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

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.