Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908851434946560 |
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| author | Eidesen, Jonas |
| author_facet | Eidesen, Jonas |
| contents | By defining projective error models we study the mathematical structure of Clifford codes and stabilizer codes using tools from projective representation theory. Furthermore, we introduce a new class of codes which we have called weak stabilizer codes and we determine some relationships between these three classes of codes. We show that the obstruction for a stabilizer code to be non-trivial is given by a class in group cohomology, and we are able to determine similar obstructions for weak stabilizer codes to be non-trivial. In the case where the projective error model corresponds to a nice error basis we give a complete characterization of when a Clifford code is a weak stabilizer code in terms of the size of the group of logical operators and the size of the group of stabilizers of the code. Lastly, we produce two infinite families of Clifford codes that are not stabilizer codes, as well as a method of combining these examples into more examples of non-stabilizer Clifford codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_01843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes Eidesen, Jonas Quantum Physics Representation Theory Primary: 81P70, Secondary: 81R99, 20C25, 20C35 By defining projective error models we study the mathematical structure of Clifford codes and stabilizer codes using tools from projective representation theory. Furthermore, we introduce a new class of codes which we have called weak stabilizer codes and we determine some relationships between these three classes of codes. We show that the obstruction for a stabilizer code to be non-trivial is given by a class in group cohomology, and we are able to determine similar obstructions for weak stabilizer codes to be non-trivial. In the case where the projective error model corresponds to a nice error basis we give a complete characterization of when a Clifford code is a weak stabilizer code in terms of the size of the group of logical operators and the size of the group of stabilizers of the code. Lastly, we produce two infinite families of Clifford codes that are not stabilizer codes, as well as a method of combining these examples into more examples of non-stabilizer Clifford codes. |
| title | Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes |
| topic | Quantum Physics Representation Theory Primary: 81P70, Secondary: 81R99, 20C25, 20C35 |
| url | https://arxiv.org/abs/2506.01843 |