Connectivity constrains quantum codes
Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It i...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2022-05-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2022-05-13-711/pdf/ |
| _version_ | 1811256788970897408 |
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| author | Nouédyn Baspin Anirudh Krishna |
| author_facet | Nouédyn Baspin Anirudh Krishna |
| author_sort | Nouédyn Baspin |
| collection | DOAJ |
| description | Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in $D$-dimensional hyperbolic space. |
| first_indexed | 2024-04-12T17:46:34Z |
| format | Article |
| id | doaj.art-ca6888e9102e47d9b625f7777b648066 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-04-12T17:46:34Z |
| publishDate | 2022-05-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-ca6888e9102e47d9b625f7777b6480662022-12-22T03:22:39ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2022-05-01671110.22331/q-2022-05-13-71110.22331/q-2022-05-13-711Connectivity constrains quantum codesNouédyn BaspinAnirudh KrishnaQuantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in $D$-dimensional hyperbolic space.https://quantum-journal.org/papers/q-2022-05-13-711/pdf/ |
| spellingShingle | Nouédyn Baspin Anirudh Krishna Connectivity constrains quantum codes Quantum |
| title | Connectivity constrains quantum codes |
| title_full | Connectivity constrains quantum codes |
| title_fullStr | Connectivity constrains quantum codes |
| title_full_unstemmed | Connectivity constrains quantum codes |
| title_short | Connectivity constrains quantum codes |
| title_sort | connectivity constrains quantum codes |
| url | https://quantum-journal.org/papers/q-2022-05-13-711/pdf/ |
| work_keys_str_mv | AT nouedynbaspin connectivityconstrainsquantumcodes AT anirudhkrishna connectivityconstrainsquantumcodes |