The XYZ$^2$ hexagonal stabilizer code
We consider a topological stabilizer code on a honeycomb grid, the "XYZ$^2$" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2022-04-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2022-04-27-698/pdf/ |
| _version_ | 1817979260651962368 |
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| author | Basudha Srivastava Anton Frisk Kockum Mats Granath |
| author_facet | Basudha Srivastava Anton Frisk Kockum Mats Granath |
| author_sort | Basudha Srivastava |
| collection | DOAJ |
| description | We consider a topological stabilizer code on a honeycomb grid, the "XYZ$^2$" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six ($XYZXYZ$) plaquette stabilizers and weight-two ($XX$) link stabilizers on a planar hexagonal grid composed of $2d^2$ qubits for code distance $d$, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure $X$, $Y$, or $Z$ noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance $d^2$ only for pure $Y$ noise, here the code distance is $2d^2$ for both pure $Z$ and pure $Y$ noise. Thresholds for noise with finite $Z$ bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding. |
| first_indexed | 2024-04-13T22:40:36Z |
| format | Article |
| id | doaj.art-11f2cbc5a3e640659b6ab07bfbb6658e |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-04-13T22:40:36Z |
| publishDate | 2022-04-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-11f2cbc5a3e640659b6ab07bfbb6658e2022-12-22T02:26:37ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2022-04-01669810.22331/q-2022-04-27-69810.22331/q-2022-04-27-698The XYZ$^2$ hexagonal stabilizer codeBasudha SrivastavaAnton Frisk KockumMats GranathWe consider a topological stabilizer code on a honeycomb grid, the "XYZ$^2$" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six ($XYZXYZ$) plaquette stabilizers and weight-two ($XX$) link stabilizers on a planar hexagonal grid composed of $2d^2$ qubits for code distance $d$, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure $X$, $Y$, or $Z$ noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance $d^2$ only for pure $Y$ noise, here the code distance is $2d^2$ for both pure $Z$ and pure $Y$ noise. Thresholds for noise with finite $Z$ bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.https://quantum-journal.org/papers/q-2022-04-27-698/pdf/ |
| spellingShingle | Basudha Srivastava Anton Frisk Kockum Mats Granath The XYZ$^2$ hexagonal stabilizer code Quantum |
| title | The XYZ$^2$ hexagonal stabilizer code |
| title_full | The XYZ$^2$ hexagonal stabilizer code |
| title_fullStr | The XYZ$^2$ hexagonal stabilizer code |
| title_full_unstemmed | The XYZ$^2$ hexagonal stabilizer code |
| title_short | The XYZ$^2$ hexagonal stabilizer code |
| title_sort | xyz 2 hexagonal stabilizer code |
| url | https://quantum-journal.org/papers/q-2022-04-27-698/pdf/ |
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