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

"Geometric Interpretation of Quantum Phase Evolution using De Moivre's Theorem"

Suman Kumar Das

Published Jun 29, 2026• 0 citations
Publisher

Abstract

This paper presents a Geometric Interpretation of Quantum Phase Evolution using De Moivre's Theorem and the Z (Complex) Plane. It connects Complex Analysis, Mathematical Physics and Engineering Mathematics through the visualization of Phase Rotation and Roots of Unity.
DOI
10.5281/zenodo.21018789

Related papers

"Quantum State Normalization in Z (Complex) Plane: A Geometric Interpretation".

Suman Kumar Das

This paper explains how Quantum State Normalization works using the Complex Plane and the Unit-Circle concept. The work looks at how the Quantum Superposition: ∣α∣^2+∣β∣^2=1 can be shown with Complex Numbers and Phase Analysis. The paper connects Quantum Mechanics to Engineering Math, Signals & Systems and Control Theory by linking Quantum Probabilities to polar and exponential forms of complex numbers. It also talks about how this relates to oscillating systems Fourier analysis and Complex Plane analysis in Engineering. This work aims help to understand principles through simple Geometric ideas, bridging Quantum Mechanics, Math and Engineering. It is meant to help understand Physics and Quantum Geometry.

•Apr 30, 2026•0 citations

Helical Representation of Complex Phase Evolution: A Geometric Visualization of Temporal Order

Krüger, Marcel

Complex phase evolution underlies the dynamics of quantum systems, yet its geometric interpretation often remains abstract. We present a formal visualization in which the time evolution of a complex phase is represented by a helical trajectory in three-dimensional space. This helical representation provides an intuitive geometric link between complex phase rotation, angular frequency, and temporal order. The construction is fully compatible with standard quantum mechanics and field theory: it does not postulate time as emergent but visualizes how temporal sequencing arises from continuous phase evolution. Applications to qubit dynamics and spiralmodulated propagators demonstrate that this geometric view offers both pedagogical clarity and physical interpretability.

•Nov 5, 2025•0 citations

Geometric phase in the G3+ quantum state evolution

Alexander Soiguine

When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.

arXiv:1511.02777•Oct 23, 2015•2 citations

Quantum-Enhanced Geometric Phase Field Analysis

Jincheng Zhang

Quantum-enhanced Geometric Phase Field Analysis presents a novel methodology for investigating complex geometric phase field dynamics. This research delves into the application of quantum algorithms to accelerate and refine the analysis of these fields, offering unprecedented insights into fundamental physics. The core mechanism centers around 'Quantum Geometric Phase Field Analysis,' a process designed to effectively navigate and understand intricate geometric phase field evolution. This work aims to provide a fundamentally new approach to solving problems currently challenging the intersection of quantum mechanics and geometric physics, potentially unlocking new discoveries and insights into complex systems.

•Aug 29, 2026•0 citations

Geometric Phase Transitions – Beyond the Classical Model

Jincheng Zhang

Geometric phase transitions, a burgeoning field in theoretical physics, seeks to bridge the gap between classical mechanics and the complex behavior of quantum systems. This paper proposes a novel mathematical framework, centered around the concept of geometric phase transitions, to address the limitations of traditional approaches. We introduce a new tool, the "Geometric Evolution Landscape" (GEL), that mathematically models the transition points of a system, moving beyond simple temperature dependence and incorporating geometric properties that significantly influence the dynamics. This approach offers a potential pathway to understanding transitions in systems characterized by intricate geometries and potentially exhibiting non-equilibrium behavior. The paper details the core mechanism, provides a preliminary mathematical model, and discusses the implications of this new perspective for various physical systems.

•Aug 31, 2026•0 citations

Hardware & Control Plane: A Mathematics & Topology Approach to Jitter-Free Quantum Phase Generation

Reginos Vlachos

Physical Quantum Processing Units (QPUs) require ultra-precise classical control systems to execute phase rotation gatesbefore quantum states suffer thermal decoherence. This paper presents a novel hybrid architecture that unifies the Arithmetic DescentAlgorithm (ADA) with a nested 2D complex root decomposition framework (3 × 3 bifurcation). By mapping higher-dimensional (9D)entangled state vectors into isolated 2D complex planes (C), we establish an “Associative Shield” that eliminates non-commutativestructural distortions. Simultaneously, the ADA cubic engine (k = 3) is injected into the classical control loop, replacing costlytranscendental De Moivre phase computations with multiplier-free finite difference subtractions. We prove that the structural invariant3! = 6 of ADA generates the exact 40◦ angular nonagon increments needed for 9th-root quantum phase compilation. This hybridapproach eliminates algorithmic timing jitter (O(1) temporal determinism), drastically reduces hardware overhead, and enablesfault-tolerant quantum control execution suitable for power-constrained or harsh avionics environments.

•Jul 25, 2026•0 citations
This paper explains how Quantum State Normalization works using the Complex Plane and the Unit-Circle concept. The work looks at how the Quantum Superposition: ∣α∣^2+∣β∣^2=1 can be shown with Complex Numbers and Phase Analysis. The paper connects Quantum Mechanics to Engineering Math, Signals & Systems and Control Theory by linking Quantum Probabilities to polar and exponential forms of complex numbers. It also talks about how this relates to oscillating systems Fourier analysis and Complex Plane analysis in Engineering. This work aims help to understand principles through simple Geometric ideas, bridging Quantum Mechanics, Math and Engineering. It is meant to help understand Physics and Quantum Geometry.
Complex phase evolution underlies the dynamics of quantum systems, yet its geometric interpretation often remains abstract. We present a formal visualization in which the time evolution of a complex phase is represented by a helical trajectory in three-dimensional space. This helical representation provides an intuitive geometric link between complex phase rotation, angular frequency, and temporal order. The construction is fully compatible with standard quantum mechanics and field theory: it does not postulate time as emergent but visualizes how temporal sequencing arises from continuous phase evolution. Applications to qubit dynamics and spiralmodulated propagators demonstrate that this geometric view offers both pedagogical clarity and physical interpretability.
When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.
Quantum-enhanced Geometric Phase Field Analysis presents a novel methodology for investigating complex geometric phase field dynamics. This research delves into the application of quantum algorithms to accelerate and refine the analysis of these fields, offering unprecedented insights into fundamental physics. The core mechanism centers around 'Quantum Geometric Phase Field Analysis,' a process designed to effectively navigate and understand intricate geometric phase field evolution. This work aims to provide a fundamentally new approach to solving problems currently challenging the intersection of quantum mechanics and geometric physics, potentially unlocking new discoveries and insights into complex systems.
Geometric phase transitions, a burgeoning field in theoretical physics, seeks to bridge the gap between classical mechanics and the complex behavior of quantum systems. This paper proposes a novel mathematical framework, centered around the concept of geometric phase transitions, to address the limitations of traditional approaches. We introduce a new tool, the "Geometric Evolution Landscape" (GEL), that mathematically models the transition points of a system, moving beyond simple temperature dependence and incorporating geometric properties that significantly influence the dynamics. This approach offers a potential pathway to understanding transitions in systems characterized by intricate geometries and potentially exhibiting non-equilibrium behavior. The paper details the core mechanism, provides a preliminary mathematical model, and discusses the implications of this new perspective for various physical systems.
Physical Quantum Processing Units (QPUs) require ultra-precise classical control systems to execute phase rotation gatesbefore quantum states suffer thermal decoherence. This paper presents a novel hybrid architecture that unifies the Arithmetic DescentAlgorithm (ADA) with a nested 2D complex root decomposition framework (3 × 3 bifurcation). By mapping higher-dimensional (9D)entangled state vectors into isolated 2D complex planes (C), we establish an “Associative Shield” that eliminates non-commutativestructural distortions. Simultaneously, the ADA cubic engine (k = 3) is injected into the classical control loop, replacing costlytranscendental De Moivre phase computations with multiplier-free finite difference subtractions. We prove that the structural invariant3! = 6 of ADA generates the exact 40◦ angular nonagon increments needed for 9th-root quantum phase compilation. This hybridapproach eliminates algorithmic timing jitter (O(1) temporal determinism), drastically reduces hardware overhead, and enablesfault-tolerant quantum control execution suitable for power-constrained or harsh avionics environments.