Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes
Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here, we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a g...
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2024-01-01
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| Series: | IEEE Transactions on Quantum Engineering |
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| Online Access: | https://ieeexplore.ieee.org/document/10531666/ |
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| author | Eric Sabo Arun B. Aloshious Kenneth R. Brown |
| author_facet | Eric Sabo Arun B. Aloshious Kenneth R. Brown |
| author_sort | Eric Sabo |
| collection | DOAJ |
| description | Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here, we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer code <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> and separates into two parts: 1) a one-time offline computation that builds a compact graphical representation of the normalizer of the code, <inline-formula><tex-math notation="LaTeX">$\mathcal {S}^{\perp}$</tex-math></inline-formula> and 2) a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. We show the utility of trellis decoding by applying it to four high-density length-20 stabilizer codes for depolarizing noise and the well-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes for <inline-formula><tex-math notation="LaTeX">$Z$</tex-math></inline-formula> only noise. Numerical simulations demonstrate a 20% improvement in the code-capacity threshold for color codes with boundaries by avoiding the mapping from color codes to surface codes. We identify trellis edge number as a key metric of difficulty of decoding, allowing us to quantify the advantage of single-axis (<inline-formula><tex-math notation="LaTeX">$X$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">$Z$</tex-math></inline-formula>) decoding for Calderbank–Steane–Shor codes and block decoding for concatenated codes. |
| format | Article |
| id | doaj-art-1d3db0be9a02454ca6448c9c8fe789f1 |
| institution | Kabale University |
| issn | 2689-1808 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Transactions on Quantum Engineering |
| spelling | doaj-art-1d3db0be9a02454ca6448c9c8fe789f12025-01-28T00:02:25ZengIEEEIEEE Transactions on Quantum Engineering2689-18082024-01-01513010.1109/TQE.2024.340185710531666Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological CodesEric Sabo0https://orcid.org/0000-0002-0998-6107Arun B. Aloshious1https://orcid.org/0000-0002-0363-8259Kenneth R. Brown2https://orcid.org/0000-0001-7716-1425School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USADepartment of Electrical and Computer Engineering, Duke University, Durham, NC, USADepartment of Electrical and Computer Engineering, Duke University, Durham, NC, USATrellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here, we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer code <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> and separates into two parts: 1) a one-time offline computation that builds a compact graphical representation of the normalizer of the code, <inline-formula><tex-math notation="LaTeX">$\mathcal {S}^{\perp}$</tex-math></inline-formula> and 2) a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. We show the utility of trellis decoding by applying it to four high-density length-20 stabilizer codes for depolarizing noise and the well-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes for <inline-formula><tex-math notation="LaTeX">$Z$</tex-math></inline-formula> only noise. Numerical simulations demonstrate a 20% improvement in the code-capacity threshold for color codes with boundaries by avoiding the mapping from color codes to surface codes. We identify trellis edge number as a key metric of difficulty of decoding, allowing us to quantify the advantage of single-axis (<inline-formula><tex-math notation="LaTeX">$X$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">$Z$</tex-math></inline-formula>) decoding for Calderbank–Steane–Shor codes and block decoding for concatenated codes.https://ieeexplore.ieee.org/document/10531666/Quantum error correctionTrellis decoding |
| spellingShingle | Eric Sabo Arun B. Aloshious Kenneth R. Brown Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes IEEE Transactions on Quantum Engineering Quantum error correction Trellis decoding |
| title | Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes |
| title_full | Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes |
| title_fullStr | Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes |
| title_full_unstemmed | Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes |
| title_short | Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes |
| title_sort | trellis decoding for qudit stabilizer codes and its application to qubit topological codes |
| topic | Quantum error correction Trellis decoding |
| url | https://ieeexplore.ieee.org/document/10531666/ |
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