |
| format | Thesis |
| id | ntu-10356/46253 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:43:16Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | ntu-10356/462532023-02-28T23:37:29Z On asymmetric quantum error-correcting codes Martianus Frederic Ezerman Ling San School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Algebra DRNTU::Science::Mathematics::Applied mathematics::Information theory This thesis focuses on the construction and analysis of the parameters of asymmetric quantum error-correcting codes (AQECCs). We use the so-called functional approach to connect AQECCs to classical codes via orthogonal arrays. Based on the functional approach, we derive pure AQECCs from CSS-like constructions that include the standard CSS construction known prior to this work. The CSS-like constructions remove both the specific choice of Euclidean inner product and the $\F_{q}$-linearity condition imposed by the standard variant. Hence, pure AQECCs can now be constructed by using pairs of nested $\F_{r}$-linear codes over $\F_{q}$ where $\F_{r}$ is any subfield of $\F_{q}$ under the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. Relationships between the various CSS-like constructions are also exhibited. A formal definition of asymmetric stabilizer codes is given and a connection between pure asymmetric stabilizer codes and AQECCs derived from the CSS-like constructions is established. It is shown that the class of pure CSS-like AQECCs forms a subset of asymmetric stabilizer codes. We show how classical linear MDS codes can be used to construct AQECCs that satisfy the quantum Singleton bound with $d_{z} \geq d_{x}\geq 2$ for all possible values of length $n$ for which linear MDS codes over $\F_{q}$ are known to exist. Beyond the lengths specified by the classical MDS conjecture, various explicit constructions of nested pairs of classical codes can be combined with linear programming to establish the optimality of pure CSS-like codes with parameters $[[n,k,d_{z}/d_{x}]]_{q}$. For $q=\{2,3,4,5,7,8,9\}$, lists of optimal asymmetric CSS-like codes for reasonable lengths are presented. DOCTOR OF PHILOSOPHY (SPMS) 2011-07-08T06:36:18Z 2011-07-08T06:36:18Z 2011 2011 Thesis Martianus, F. E. (2011). On asymmetric quantum error-correcting codes. Doctoral thesis, Nanyang Technological University, Singapore. https://hdl.handle.net/10356/46253 10.32657/10356/46253 en 190 p. application/pdf |
| spellingShingle | DRNTU::Science::Mathematics::Algebra DRNTU::Science::Mathematics::Applied mathematics::Information theory Martianus Frederic Ezerman On asymmetric quantum error-correcting codes |
| title | On asymmetric quantum error-correcting codes |
| title_full | On asymmetric quantum error-correcting codes |
| title_fullStr | On asymmetric quantum error-correcting codes |
| title_full_unstemmed | On asymmetric quantum error-correcting codes |
| title_short | On asymmetric quantum error-correcting codes |
| title_sort | on asymmetric quantum error correcting codes |
| topic | DRNTU::Science::Mathematics::Algebra DRNTU::Science::Mathematics::Applied mathematics::Information theory |
| url | https://hdl.handle.net/10356/46253 |
| work_keys_str_mv | AT martianusfredericezerman onasymmetricquantumerrorcorrectingcodes |