Perez Hourglass Fractal Quantum Computing - A Topological Blueprint for Fault-Tolerant Scalable Quantum Processors

Six Exceptional Properties of the “Perez Hourglass”:
Perspectives toward New Types of Artificial Intelligence
and Quantum Computers
Appendix : 
Perez Hourglass Fractal Quantum Computing - A Topological Blueprint for
Fault-Tolerant Scalable Quantum Processors
“Where there is matter, there is geometry.” — Johannes Kepler
“The universe is a mirror. The Hourglass is its frame. We are the reflection — and the
gaze.” — Jean-Claude Perez
Jean-Claude Perez
PhD Mathematics & Computer Science, Bordeaux University
Retired IBM Artificial Intelligence European Research Centre, Montpellier
Luc Montagnier Foundation
jeanclaudeperez2@gmail.com ( mailto:jeanclaudeperez2@gmail.com)
Abstract
More than thirty-five years after the first intuitions linking self-organization, neuronal
networks and the golden ratio (Perez, 1988), a remarkable numerical structure has finally
emerged from the depths of Pascal’s triangle: the “Perez Hourglass”. This fractal, self-
similar pattern, simultaneously antimatter of the famous Pascal triangle and digital
incarnation of the Fibonacci sequence (OEIS A000975), reveals a perfect hourglass-
shaped distribution of parity across the rows of the binomial triangle when analyzed
through a specific recursive filtering method. Here we demonstrate for the first time that
this structure is not merely a mathematical curiosity but constitutes an ideal topological
substrate for a radically new generation of quantum computers: fractal quantum
computers based on the Hourglass attractor.
In Appendix, we present the first rigorous formulation of a quantum computing architecture
whose topological backbone is the recently discovered “Perez Hourglass” fractal attractor
embedded in Pascal’s triangle modulo 2 (Perez, 2025a,b). The Hourglass defines a family
of sparse, exactly self-similar qubit lattices whose connectivity graph is governed by
Fibonacci numbering and whose bond angles are multiples of the golden angle 2πφ ². We ⁻
prove that this geometry yields:
1/ a distance-3 CSS quantum code with parameters [[F_{2n+1}, 1, F_n]] exceeding the
Bravyi-Poulin-Terhal bound,
2/ native implementation of golden-phase gates e^{iπφ ²} that are optimal for quadratic ⁻
speedup in quantum phase estimation,
3/ magic-state distillation factories with overhead reduced to O(log log N).
Full open-source implementations in Qiskit, Cirq, and Quantify are provided.