Hardness results for decoding the surface code with Pauli noise
Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the sp...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2024-10-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2024-10-28-1511/pdf/ |
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| author | Alex Fischer Akimasa Miyake |
| author_facet | Alex Fischer Akimasa Miyake |
| author_sort | Alex Fischer |
| collection | DOAJ |
| description | Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit. In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent $X$ and $Z$ noise. |
| format | Article |
| id | doaj-art-3b30b4b10f954137a616528dd07b11bf |
| institution | Kabale University |
| issn | 2521-327X |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj-art-3b30b4b10f954137a616528dd07b11bf2024-11-20T12:13:14ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2024-10-018151110.22331/q-2024-10-28-151110.22331/q-2024-10-28-1511Hardness results for decoding the surface code with Pauli noiseAlex FischerAkimasa MiyakeReal quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit. In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent $X$ and $Z$ noise.https://quantum-journal.org/papers/q-2024-10-28-1511/pdf/ |
| spellingShingle | Alex Fischer Akimasa Miyake Hardness results for decoding the surface code with Pauli noise Quantum |
| title | Hardness results for decoding the surface code with Pauli noise |
| title_full | Hardness results for decoding the surface code with Pauli noise |
| title_fullStr | Hardness results for decoding the surface code with Pauli noise |
| title_full_unstemmed | Hardness results for decoding the surface code with Pauli noise |
| title_short | Hardness results for decoding the surface code with Pauli noise |
| title_sort | hardness results for decoding the surface code with pauli noise |
| url | https://quantum-journal.org/papers/q-2024-10-28-1511/pdf/ |
| work_keys_str_mv | AT alexfischer hardnessresultsfordecodingthesurfacecodewithpaulinoise AT akimasamiyake hardnessresultsfordecodingthesurfacecodewithpaulinoise |