Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0512041 · Physical Review A

Trellises for stabilizer codes: Definition and uses

Harold Ollivier, Jean–Pierre Tillich

Published Sep 7, 2006• 10 citations
arXivHTMLPDF

Abstract

Trellises play an important theoretical and practical role for classical codes. Their main utility is to devise complexity-efficient error estimation algorithms. Here, we describe trellis representations for quantum stabilizer codes. We show that they share the same properties as their classical analogs. In particular, for any stabilizer code it is possible to find a minimal trellis representation. Our construction is illustrated by two fundamental error estimation algorithms.

DOI
10.1103/physreva.74.032304

Related papers

Construction of minimal trellises for quantum stabilizer codes

Xiao Fang

The size of vertex set of quantum trellises affects the efficiency of decoding algorithms,which can be improved by reducing the number of vertices.Based on the standard check matrix of stabilizer codes,an algorithm to construct trellis-oriented generators for quantum stabilizer codes is presented.By using this algorithm,the trellises with minimal vertex set can be constructed.In addition,an algorithm to construct trellises iteratively for quantum stabilizer codes is also introduced.The algorithm proposed in this paper is more efficient and less complex than the one proposed by Olliver,and so is more suitable for the applications of larger scale and stricter timeliness.

Science China Information Sciences•Jan 1, 2013•0 citations

Minimal Trellises for Degenerate Decoding of Quantum Stabilizer Codes

Evagoras Stylianou, Vladimir R. Sidorenko, Christian Deppe +1 more

This paper introduces several techniques for minimal trellis construction for degenerate decoding of quantum stabilizer codes, specifically the minimal multi-goal trellis for the cosets of the stabilizer group S in the normalizer group N. The methods include a merging algorithm, a Shannon-product approach, and the BCJR-Wolf method. The study establishes the necessary properties of multi-goal trellises and bounds on the decoding complexity of the minimal multi-goal trellis using the sum-product Viterbi algorithm. The proposed multi-goal trellises decrease the decoding complexity by a factor O(n), where n is the code length.

Globecom•Dec 8, 2024•1 citation

Trellis Decoding for Qudit Stabilizer Codes and Its Application to Qubit Topological Codes

Eric Sabo, Arun B. Aloshious, Kenneth R. Brown

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$S$and separates into two parts: 1) a one-time offline computation that builds a compact graphical representation of the normalizer of the code,$\mathcal {S}^{\perp}$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$Z$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 ($X$or$Z$) decoding for Calderbank–Steane–Shor codes and block decoding for concatenated codes.

arXiv:2106.08251•IEEE Transactions on Quantum Engineering•Jan 1, 2024•12 citations

Stabilizer codes from modified symplectic form

Tejas Gandhi, Piyush P. Kurur, Mittal, Rajat

Stabilizer codes form an important class of quantum error correcting codes which have an elegant theory, efficient error detection, and many known examples. Constructing stabilizer codes of length $n$ is equivalent to constructing subspaces of $\mathbb{F}_p^n \times \mathbb{F}_p^n$ which are "isotropic" under the symplectic bilinear form defined by $\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}$. As a result, many, but not all, ideas from the theory of classical error correction can be translated to quantum error correction. One of the main theoretical contribution of this article is to study stabilizer codes starting with a different symplectic form. In this paper, we concentrate on cyclic codes. Modifying the symplectic form allows us to generalize the previous known construction for linear cyclic stabilizer codes, and in the process, circumvent some of the Galois theoretic no-go results proved there. More importantly, this tweak in the symplectic form allows us to make use of well known error correcting algorithms for cyclic codes to give efficient quantum error correcting algorithms. Cyclicity of error correcting codes is a "basis dependent" property. Our codes are no more "cyclic" when they are derived using the standard symplectic forms (if we ignore the error correcting properties like distance, all such symplectic forms can be converted to each other via a basis transformation). Hence this change of perspective is crucial from the point of view of designing efficient decoding algorithm for these family of codes. In this context, recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true.

arXiv:1708.00617•Aug 2, 2017•1 citation

Minimal error-trellises for linear block codes

M. Ariel, Jakov Snyders

An error-trellis is an efficient means to accomplish maximum likelihood decoding of a linear (n,k) block code C over GF(q). It represents the coset of C that contains the symbol-by-symbol detected version of the received vector. The trellis we construct has only n-k sections and the structure of each section is determined solely by the value of an associated q-ary symbol of the syndrome. Hence, the entire set of q/sup n-k/ trellises for all the cosets of C is formed by a set of q(n-k) distinct sections. The error-trellis is invariant under permutation of coordinates and attains the vertex count of a minimal code-trellis with the same sectionalization.

IEEE International Symposium on Information Theory (ISIT)•Nov 27, 2002•1 citation

Quantum stabilizer codes and beyond

Sarvepalli, Pradeep Kiran

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.

arXiv:0810.2574•Oct 14, 2008•8 citations
Trellises play an important theoretical and practical role for classical codes. Their main utility is to devise complexity-efficient error estimation algorithms. Here, we describe trellis representations for quantum stabilizer codes. We show that they share the same properties as their classical analogs. In particular, for any stabilizer code it is possible to find a minimal trellis representation. Our construction is illustrated by two fundamental error estimation algorithms.
The size of vertex set of quantum trellises affects the efficiency of decoding algorithms,which can be improved by reducing the number of vertices.Based on the standard check matrix of stabilizer codes,an algorithm to construct trellis-oriented generators for quantum stabilizer codes is presented.By using this algorithm,the trellises with minimal vertex set can be constructed.In addition,an algorithm to construct trellises iteratively for quantum stabilizer codes is also introduced.The algorithm proposed in this paper is more efficient and less complex than the one proposed by Olliver,and so is more suitable for the applications of larger scale and stricter timeliness.
This paper introduces several techniques for minimal trellis construction for degenerate decoding of quantum stabilizer codes, specifically the minimal multi-goal trellis for the cosets of the stabilizer group S in the normalizer group N. The methods include a merging algorithm, a Shannon-product approach, and the BCJR-Wolf method. The study establishes the necessary properties of multi-goal trellises and bounds on the decoding complexity of the minimal multi-goal trellis using the sum-product Viterbi algorithm. The proposed multi-goal trellises decrease the decoding complexity by a factor O(n), where n is the code length.
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 codeSSSand separates into two parts: 1) a one-time offline computation that builds a compact graphical representation of the normalizer of the code,S⊥\mathcal {S}^{\perp}S⊥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 forZZZonly 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 (XXXorZZZ) decoding for Calderbank–Steane–Shor codes and block decoding for concatenated codes.
Stabilizer codes form an important class of quantum error correcting codes which have an elegant theory, efficient error detection, and many known examples. Constructing stabilizer codes of length nnn is equivalent to constructing subspaces of Fpn×Fpn\mathbb{F}_p^n \times \mathbb{F}_p^nFpn​×Fpn​ which are "isotropic" under the symplectic bilinear form defined by ⟨(a,b),(c,d)⟩=aTd−bTc\left\langle (\mathbf{a},\mathbf{b}),(\mathbf{c},\mathbf{d}) \right\rangle = \mathbf{a}^{\mathrm{T}} \mathbf{d} - \mathbf{b}^{\mathrm{T}} \mathbf{c}⟨(a,b),(c,d)⟩=aTd−bTc. As a result, many, but not all, ideas from the theory of classical error correction can be translated to quantum error correction. One of the main theoretical contribution of this article is to study stabilizer codes starting with a different symplectic form. In this paper, we concentrate on cyclic codes. Modifying the symplectic form allows us to generalize the previous known construction for linear cyclic stabilizer codes, and in the process, circumvent some of the Galois theoretic no-go results proved there. More importantly, this tweak in the symplectic form allows us to make use of well known error correcting algorithms for cyclic codes to give efficient quantum error correcting algorithms. Cyclicity of error correcting codes is a "basis dependent" property. Our codes are no more "cyclic" when they are derived using the standard symplectic forms (if we ignore the error correcting properties like distance, all such symplectic forms can be converted to each other via a basis transformation). Hence this change of perspective is crucial from the point of view of designing efficient decoding algorithm for these family of codes. In this context, recall that for general codes, efficient decoding algorithms do not exist if some widely believed complexity theoretic assumptions are true.
An error-trellis is an efficient means to accomplish maximum likelihood decoding of a linear (n,k) block code C over GF(q). It represents the coset of C that contains the symbol-by-symbol detected version of the received vector. The trellis we construct has only n-k sections and the structure of each section is determined solely by the value of an associated q-ary symbol of the syndrome. Hence, the entire set of q/sup n-k/ trellises for all the cosets of C is formed by a set of q(n-k) distinct sections. The error-trellis is invariant under permutation of coordinates and attains the vertex count of a minimal code-trellis with the same sectionalization.
The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.