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

IEEE International Symposium on Information Theory (ISIT)

Subsystem codes over nice nearrings

Sang-Jun Lee, Andreas Klappenecker

Published Jul 1, 2013• 2 citations
Publisher

Abstract

Subsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. Subsystem codes were most commonly based on the generalized Pauli basis with dimensions of a power of prime. Recently, a class of nice error bases indexed by a nearring were introduced by the second author. We give a construction of subsystem codes over nice nearrings. Furthermore, we show that free subsystem codes over a finite chain ring cannot perform better than those over a finite field.

DOI
10.1109/isit.2013.6620359

Related papers

Quantum Error Correcting Codes and Fault-Tolerant Quantum Computation over Nice Rings

Sang-Jun Lee

Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime.\n\nSubsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code.\n\nFault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.\n\nFinally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks.

•Jun 29, 2017•0 citations

Clifford subsystem codes

Andreas Klappenecker

Subsystem codes are a generalization of decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. The known constructions of subsystem codes from classical codes are limited to quantum systems that all have the same dimension, and this dimension must be a power of a prime. It is shown that one can remove these restrictions and obtain subsystem codes in quantum systems of arbitrary finite dimension from classical codes that are subgroups of an abelian group. The constructions are derived from Clifford codes over abstract error groups with abelian index groups.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2010•0 citations

Nice nearrings

Andreas Klappenecker

Nice error bases are a fundamental primitive of quantum information processing. For example, they govern the discretization of errors in quantum error-correcting codes. It is show that the generalized Pauli basis, the most widely used example of nice error bases, has some remarkable structural properties. However, the generalized Pauli basis is limited to dimensions that are a power of a prime, since it is constructed with the help of a finite field. A wider class of nice error bases is introduced that shares many features of the generalized Pauli basis, yet allows one to remove the restriction to prime power dimensions. The nice error bases are indexed by nearrings. Nearrings that support the construction of nice error bases are called nice. It is shown that all finite nearfields are nice. It is shown that a finite ring is nice if and only if it finite Frobenius ring. Several fundamental properties of nice nearrings are established.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2012•6 citations

On subsystem codes beating the quantum Hamming or Singleton bound

Andreas Klappenecker, Pradeep Kiran Sarvepalli

Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

arXiv:quant-ph/0703213•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 21, 2007•26 citations

Entanglement-assisted quantum error correcting codes over nice rings

Sang-Jun Lee, Andreas Klappenecker

Entanglement-assisted quantum error correcting codes have the nice feature that they can be constructed from classical additive codes that do not need to be self-orthogonal - an advantage over stabilizer codes. In the literature, these codes were investigated for finite fields, mostly binary fields. A generalization of the Pauli basis to nice error bases indexed by rings allows one to consider alphabet sizes that are not restricted to powers of a prime. The main goal of this paper is to show how entanglement-assisted quantum error correcting codes over nice rings can be constructed. We develop the rudiments of symplectic geometry over rings and prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs.

Allerton Conference on Communication, Control, and Computing•Sep 1, 2014•2 citations

Beyond stabilizer codes .I. Nice error bases

Andreas Klappenecker, Martin Rötteler

Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.

IEEE Transactions on Information Theory•Aug 1, 2002•81 citations
Subsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. Subsystem codes were most commonly based on the generalized Pauli basis with dimensions of a power of prime. Recently, a class of nice error bases indexed by a nearring were introduced by the second author. We give a construction of subsystem codes over nice nearrings. Furthermore, we show that free subsystem codes over a finite chain ring cannot perform better than those over a finite field.
Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime.\n\nSubsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code.\n\nFault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.\n\nFinally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks.
Subsystem codes are a generalization of decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. The known constructions of subsystem codes from classical codes are limited to quantum systems that all have the same dimension, and this dimension must be a power of a prime. It is shown that one can remove these restrictions and obtain subsystem codes in quantum systems of arbitrary finite dimension from classical codes that are subgroups of an abelian group. The constructions are derived from Clifford codes over abstract error groups with abelian index groups.
Nice error bases are a fundamental primitive of quantum information processing. For example, they govern the discretization of errors in quantum error-correcting codes. It is show that the generalized Pauli basis, the most widely used example of nice error bases, has some remarkable structural properties. However, the generalized Pauli basis is limited to dimensions that are a power of a prime, since it is constructed with the help of a finite field. A wider class of nice error bases is introduced that shares many features of the generalized Pauli basis, yet allows one to remove the restriction to prime power dimensions. The nice error bases are indexed by nearrings. Nearrings that support the construction of nice error bases are called nice. It is shown that all finite nearfields are nice. It is shown that a finite ring is nice if and only if it finite Frobenius ring. Several fundamental properties of nice nearrings are established.
Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.
Entanglement-assisted quantum error correcting codes have the nice feature that they can be constructed from classical additive codes that do not need to be self-orthogonal - an advantage over stabilizer codes. In the literature, these codes were investigated for finite fields, mostly binary fields. A generalization of the Pauli basis to nice error bases indexed by rings allows one to consider alphabet sizes that are not restricted to powers of a prime. The main goal of this paper is to show how entanglement-assisted quantum error correcting codes over nice rings can be constructed. We develop the rudiments of symplectic geometry over rings and prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs.