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 flow of Bloch sphere manifolds and Quantum error corrections

Genji Fujii

Published Dec 17, 2025• 0 citations
OpenAlex

Abstract

It is well known that quantum states, particularly qubits and BECs qubits, can be represented by a spherical structure known as the Bloch sphere. Several error models describe errors that effectively shrink the Bloch sphere, such as the depolarizing channel and particle loss, which can be mitigated through redundancy error correction models. In contrast, it may be possible to realize error correction by directly deforming the Bloch sphere itself. Geometric flows, particularly the Ricci flow, describe the evolution of a metric over time. In the Ricci flow equation, the metric contracts along directions of higher curvature, leading to a uniformization of curvature across the manifold. In our approach, we utilize such geometric flows to achieve error correction by allowing the metric of the Bloch sphere naturally defined by geometric quantities such as the Fubini-Study or Bures metric to evolve. To employ these geometric quantities and equations, it is necessary to endow the Bloch sphere with a differentiable manifold structure, which can indeed be done. Moreover, our error correction scheme is distinctive in that it does not rely on redundancy.

Related papers

Ricci flow for the Bures--Helstrom qubit metric

Andrew Lesniewski

The Bures--Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three--sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton--DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function $Ψ=Φ^2$ satisfies the linear forced heat equation \begin{equation*} D_tΨ=Ψ_{ss}-2, \end{equation*} while the fixed-lapse coordinate form contains the associated transport term. Since the Bures--Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \begin{equation*} g(t)=(1-4t)g_{\mathrm{BH}}, \end{equation*} with scalar curvature $6/(1-4t)$ and extinction time $T=1/4$. Thus the metric remains inside the monotone cone for all $t<T$ and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume--normalized flow, for which the Bures--Helstrom metric is a fixed point. Its linearization is the shifted round--sphere Laplacian $Δ_{\mathbb S^3}+3$, with spectrum \begin{equation*} σ_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap $5$ after removal of the scaling mode.

cs.ITCross-listedarXiv:2606.19493•Jun 19, 2026

Stereographic Projection of Bloch Sphere and Quantum Error Correction

Unnathi Venkatesh

The Bloch sphere is a powerful geometrical representation of qubit states, fundamental to understanding quantum information theory. This paper explores the stereographic projection of the Bloch sphere, focusing on its utility in addressing decoherence and quantum error correction, two major challenges in quantum computing. By mapping the complex quantum states onto a two-dimensional plane, the stereographic projection simplifies the visualization and manipulation of qubit transformations. We investigate how this projection aids in identifying and correcting errors that arise from decoherence---such as phase flips and bit flips---while maintaining the integrity of quantum states. Additionally, we investigate the influence of decoherence on qubits through this projection, demonstrating how error correction protocols can be optimized by mapping errors onto the complex plane. The analysis highlights the potential of stereographic projection in developing more robust strategies for mitigating decoherence, thereby advancing the overall performance of quantum systems.

•Oct 30, 2024•0 citations

Quantum State Curvature Flow (QSCF)

Jincheng Zhang

Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.

•Aug 17, 2026•0 citations

Quantum Geometrization B: Quantum Geometrization Phase Transition Induced by Discrete Ricci Flow

changzheng zhou, ZIQING zhou

This paper proposes a novel theoretical framework based on discrete geometricflow and quantum state evolution to study topological phase transitions and theemergence of low-energy effective physics in quantum geometry. We construct discrete geometric dynamics on a three-dimensional periodic point cloud, driving edgelength evolution via the discrete Ricci flow equation, and attach a four-dimensionallocal Hilbert space to each edge, representing the quantum state of the system usingmatrix product states (MPS). Numerical simulations show that during the evolution of the discrete Ricci flow, the system undergoes a topological phase transitionat a specific curvature threshold, manifested as a jump in Betti numbers detectedby persistent homology. After the phase transition, the geometric mean curvature and the low-energy spectral gap of the quantum system synchronously entera plateau region, exhibiting a stable inverse relationship: ¯ ∆ = Λ/∥ ¯R∥+δ, whereΛ is an O(1) constant and δ is a small correction. This result, for the first timeat the numerical level, realizes a complete causal chain from pure geometric evolution to the generation of quantum low-energy parameters, providing a computableexample for the idea that ”geometric flow is renormalization group flow”.

•Dec 22, 2025•0 citations

A Bloch-sphere-type model for two qubits in the geometric algebra of a 6D Euclidean vector space

Timothy F. Havel, Chris Doran

Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.

arXiv:quant-ph/0403136•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 24, 2004•14 citations

Geometric Quantum Mechanics in a Symplectic Framework: Metric-Affine Extensions and Deformed Quantum Dynamics

Hoshang Heydari

We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard Kähler framework, we introduce an extension in which the symplectic structure is allowed to couple to a metric-affine background geometry, leading to a deformation of the Hamiltonian flow on the state space. We show that, under suitable conditions, the deformed structure remains symplectic and defines a well-posed Hamiltonian system. The formulation reduces to standard Schrödinger dynamics in the limit where the geometric deformation vanishes. Explicit analytical examples are constructed to illustrate the effect of the deformation. In particular, curvature-dependent deformations lead to a rescaling of Hamiltonian flows, while torsion-induced contributions produce direction-dependent corrections. In addition, geometric phases acquire corrections determined by the deformed symplectic structure. These results provide a mathematically consistent framework for exploring geometric modifications of quantum evolution induced by background curvature and affine structure.

quant-pharXiv:2603.22354•Mar 24, 2026•0 citations
It is well known that quantum states, particularly qubits and BECs qubits, can be represented by a spherical structure known as the Bloch sphere. Several error models describe errors that effectively shrink the Bloch sphere, such as the depolarizing channel and particle loss, which can be mitigated through redundancy error correction models. In contrast, it may be possible to realize error correction by directly deforming the Bloch sphere itself. Geometric flows, particularly the Ricci flow, describe the evolution of a metric over time. In the Ricci flow equation, the metric contracts along directions of higher curvature, leading to a uniformization of curvature across the manifold. In our approach, we utilize such geometric flows to achieve error correction by allowing the metric of the Bloch sphere naturally defined by geometric quantities such as the Fubini-Study or Bures metric to evolve. To employ these geometric quantities and equations, it is necessary to endow the Bloch sphere with a differentiable manifold structure, which can indeed be done. Moreover, our error correction scheme is distinctive in that it does not rely on redundancy.
The Bures--Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three--sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton--DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function Ψ=Φ2Ψ=Φ^2Ψ=Φ2 satisfies the linear forced heat equation \begin{equation*} D_tΨ=Ψ_{ss}-2, \end{equation*} while the fixed-lapse coordinate form contains the associated transport term. Since the Bures--Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \begin{equation*} g(t)=(1-4t)g_{\mathrm{BH}}, \end{equation*} with scalar curvature 6/(1−4t)6/(1-4t)6/(1−4t) and extinction time T=1/4T=1/4T=1/4. Thus the metric remains inside the monotone cone for all t<Tt<Tt<T and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume--normalized flow, for which the Bures--Helstrom metric is a fixed point. Its linearization is the shifted round--sphere Laplacian ΔS3+3Δ_{\mathbb S^3}+3ΔS3​+3, with spectrum \begin{equation*} σ_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap 555 after removal of the scaling mode.
The Bloch sphere is a powerful geometrical representation of qubit states, fundamental to understanding quantum information theory. This paper explores the stereographic projection of the Bloch sphere, focusing on its utility in addressing decoherence and quantum error correction, two major challenges in quantum computing. By mapping the complex quantum states onto a two-dimensional plane, the stereographic projection simplifies the visualization and manipulation of qubit transformations. We investigate how this projection aids in identifying and correcting errors that arise from decoherence---such as phase flips and bit flips---while maintaining the integrity of quantum states. Additionally, we investigate the influence of decoherence on qubits through this projection, demonstrating how error correction protocols can be optimized by mapping errors onto the complex plane. The analysis highlights the potential of stereographic projection in developing more robust strategies for mitigating decoherence, thereby advancing the overall performance of quantum systems.
Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.
This paper proposes a novel theoretical framework based on discrete geometricflow and quantum state evolution to study topological phase transitions and theemergence of low-energy effective physics in quantum geometry. We construct discrete geometric dynamics on a three-dimensional periodic point cloud, driving edgelength evolution via the discrete Ricci flow equation, and attach a four-dimensionallocal Hilbert space to each edge, representing the quantum state of the system usingmatrix product states (MPS). Numerical simulations show that during the evolution of the discrete Ricci flow, the system undergoes a topological phase transitionat a specific curvature threshold, manifested as a jump in Betti numbers detectedby persistent homology. After the phase transition, the geometric mean curvature and the low-energy spectral gap of the quantum system synchronously entera plateau region, exhibiting a stable inverse relationship: ¯ ∆ = Λ/∥ ¯R∥+δ, whereΛ is an O(1) constant and δ is a small correction. This result, for the first timeat the numerical level, realizes a complete causal chain from pure geometric evolution to the generation of quantum low-energy parameters, providing a computableexample for the idea that ”geometric flow is renormalization group flow”.
Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.
We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard Kähler framework, we introduce an extension in which the symplectic structure is allowed to couple to a metric-affine background geometry, leading to a deformation of the Hamiltonian flow on the state space. We show that, under suitable conditions, the deformed structure remains symplectic and defines a well-posed Hamiltonian system. The formulation reduces to standard Schrödinger dynamics in the limit where the geometric deformation vanishes. Explicit analytical examples are constructed to illustrate the effect of the deformation. In particular, curvature-dependent deformations lead to a rescaling of Hamiltonian flows, while torsion-induced contributions produce direction-dependent corrections. In addition, geometric phases acquire corrections determined by the deformed symplectic structure. These results provide a mathematically consistent framework for exploring geometric modifications of quantum evolution induced by background curvature and affine structure.