Quantum Riemannian Geometry

QNFO Research

Quantum Riemannian Geometry

Published: 2026-07-04

Quantum Riemannian Geometry

Curved

Geometry of Quantum State Manifolds and its Physical Implications

Author: Rowan Brad Quni-Gudzinas

Contact: rowan.quni@outlook.com ORCID:

0009-0002-4317-5604 ISNI: 0000000526456062

DOI: 10.5281/zenodo.18441016 Date:

2026-01-31 Version: 1.0

Abstract: This manuscript explores the Riemannian

geometry of quantum state manifolds and its physical implications. We

establish that quantum state manifolds, as submanifolds of Hilbert

space, possess intrinsic Riemannian geometry characterized by a metric

tensor derived from the Hilbert space inner product. This geometric

framework reveals fundamental connections: the distance between nearby

states scales with quantum fluctuations, curvature indicates state

robustness, and geometric phase transitions occur in many-body systems.

We review historical developments from early quantum geometry to modern

quantum information geometry, and present methodologies for computing

geometric quantities and their physical interpretations. Results include

distance-fluctuation relations for coherent states, curvature

calculations for SU(2) manifolds, and applications to quantum

optimization and metrology via the quantum Fisher information metric.

The discussion interprets geometric quantities in terms of uncertainty

principles, state robustness, and quantum criticality, while outlining

future research directions in quantum chaos-geometry correlations,

non-associative geometry, and geometric quantum algorithm design. We

conclude that Riemannian geometry provides a powerful framework for

understanding and exploiting the structure of quantum state space, with

applications across quantum information processing, metrology, and

foundational physics.

Keywords: Quantum Geometry, Riemannian Manifolds,

Quantum Information, Metric Tensor, Quantum Fluctuations, Quantum

Metrology, Geometric Quantum Mechanics

1.0 Introduction: The Geometric Framework of Quantum States |

References |

Beggs, E., & Majid, S. (2020). *Quantum Riemannian

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https://doi.org/10.1007/978-3-030-30294-8 - Beggs, E. J., & Majid,

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uncertainty relations, and the geometry of quantum state manifolds.

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88(6). https://doi.org/10.1103/PhysRevB.88.064304 - Luchnikov,

I. A., Fistul, M. V., & Ustinov, S. V. (2021). Riemannian geometry

and automatic differentiation for optimization problems of quantum

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23(7). https://doi.org/10.1088/1367-2630/ac0b02 - Mrugała, R.

(1990). Riemannian geometry and stability of ideal quantum gases.

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https://doi.org/10.1088/0305-4470/23/4/016 - Oikonomou, F. D. (2025).

*Product-State Manifolds for M Quantum Systems with N Levels using

the Fano form and the Induced Euclidean Metric*. arXiv preprint.

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(1980). Riemannian structure on manifolds of quantum states.

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https://doi.org/10.1007/bf02193559 |

Appendices

Appendix A: Formal Derivations

*Symbolic Derivation of the Fubini-Study Metric for a

Qubit*

The metric tensor is derived from the formula \(g_{\mu\nu} =

\text{Re}(\langle\partial\mu\psi|\partial\nu\psi\rangle -

\langle\partial\mu\psi|\psi\rangle\langle\psi|\partial\nu\psi\rangle)\).

For a general qubit state parameterized by spherical coordinates \((\theta, \phi)\): \(|\psi(\theta, \phi)\rangle =

\cos(\frac{\theta}{2})|0\rangle +

e^{i\phi}\sin(\frac{\theta}{2})|1\rangle\)

The symbolic computation of the metric tensor components yields: -

\(g{\theta\theta} = 1/4\) - \(g{\phi\phi} = \frac{1}{4}\sin^2(\theta)\)

\(g_{\theta\phi} = 0\)

This results in the line element \(ds^2 =

\frac{1}{4}(d\theta^2 + \sin^2(\theta)d\phi^2)\), which is the

metric for a 2-sphere of radius \(r=1/2\). This confirms that the state space

of a single qubit is geometrically equivalent to the surface of the

Bloch sphere, and that it is a curved manifold with constant positive

curvature.

Appendix B: Code Implementation *Python Functions

for Geometric Calculations*

[](#cb1-1)import sympy as sp

[](#cb1-2)

[](#cb1-3)def getqubitmetric_symbolic():

[](#cb1-4) """

[](#cb1-5) Calculates the symbolic components of the Fubini-Study metric for a qubit.

[](#cb1-6) """

[](#cb1-7) # Define symbols

[](#cb1-8) theta, phi = sp.symbols('theta phi', real=True)

[](#cb1-9)

[](#cb1-10) # Define the qubit state vector

[](#cb1-11) psi = sp.Matrix([sp.cos(theta/2), sp.exp(sp.I phi) sp.sin(theta/2)])

[](#cb1-12)

[](#cb1-13) # Calculate partial derivatives

[](#cb1-14) d_theta = sp.diff(psi, theta)

[](#cb1-15) d_phi = sp.diff(psi, phi)

[](#cb1-16)

[](#cb1-17) # Helper function to calculate a single metric component

[](#cb1-18) def calcgcomponent(d1, d2, state):

[](#cb1-19) # Implements guv = Re(<du|dv> - <du|psi><psi|d_v>)

[](#cb1-20) term1 = (d1.H * d2)[0] # .H is Hermitian conjugate (dagger)

[](#cb1-21) term2 = (d1.H state)[0] (state.H * d2)[0]

[](#cb1-22) return sp.re(term1 - term2)

[](#cb1-23)

[](#cb1-24) # Calculate the metric tensor components

[](#cb1-25) gtt = calcgcomponent(dtheta, d_theta, psi)

[](#cb1-26) gpp = calcgcomponent(dphi, d_phi, psi)

[](#cb1-27) gtp = calcgcomponent(dtheta, d_phi, psi)

[](#cb1-28)

[](#cb1-29) # Simplify and return the results

[](#cb1-30) return {

[](#cb1-31) "gthetatheta": sp.simplify(g_tt),

[](#cb1-32) "gphiphi": sp.simplify(g_pp),

[](#cb1-33) "gthetaphi": sp.simplify(g_tp)

[](#cb1-34) }