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:cs/0703129

A theorem on the quantum evaluation of Weight Enumerators for a certain class of Cyclic Codes with a note on Cyclotomic cosets

Joseph Geraci, Frank Van Bussel

Published Mar 26, 2007• 2 citations
arXivHTMLPDF

Abstract

This note is a stripped down version of a published paper on the Potts partition function, where we concentrate solely on the linear coding aspect of our approach. It is meant as a resource for people interested in coding theory but who do not know much of the mathematics involved and how quantum computation may provide a speed up in the computation of a very important quantity in coding theory. We provide a theorem on the quantum computation of the Weight Enumerator polynomial for a restricted family of cyclic codes. The complexity of obtaining an exact evaluation is $O(k^{2s}(\log q)^{2})$, where $s$ is a parameter which determines the class of cyclic codes in question, $q$ is the characteristic of the finite field over which the code is defined, and $k$ is the dimension of the code. We also provide an overview of cyclotomic cosets and discuss applications including how they can be used to speed up the computation of the weight enumerator polynomial (which is related to the Potts partition function). We also give an algorithm which returns the coset leaders and the size of each coset from the list $\{0,1,2,...,N-1\}$, whose time complexity is soft-O(N). This algorithm uses standard techniques but we include it as a resource for students. Note that cyclotomic cosets do not improve the asymptotic complexity of the computation of weight enumerators.

DOI
10.48550/arxiv.cs/0703129

Related papers

On the relation between quantum computation and classical statistical mechanics

Joseph Geraci

We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. \n\nWe accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. \n\nThe result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.

•Jan 1, 2008•0 citations

Families of Cyclic Codes over Finite Chain Rings

Andrew Nemec

A major difficulty in quantum computation and communication is preventing and correcting errors in the quantum bits. Most of the research in this area has focused on stabilizer codes derived from self-orthogonal cyclic error-correcting codes over finite fields. Our goal is to develop a similar theory for self-orthogonal cyclic codes over the class of finite chain rings which have been proven to also produce stabilizer codes. We also will discuss these restrictions on families of cyclic codes, including, but not limited to quadratic residue codes and Bose-Chaudhuri-Hocquenghem codes. Finally, we will extend the concepts of weight enumerators to the class of Frobenius rings and use them to derive bounds for our codes.

•Sep 29, 2015•0 citations

Quantum Lego Expansion Pack: Enumerators from Tensor Networks

ChunJun Cao, Michael J. Gullans, Brad Lackey +1 more

We provide the first tensor-network method for computing quantum weight enumerator polynomials in the most general form. If a quantum code has a known tensor-network construction of its encoding map, our method is far more efficient, and in some cases exponentially faster than the existing approach. As a corollary, it produces decoders and an algorithm that computes the code distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can be substantially faster compared to the current methods. We also introduce novel weight enumerators and their applications. In particular, we show that these enumerators can be used to compute logical error rates exactly and thus construct (optimal) decoders for any independent and identically distributed single qubit or qudit error channels. The enumerators also provide a more efficient method for computing nonstabilizerness in quantum many-body states. As the power for these speedups rely on a quantum Lego decomposition of quantum codes, we further provide systematic methods for decomposing quantum codes and graph states into a modular construction for which our technique applies. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the two-dimensional Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force. Published by the American Physical Society 2024

arXiv:2308.05152•PRX Quantum•Jul 22, 2024•18 citations

On Quantum Codes Obtained From Cyclic Codes Over $\F_2+u\F_2+u^2\F_2$

Sukhamoy Pattanayak, Abhay Kumar Singh, Pratyush Kumar

•Jan 11, 2016•0 citations

A Note on Cyclotomic Cosets, an Algorithm for finding Coset Representatives and Size, and a Theorem on the Quantum evaluation of Weight Enumerators for a Certain Class of Cyclic Codes

Joseph Geraci, Frank Van Bussel

We provide an overview of cyclotomic cosets and discuss applications including how they can be used to speed up the computation of the Weight Enumerator polynomial (which is related to the Potts partition function). We also give an algorithm which returns the coset leaders and the size of each coset from the list $\{0,1,2,...,N-1\}$, whose time complexity is O(N). We also provide a theorem on the quantum computation of the Weight Enumerator polynomial for a certain family of cyclic codes.

•Mar 26, 2007•2 citations

Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length 4ps Over $\mathbb {F}_{p^{m}}$

Saroj Rani, Ram Krishna Verma, Om Prakash

International Journal of Theoretical Physics•Mar 15, 2021•4 citations
This note is a stripped down version of a published paper on the Potts partition function, where we concentrate solely on the linear coding aspect of our approach. It is meant as a resource for people interested in coding theory but who do not know much of the mathematics involved and how quantum computation may provide a speed up in the computation of a very important quantity in coding theory. We provide a theorem on the quantum computation of the Weight Enumerator polynomial for a restricted family of cyclic codes. The complexity of obtaining an exact evaluation is O(k2s(log⁡q)2)O(k^{2s}(\log q)^{2})O(k2s(logq)2), where sss is a parameter which determines the class of cyclic codes in question, qqq is the characteristic of the finite field over which the code is defined, and kkk is the dimension of the code. We also provide an overview of cyclotomic cosets and discuss applications including how they can be used to speed up the computation of the weight enumerator polynomial (which is related to the Potts partition function). We also give an algorithm which returns the coset leaders and the size of each coset from the list {0,1,2,...,N−1}\{0,1,2,...,N-1\}{0,1,2,...,N−1}, whose time complexity is soft-O(N). This algorithm uses standard techniques but we include it as a resource for students. Note that cyclotomic cosets do not improve the asymptotic complexity of the computation of weight enumerators.
We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. \n\nWe accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. \n\nThe result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.
A major difficulty in quantum computation and communication is preventing and correcting errors in the quantum bits. Most of the research in this area has focused on stabilizer codes derived from self-orthogonal cyclic error-correcting codes over finite fields. Our goal is to develop a similar theory for self-orthogonal cyclic codes over the class of finite chain rings which have been proven to also produce stabilizer codes. We also will discuss these restrictions on families of cyclic codes, including, but not limited to quadratic residue codes and Bose-Chaudhuri-Hocquenghem codes. Finally, we will extend the concepts of weight enumerators to the class of Frobenius rings and use them to derive bounds for our codes.
We provide the first tensor-network method for computing quantum weight enumerator polynomials in the most general form. If a quantum code has a known tensor-network construction of its encoding map, our method is far more efficient, and in some cases exponentially faster than the existing approach. As a corollary, it produces decoders and an algorithm that computes the code distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can be substantially faster compared to the current methods. We also introduce novel weight enumerators and their applications. In particular, we show that these enumerators can be used to compute logical error rates exactly and thus construct (optimal) decoders for any independent and identically distributed single qubit or qudit error channels. The enumerators also provide a more efficient method for computing nonstabilizerness in quantum many-body states. As the power for these speedups rely on a quantum Lego decomposition of quantum codes, we further provide systematic methods for decomposing quantum codes and graph states into a modular construction for which our technique applies. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the two-dimensional Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force. Published by the American Physical Society 2024
We provide an overview of cyclotomic cosets and discuss applications including how they can be used to speed up the computation of the Weight Enumerator polynomial (which is related to the Potts partition function). We also give an algorithm which returns the coset leaders and the size of each coset from the list {0,1,2,...,N−1}\{0,1,2,...,N-1\}{0,1,2,...,N−1}, whose time complexity is O(N). We also provide a theorem on the quantum computation of the Weight Enumerator polynomial for a certain family of cyclic codes.
Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length 4ps Over Fpm\mathbb {F}_{p^{m}}Fpm​