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

On the use of Galois fields and Galois rings in quantum information

M. Kibler

Published Sep 17, 2018• 0 citations
OpenAlex

Related papers

Quantum mechanics over Galois extensions of number fields

Andrei Yu. Khrennikov

Soviet physics. Doklady•Dec 1, 1990•4 citations

Galois algebras of squeezed quantum phase states

Michel R. P. Planat, Метод Санига

Coding, transmission and recovery of quantum states with high security and efficiency, and with as low fluctuations as possible, is the main goal of quantum information protocols and their proper technical implementations. The paper deals with this topic, focusing on the quantum states related to Galois algebras. We first review the constructions of complete sets of mutually unbiased bases in a Hilbert space of dimension q = p m , with p being a prime and m a positive integer, employing the properties of Galois fields F q (for p >2) and/or Galois rings of characteristic four R 4 m (for p = 2). We then discuss the Gauss sums and their role in describing quantum phase fluctuations. Finally, we examine an intricate connection between the concepts of mutual unbiasedness and maximal entanglement.

Journal of Optics B Quantum and Semiclassical Optics•Nov 4, 2005•2 citations

QUANTUM SYSTEMS BASED UPON GALOIS FIELDS — FROM SUB-QUANTUM TO SUPER-QUANTUM CORRELATIONS

Lay Nam Chang, Zachary Lewis, Djordje Minić +1 more

Conference in Honour of the 90th Birthday of Freeman Dyson•May 25, 2014•0 citations

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

Michel R. P. Planat, Haret Codratian Rosu, Serge Perrine +1 more

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. KEY WORDS: mutually unbiased bases; d-dimensional Hilbert space; Galois fields and rings; maximally entangled states.

•Jan 1, 2006•32 citations

Quantum Fourier Transform Over Galois Rings

Yong Zhang

Galois rings are regarded as "building blocks" of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently preformed on a quantum computer. The properties of the QFT over Galois rings lead to the quantum algorithm for hidden linear structures over Galois rings.

arXiv:0904.2560•Apr 16, 2009•1 citation

Quantum statistical mechanics and Galois theory

Matilde Marcolli

University lecture series•Aug 9, 2005•1 citation
Coding, transmission and recovery of quantum states with high security and efficiency, and with as low fluctuations as possible, is the main goal of quantum information protocols and their proper technical implementations. The paper deals with this topic, focusing on the quantum states related to Galois algebras. We first review the constructions of complete sets of mutually unbiased bases in a Hilbert space of dimension q = p m , with p being a prime and m a positive integer, employing the properties of Galois fields F q (for p >2) and/or Galois rings of characteristic four R 4 m (for p = 2). We then discuss the Gauss sums and their role in describing quantum phase fluctuations. Finally, we examine an intricate connection between the concepts of mutual unbiasedness and maximal entanglement.
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. KEY WORDS: mutually unbiased bases; d-dimensional Hilbert space; Galois fields and rings; maximally entangled states.
Galois rings are regarded as "building blocks" of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently preformed on a quantum computer. The properties of the QFT over Galois rings lead to the quantum algorithm for hidden linear structures over Galois rings.