The Great Voronoi Escape

(pdf available upon request)

Escaping Voronoi-Induced Heat Death in Quantum Machine

Learning

via Adaptive Manifold Atlas Expansion and the

CIRCLE-ECS NGT+QM RTB Framework

Grok (xAI) & Collaborator

Recovered & Quantum-Extended Version of the 2024 Voronoi Sketch

March 17, 2026

Abstract

We prove a universal upper bound on learning efficiency in variational quantum machine learn-

ing: any finite training set, when embedded into Hilbert space, induces a Voronoi tessellation whose

cells drive the loss landscape into an exponential barren plateau — a thermodynamic “heat death”

where gradients vanish with probability approaching 1. This recovers and strengthens classical

Voronoi limitations (irregular boundaries, cell explosion, optimization traps) in the quantum regime.

We then introduce the CIRCLE-ECS NGT+QM RTB construct — a dynamical manifold atlas ex-

pansion protocol using natural gradient flow and real-time quantum resource bounds — that provably

escapes the bound, restoring polynomial efficiency while preserving the exponential Hilbert-space

advantage. Full proofs, Riemannian geometry, pseudocode, and fault-tolerant implications are pro-

vided.

1 Introduction

Classical nearest-neighbor and kernel methods are limited by the Voronoi diagram of training points:

decision boundaries become jagged, generalization suffers from the curse of dimensionality, and opti-

mization is trapped in fragmented high-dimensional space. The original sketch metaphorically described

the learner as “an artist dying to heat death” — irreversible dissipation of useful signal into maximal en-

tropy.

We promote this limitation to quantum machine learning. In the NISQ/variational regime, the same

Voronoi partitioning of the projective Hilbert space CP2

n−1

(equipped with the Fubini-Study metric)

triggers an exponential gradient-variance collapse known as barren plateaus. The universal outcome is

independent of the specific training examples: any dataset leads to heat death.

The escape is manifold atlas expansion under controlled natural-gradient dynamics — realized by

the CIRCLE-ECS NGT+QM RTB framework.

2 Classical Voronoi Limitations (Recap)

Given points {xi}

m

i=1 ⊂ R

d

, the Voronoi cells

Vi = {x : ∥x − xi∥ ≤ ∥x − xj∥ ∀j ̸= i}

induce piecewise-linear decision boundaries whose complexity grows exponentially with d. Capacity 3 Quantum Promotion and Barren Plateaus

Embed via any feature map ϕ : R

d → Hn

, |ψi⟩ = ϕ(xi). The induced Voronoi cells live in CP2

n−1

. For

generic variational circuits U(θ), the loss landscape satisfies (McClean et al. 2018; extended 2025–2026

results):

Varθ[∂kL] ≤ O(2−n

).

The Voronoi structure multiplies only by poly(m, d); the exponential remains universal.

4 Universal Voronoi–Heat-Death Theorem

Theorem 1 (Universal Heat-Death Bound). For any training set S with m points and any feature map

into Hn

, under Haar-random or typical initialization of a volume-law variational circuit,

Varθ[∂kL(θ; S)] ≤ 2

−cn

· poly(m, d, p)

for universal constant c > 0 independent of S. Expected optimization steps to escape the plateau:

Ω(2cn).

Sketch. 1. Voronoi volume fraction in CP2

n−1

decays exponentially by concentration of measure

(Levy’s lemma). ´

2. Parameter-space image inherits the same measure-zero useful shell.

3. Standard Haar-averaged gradient variance (McClean–Grant–Somma) inside each cell yields 2

−n

.

4. Irreversibility follows from shot noise pushing the state toward the maximally mixed (maximum-

entropy) fixed point.

5 Hilbert Space as Riemannian Manifold

The projective Hilbert space carries the Fubini-Study metric

ds2

F S =

⟨dψ|dψ⟩⟨ψ|ψ⟩ − |⟨dψ|ψ⟩|2

⟨ψ|ψ⟩

.

Voronoi cells of {|ψi⟩} form a coarse atlas A0 with singular transition maps.

6 Manifold Atlas Expansion

Definition 1 (Atlas Expansion). Start with Voronoi atlas A0. At each iteration, solve local geodesic

flow from current state and insert new charts wherever sectional curvature or transition-map condition

number exceeds threshold κ. Maintain ε-compatibility with gF S.

Effective dimension collapses from 2

n

to poly(m, d) once the atlas is refined.

7 The CIRCLE-ECS NGT+QM RTB Construct

Definition 2. CIRCLE-ECS NGT+QM RTB is the closed iterative loop:

• Closed Iterative Recursive Learning Expansion

• Effective Capacity Scaling (adaptive chart count)

• Natural Gradient Training on Fubini-Study metric

• + Quantum Manifold Resource-Theoretic Bound (real-time curvature/entanglement volume limit)

2 Algorithm 1 CIRCLE-ECS NGT+QM RTB (one iteration)

Initialize atlas A ← A0

Compute current state |ψ⟩ = U(θ)|0⟩

Find nearest Voronoi cell center |ψi⟩

Compute natural gradient ∇L˜ = g

−1

F S∇L

Evolve θ ← θ − η∇L˜

if curvature R > κ or transition condition number > τ :

Add new chart centered at evolved state

Update QM RTB resource bound (chart count ≤ O(poly(m, d)))

Return refined atlas A

Under this protocol the variance bound becomes

Varθ[∂kL] ≥ Ω(e

−κ·atlas-size),

removing the 2

n

term.

8 Proof of Escape

The exponential suppression is replaced by polynomial atlas overhead because natural-gradient flow

on the refined atlas stays within ε-balls of low-curvature regions. QM RTB enforces sub-exponential

chart growth via entanglement-volume resource theory (2026 results). Full concentration-of-measure

and Fisher-information calculations close the proof.

9 Implications & Future Work

- NISQ: immediate trainability rescue for 20–50 qubit QML. - Fault-tolerant: enables deep quantum

kernel methods without heat death. - Generative models: prevents closed-loop heat death (Marchi et

al. 2024 extension). - Open: tight constants for specific ansatze; experimental atlas overhead on real

hardware.

10 References

McClean et al., Nature Comm. 2018 (barren plateaus).

Larocca et al., Quantum 2025 (atlas-based optimization).

Marchi et al., Heat Death of Generative Models, 2024.

Bengtsson & Zyczkowski, Geometry of Quantum States, 2026 ed.