The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.

Journal of Xidian UniversityJan 1, 20070 citations

Towards the exact spectrum of the three-anyon problem

Stefan V. Mashkevich

Physics Letters BDec 1, 19926 citations

NONBINARY QUANTUM CODES DERIVED FROM REPEATED-ROOT CYCLIC CODES

Jianfa Qian, LINA ZHANG

Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.

Modern Physics Letters BMar 4, 20137 citations

On the error-correcting capabilities of iterative error correction codes

Vasyl Semerenko

The influence of the theory of information on development of the error correcting coding theory has been studied. Main differences between the probabilistic approach and the deterministic approach in the analysis of error-correcting capabilities of different classes of linear codes have been demonstrated.The automaton hierarchical models for analysis of permutation decoding of cyclic codes have been developed and a cyclic permutation generator based on two Moore automata has been proposed.A study has been carried out into the regular and irregular states of linear finite-state machines (LFSM) based on the automaton representation of cyclic codes. A possibility of significant simplification of decoding of cyclic codes based on conversion of irregular LFSM syndromes into regular ones using permutations has been shown.The formalized methods for determination of error-correcting capabilities of iteratively decoded cyclic codes (IDCC) have been devised. They imply the replacementof traditional complete checking of all possible options for comparison of code words to directional search for the solution of the assigned problem, which leads to a significant time saving for calculations. The algorithm for determination of error-correcting capabilities of IDCC with respect to double errors is given.It has been shown that all iterative codes increase their error-correcting capabilities with an increase in the number of iterations and one can set it as a percentage for errors of various multiplicities. A distribution of error syndromes to separate iterations has been performed, which makes it possible to reduce the length of a check word in a code. As a result, this leads to an increase in a rate of iterative codes in comparison with the traditional correction codes.A comparative analysis of IDCC and LDPC codes has been carried out to determine a scope of their optimal useThe influence of the theory of information on development of the error correcting coding theory has been studied. Main differences between the probabilistic approach and the deterministic approach in the analysis of error-correcting capabilities of different classes of linear codes have been demonstrated.The automaton hierarchical models for analysis of permutation decoding of cyclic codes have been developed and a cyclic permutation generator based on two Moore automata has been proposed.A study has been carried out into the regular and irregular states of linear finite-state machines (LFSM) based on the automaton representation of cyclic codes. A possibility of significant simplification of decoding of cyclic codes based on conversion of irregular LFSM syndromes into regular ones using permutations has been shown.The formalized methods for determination of error-correcting capabilities of iteratively decoded cyclic codes (IDCC) have been devised. They imply the replacementof traditional complete checking of all possible options for comparison of code words to directional search for the solution of the assigned problem, which leads to a significant time saving for calculations. The algorithm for determination of error-correcting capabilities of IDCC with respect to double errors is given.It has been shown that all iterative codes increase their error-correcting capabilities with an increase in the number of iterations and one can set it as a percentage for errors of various multiplicities. A distribution of error syndromes to separate iterations has been performed, which makes it possible to reduce the length of a check word in a code. As a result, this leads to an increase in a rate of iterative codes in comparison with the traditional correction codes.A comparative analysis of IDCC and LDPC codes has been carried out to determine a scope of their optimal use

Eastern-European Journal of Enterprise TechnologiesFeb 21, 20191 citation

Integer Linear Programming Decoder for Abelian and Non-Abelian Topological Codes

Dian Jing, Aubrey Zhang, Liang Jiang +1 more

Topological orders (TOs) are widely used as quantum error-correcting codes, with anyon excitations serving as error syndromes. For certain Abelian TOs, decoding can be performed by independently matching particle-antiparticle pairs of each species. However, matching-based decoders cannot handle more general fusion rules in either Abelian or non-Abelian TOs, nor account for noise that correlates different anyon species. While clustering decoders are more broadly applicable, they typically neglect anyon data and fusion properties, leading to poor performance in practice. In this work, we introduce a fundamentally different decoder for arbitrary TOs based on integer linear programming (ILP). The ILP formulation linearizes the error-correction problem through the introduction of auxiliary variables and encodes fusion rules as linear constraints. Classical optimization then identifies the minimum-weight error configuration. As concrete examples, we determine error-correction thresholds for three TOs: the Abelian $\mathbb{Z}_2$ TO under depolarizing noise, where charge and flux errors are correlated; the Abelian $\mathbb{Z}_3$ TO, which does not admit a pairwise matching decoder; and the non-Abelian $D_4$ TO under noise channels that generate all anyon species. We demonstrate the versatility of the ILP decoder by showing a clear performance advantage over most existing decoders in all three cases. We further extend the method to incorporate noisy syndrome measurements and propose a just-in-time variant for continuous error correction. Our results establish ILP as a natural framework for handling correlated errors and general anyon fusion rules, and as a powerful and flexible general-purpose decoder for incoherent anyon noise in arbitrary TOs, with applications to fault-tolerant quantum computation.Topological orders (TOs) are widely used as quantum error-correcting codes, with anyon excitations serving as error syndromes. For certain Abelian TOs, decoding can be performed by independently matching particle-antiparticle pairs of each species. However, matching-based decoders cannot handle more general fusion rules in either Abelian or non-Abelian TOs, nor account for noise that correlates different anyon species. While clustering decoders are more broadly applicable, they typically neglect anyon data and fusion properties, leading to poor performance in practice. In this work, we introduce a fundamentally different decoder for arbitrary TOs based on integer linear programming (ILP). The ILP formulation linearizes the error-correction problem through the introduction of auxiliary variables and encodes fusion rules as linear constraints. Classical optimization then identifies the minimum-weight error configuration. As concrete examples, we determine error-correction thresholds for three TOs: the Abelian $\mathbb{Z}_2$ TO under depolarizing noise, where charge and flux errors are correlated; the Abelian $\mathbb{Z}_3$ TO, which does not admit a pairwise matching decoder; and the non-Abelian $D_4$ TO under noise channels that generate all anyon species. We demonstrate the versatility of the ILP decoder by showing a clear performance advantage over most existing decoders in all three cases. We further extend the method to incorporate noisy syndrome measurements and propose a just-in-time variant for continuous error correction. Our results establish ILP as a natural framework for handling correlated errors and general anyon fusion rules, and as a powerful and flexible general-purpose decoder for incoherent anyon noise in arbitrary TOs, with applications to fault-tolerant quantum computation.

quant-pharXiv:2608.18512Aug 20, 20260 citations

\mathbb{Z}_2

TO under depolarizing noise, where charge and flux errors are correlated; the Abelian

\mathbb{Z}_3

TO, which does not admit a pairwise matching decoder; and the non-Abelian

D_4

TO under noise channels that generate all anyon species. We demonstrate the versatility of the ILP decoder by showing a clear performance advantage over most existing decoders in all three cases. We further extend the method to incorporate noisy syndrome measurements and propose a just-in-time variant for continuous error correction. Our results establish ILP as a natural framework for handling correlated errors and general anyon fusion rules, and as a powerful and flexible general-purpose decoder for incoherent anyon noise in arbitrary TOs, with applications to fault-tolerant quantum computation.