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

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Fast quantum Fourier-Weyl-Heisenberg transforms

Valeri G. Labunets, Ekaterina V. Labunets-Rundblad, Jaakko T. Astola

Published Jul 26, 2001• 1 citation
Publisher

Abstract

We study the Fourier harmonic analysis of a functions on discrete 1D and nD Heisenberg-Weyl groups HW3 and HW2n+1, where K equals GF(2), GF(2m), GF(p), GF(pm) are the Galois fields, and develop fast quantum Fourier- Heisenberg-Weyl transforms on this groups.
DOI
10.1117/12.434210

Related papers

Fast Quantum Fourier-Heisenberg-Weyl Transforms

Valery Labunets, E. Rundblad, Jaakko T. Astola

We study the Fourier harmonic analysis of functions on discrete Heisenberg-Weyl groups and develop fast quantum Fourier-Heisenberg-Weyl transforms on these groups.

Birkhäuser Boston eBooks•Jan 1, 2002•1 citation

Quantum multiresolution analysis via fast Fourier transform on Heisenberg group

E. Rundblad, Valeriy G. Labunets, Peter Novak

We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW2N+1(ZNmn, ZNmn, Zmn) over the ring Zmn. This group is the discrete version of the real Heisenberg group HW2N+1(RN, RN, R), where R is the real field. These functions have one dimensional and m1, m2,...,mn-dimensions matrix valued spectral components (for irreducible representations of HW2N+1. The family of all 1D representations gives classical world world (CW). Various mi-dimension representations (i=1,2,...,n) map classical world (CW) into quantum worlds QW}mi of ith resolution i=1,2,...,n). Worlds QW(m1) and QW(mn) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Mar 28, 2005•0 citations

Quantum multiresolution analysis via fast Fourier transforms on Heisenbrg finite group

E. Rundblad, Valery Labunets, Péter Novák

We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW 2N+1 (Z N m n,Z N m n, Z m n) over the ring Z m n. This group is the discrete version of the real Heisenberg group HW 2N+1 (R N , R N , R), where R is the real field. These functions have one dimensional and m 1 , m 2 ,..., m n -dimensions matrix valued spectral components (for irreducible representations of HW 2N+1 . The family of all ID representations gives classical world world (CW). Various m i -dimension representations (i = 1,2,...,n) map classical world (CW) into quantum worlds QW(m i ) of ith resolution (i = 1,2,....n). Worlds QW(m 1 ) and QW(m n ) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.

•Jan 1, 2005•0 citations

Fast Quantum n-D Fourier and Radon Transforms

Vladimir G. Labunets, E. Rundblad, Jaakko T. Astola

Fast classical and quantum algorithms are introduced for a wide class of nonseparable nD discrete unitary K-transforms (DKT) $$ {\cal K}_{N^n } $$ .

Birkhäuser Boston eBooks•Jan 1, 2002•1 citation

Chapter 41 Fast Quantum n-D Fourier and Radon Transforms

E. Rundblad, Jaakko T. Astola

Fast classical and quantum algorithms are introduced for a wide class of nonseparable nD discrete unitary K-transforms (DKT) KNn.

•Jan 1, 2002•0 citations

Galois quantum systems

Apostol Vourdas

A finite quantum system in which the position and momentum take values in the Galois field GF( p ℓ ) is constructed from a smaller quantum system in which the position and momentum take values in , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg–Weyl group of displacements and the Sp(2, GF( p ℓ )) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF( p ℓ ) is discussed.

Journal of Physics A Mathematical and General•Sep 14, 2005•40 citations
We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW2N+1(ZNmn, ZNmn, Zmn) over the ring Zmn. This group is the discrete version of the real Heisenberg group HW2N+1(RN, RN, R), where R is the real field. These functions have one dimensional and m1, m2,...,mn-dimensions matrix valued spectral components (for irreducible representations of HW2N+1. The family of all 1D representations gives classical world world (CW). Various mi-dimension representations (i=1,2,...,n) map classical world (CW) into quantum worlds QW}mi of ith resolution i=1,2,...,n). Worlds QW(m1) and QW(mn) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.
We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW 2N+1 (Z N m n,Z N m n, Z m n) over the ring Z m n. This group is the discrete version of the real Heisenberg group HW 2N+1 (R N , R N , R), where R is the real field. These functions have one dimensional and m 1 , m 2 ,..., m n -dimensions matrix valued spectral components (for irreducible representations of HW 2N+1 . The family of all ID representations gives classical world world (CW). Various m i -dimension representations (i = 1,2,...,n) map classical world (CW) into quantum worlds QW(m i ) of ith resolution (i = 1,2,....n). Worlds QW(m 1 ) and QW(m n ) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.
Fast classical and quantum algorithms are introduced for a wide class of nonseparable nD discrete unitary K-transforms (DKT) KNn {\cal K}_{N^n } KNn​ .
A finite quantum system in which the position and momentum take values in the Galois field GF( p ℓ ) is constructed from a smaller quantum system in which the position and momentum take values in , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg–Weyl group of displacements and the Sp(2, GF( p ℓ )) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF( p ℓ ) is discussed.