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:0809.0579 · Journal of Physics A Mathematical and Theoretical

Teleportation of geometric structures in 3D

Diederik Aerts, Marek Czachor, Łukasz Orłowski

Published Mar 12, 2009• 6 citations
arXivHTMLPDF

Abstract

The simplest quantum teleportation algorithms can be represented in geometric terms in spaces of dimensions 3 (for real state vectors) and 4 (for complex state vectors). The geometric representation is based on geometric-algebra coding, a geometric alternative to the tensor-product coding typical of quantum mechanics. We discuss all the elementary ingredients of the geometric version of the algorithm: geometric analogs of states and controlled Pauli gates. A fully geometric presentation is possible if one employs a nonstandard representation of directed magnitudes, formulated in terms of colors defined via stereographic projection of a color wheel, and not by means of directed volumes.

DOI
10.1088/1751-8113/42/13/135307

Related papers

Tensor-product versus geometric-product coding

Diederik Aerts, Marek Czachor

Quantum computation is based on tensor products and entangled states. We discuss an alternative to the quantum framework where tensor products are replaced by geometric products and entangled states by multivectors. The resulting theory is analogous to quantum computation but does not involve quantum mechanics. We discuss in detail similarities and differences between the two approaches and illustrate the formulas by explicit geometric objects where multivector versions of the Bell-basis, Greenberger-Horne-Zeilinger, and Hadamard states are visualized by means of colored oriented polylines.

arXiv:0709.1268•Physical Review A•Jan 15, 2008•17 citations

The Geometry of Quantum Teleportation in Phase Space

Jasel Berra-Montiel, Alberto Molgado, Mar Sánchez-Córdova

We formulate the standard quantum teleportation protocol within the framework of deformation quantization, employing the Stratonovich--Weyl (SW) correspondence and the associated star product on the coadjoint orbits of $SU(2)$. Each qubit is represented by a spherical phase space $S^2$ endowed with the Kirillov--Kostant--Souriau (KKS) symplectic structure. Within this setting, we explicitly construct the phase space representations of the entangled Bell resource, the controlled-NOT and Hadamard operators, the projective measurements, and the conditional Pauli corrections. We demonstrate how the quantum information initially encoded in the Wigner function for the input qubit is redistributed through bipartite and tripartite phase space correlations. Upon consideration of a conditional unitary transformation, the measurement outcome produces a Wigner function on the receiver's coadjoint orbit that is related to the input by an outcome-dependent rotation, which can subsequently be inverted through the implementation of an appropriate Pauli correction. The covariance of the SW correspondence provides a geometric interpretation of this process, in which quantum information is redistributed and reconstructed without transferring the underlying symplectic geometry. Our formulation can also be extended to study quantum teleportation in more general phase space geometries equipped with suitable star products.

quant-pharXiv:2610.37861•Oct 29, 2026•0 citations

Notes on Geometric-Algebra Quantum-Like Algorithms

Diederik Aerts, Marek Czachor

In these notes we present preliminary results on quantum-like algorithms where tensor product is replaced by geometric product. Such algorithms possess the essential properties typical of quantum computation (entanglement, parallelism) but employ additional algebraic structures typical of geometric algebra -- structures absent in standard quantum computation. As a test we reformulate in Geometric Algebra terms the Deutsch-Jozsa problem.

arXiv:quant-ph/0610187•Oct 23, 2006•1 citation

Geometric Algebra Quantum Gate Decomposition

Youssef Amraoui, Zeno Toffano

Quantum gates are traditionally described using matrix and tensor-product formalisms, representations that provide limited geometric intuition. In this work, we develop a formulation of the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework in order to obtain useful quantum gate decompositions. We show that the Pauli group is naturally identified with the group of blades up to a global phase, providing an intuitive geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical performance reveals remarkably compact decompositions of Clifford operators. Finally, we show that Clifford+T universality also acquires a natural geometric interpretation through π/8-rotors within this framework. This work highlights Geometric Algebra as a potential geometric tool for quantum computation applications.

Axioms•Jul 13, 2026•0 citations

Cartoon Computation: Quantum-like computing without quantum mechanics

Diederik Aerts, Marek Czachor

We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.

•Nov 28, 2006•13 citations

Cartoon computation: quantum-like computing without quantum mechanics

Diederik Aerts, Marek Czachor

We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.

arXiv:quant-ph/0611279•Journal of Physics A Mathematical and Theoretical•Mar 14, 2007•16 citations
The simplest quantum teleportation algorithms can be represented in geometric terms in spaces of dimensions 3 (for real state vectors) and 4 (for complex state vectors). The geometric representation is based on geometric-algebra coding, a geometric alternative to the tensor-product coding typical of quantum mechanics. We discuss all the elementary ingredients of the geometric version of the algorithm: geometric analogs of states and controlled Pauli gates. A fully geometric presentation is possible if one employs a nonstandard representation of directed magnitudes, formulated in terms of colors defined via stereographic projection of a color wheel, and not by means of directed volumes.
Quantum computation is based on tensor products and entangled states. We discuss an alternative to the quantum framework where tensor products are replaced by geometric products and entangled states by multivectors. The resulting theory is analogous to quantum computation but does not involve quantum mechanics. We discuss in detail similarities and differences between the two approaches and illustrate the formulas by explicit geometric objects where multivector versions of the Bell-basis, Greenberger-Horne-Zeilinger, and Hadamard states are visualized by means of colored oriented polylines.
We formulate the standard quantum teleportation protocol within the framework of deformation quantization, employing the Stratonovich--Weyl (SW) correspondence and the associated star product on the coadjoint orbits of SU(2)SU(2)SU(2). Each qubit is represented by a spherical phase space S2S^2S2 endowed with the Kirillov--Kostant--Souriau (KKS) symplectic structure. Within this setting, we explicitly construct the phase space representations of the entangled Bell resource, the controlled-NOT and Hadamard operators, the projective measurements, and the conditional Pauli corrections. We demonstrate how the quantum information initially encoded in the Wigner function for the input qubit is redistributed through bipartite and tripartite phase space correlations. Upon consideration of a conditional unitary transformation, the measurement outcome produces a Wigner function on the receiver's coadjoint orbit that is related to the input by an outcome-dependent rotation, which can subsequently be inverted through the implementation of an appropriate Pauli correction. The covariance of the SW correspondence provides a geometric interpretation of this process, in which quantum information is redistributed and reconstructed without transferring the underlying symplectic geometry. Our formulation can also be extended to study quantum teleportation in more general phase space geometries equipped with suitable star products.
Quantum gates are traditionally described using matrix and tensor-product formalisms, representations that provide limited geometric intuition. In this work, we develop a formulation of the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework in order to obtain useful quantum gate decompositions. We show that the Pauli group is naturally identified with the group of blades up to a global phase, providing an intuitive geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical performance reveals remarkably compact decompositions of Clifford operators. Finally, we show that Clifford+T universality also acquires a natural geometric interpretation through π/8-rotors within this framework. This work highlights Geometric Algebra as a potential geometric tool for quantum computation applications.
We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.
We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.