What question did this study set out to answer?

This work aims to derive the analytical phase and demonstrate dissipation stabilization in topological quantum systems. It explores implications for fault-tolerant quantum cryptography.

May 29, 2026Open Access

Topological Z/6Z Superselection: Evasion of Ergodic Thermalization and Analytic Phase Derivation in Open Quantum Systems

Key Points

This work aims to derive the analytical phase and demonstrate dissipation stabilization in topological quantum systems. It explores implications for fault-tolerant quantum cryptography.
Constructed a topological superselection model using a Z/6Z framework.
Applied the Lindblad Master Equation to evaluate dynamics of up to 60 qubits.
Validated theoretical findings using mechanized formal verification and extensive simulations.
Found that the optimal initialization phase ϕ2=π stabilizes the system and enhances efficiency in quantum computations.
Demonstrated that the system enters a Non-Ergodic Extended phase with a rapid mixing bound of Δ=Ω(1).
Achieved a significant reduction in the Toffoli gate count from 2.83×10^11 to 9.45×10^10 gates for RSA-2048.

Abstract

🌀 Phase-Pi-Quantum-Prior Topological Superselection Z/6Z: Analytical Phase Derivation and Dissipative Stabilization for FTQC Cryptanalysis The analytic core of the Z/6Z topological superselection. The perfect geometric duality between the resonant channels (blue and red) demonstrates the Zero-Leakage adaptive strategy driven by the exact holonomic phase shift Δϕ=π.* 🎯 TL;DR – The Essentials 🔬 Theoretical Breakthroughs 📐 Analytical Phase Discovery: Proof that the optimal initialization phases (ϕ1,ϕ2) are not heuristic. ϕ2=π emerges from the unit group isomorphism (Z/6Z)×≅Z/2Z acting as a non-commutative Berry Holonomy. 🔄 Renormalization Group (RG) Origin: The thermodynamic phase ϕ1≈Rfund/10 is strictly derived as an infrared fixed point of the Callan-Symanzik beta function, coupled to the Euler-Kronecker invariant. ⚡ Polynomial Complexity: Exact state preparation via Matrix Product States (MPS) with constant topological bond dimension χ≤6, avoiding the exponential O(2n) overhead of arbitrary distributions. 🛡️ Parent Lindbladian it is the geometric penalty for trying to squeeze the continuous vacuum into a discrete binary register." This work establishes that arithmetic is not a random sequence, but a deterministic wave structure encoded in the Z/6Z topology. Recognizing this allows us to fundamentally rewrite the rules of quantum cryptanalysis and hardware initialization for the FTQC era. Last Update: May, 2026 | Status: Under Review at Journal of Physics A: Mathematical and Theoretical JPhysA-124899 | Built with ⚛️, 🐍 & 🛡️ Lean 4

Topological Z/6Z Superselection: Evasion of Ergodic Thermalization and Analytic Phase Derivation in Open Quantum Systems

Key Points

Abstract

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