Quantum XYZ Product Codes
We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum dist...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2022-07-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2022-07-14-766/pdf/ |
| _version_ | 1811223525709578240 |
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| author | Anthony Leverrier Simon Apers Christophe Vuillot |
| author_facet | Anthony Leverrier Simon Apers Christophe Vuillot |
| author_sort | Anthony Leverrier |
| collection | DOAJ |
| description | We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.
Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as $\Theta(N^{2/3})$. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier. |
| first_indexed | 2024-04-12T08:34:09Z |
| format | Article |
| id | doaj.art-86ae33af33a444bcaa47625190e1c63f |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-04-12T08:34:09Z |
| publishDate | 2022-07-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-86ae33af33a444bcaa47625190e1c63f2022-12-22T03:40:03ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2022-07-01676610.22331/q-2022-07-14-76610.22331/q-2022-07-14-766Quantum XYZ Product CodesAnthony LeverrierSimon ApersChristophe VuillotWe study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved. Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as $\Theta(N^{2/3})$. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.https://quantum-journal.org/papers/q-2022-07-14-766/pdf/ |
| spellingShingle | Anthony Leverrier Simon Apers Christophe Vuillot Quantum XYZ Product Codes Quantum |
| title | Quantum XYZ Product Codes |
| title_full | Quantum XYZ Product Codes |
| title_fullStr | Quantum XYZ Product Codes |
| title_full_unstemmed | Quantum XYZ Product Codes |
| title_short | Quantum XYZ Product Codes |
| title_sort | quantum xyz product codes |
| url | https://quantum-journal.org/papers/q-2022-07-14-766/pdf/ |
| work_keys_str_mv | AT anthonyleverrier quantumxyzproductcodes AT simonapers quantumxyzproductcodes AT christophevuillot quantumxyzproductcodes |