Ultrametric Relaxation Dynamics in Topological Quantum Memory

In the paper's words

The scalability of Topological Quantum Computing (TQC) is currently impeded by a “Thermodynamic Wall,” where the entropy generation from active error correction cycles scales exponentially with logical qubit count. This paper proposes a paradigm shift from active gate synthesis to passive topological relaxation, grounded in the Quni-Gudzinas hypothesis (2025). By establishing a rigorous functorial isomorphism between the inverse limit of abelian anyonic braid groups and the p-adic solenoid (Σ_p), we demonstrate that the vacuum structure of specific strain-engineered materials can encode topological quantum information.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. For $N=1000$, the heat density exceeds the cooling power of standard dilution fridges (typically $\mu W$ at 10mK).

   from 4.1 The

2. We propose a 2D Frenkel-Kontorova-like model: $$ H = \int d^2x \left[ \frac{1}{2}(\partial_t \phi)^2 + \frac{1}{2}(\nabla \phi)^2 + V_{strain}(\phi, x, y) \right] $$ Here, the potential is defined as: $$ V_{strain}(\phi, x, y) = \sum_{k=0}^{\infty} \Delta_k \cos(p^k \phi + \vec{q}_k \cdot \vec{x}) $$ where $\Delta_k = \Delta_0 p^{-\alpha k}$ represents the barrier height at hierarchy level $k$, and $\vec{q}_k$ are wavevectors corresponding to the hierarchical strain superlattice (e.g., from a Moiré pattern).

   from 5.1 Derivation of the

3. However, as noted in peer review, at millikelvin temperatures (10mK), the phonon density of states is low, potentially leading to a “phonon bottleneck” where relaxation becomes too slow to be practical for system initialization or reset.

   from 6.3

4. The active architecture hits the Thermodynamic Wall before reaching 1,000 qubits.

   from 6.5 Scalability Analysis

5. According to the Quni-Gudzinas hypothesis (2025), the thermodynamic cost of this process scales non-linearly.

   from 4.1 The

6. At $N=1000$, the passive system generates orders of magnitude less entropy than the active equivalent.

   from 4.5 Comparative

How it is done

1. We derive a 2D Hamiltonian for a hierarchical “synthetic vacuum” where relaxation dynamics follow an ultrametric trajectory, effectively freezing the system into a protected topological sector without external intervention.

   from Abstract

2. While theoretically sound for small-scale systems, this approach faces a fundamental “Thermodynamic Wall” at macroscopic scales (Quni-Gudzinas, 2025).

   from 1.1 The Active

3. This approach bypasses the active control bottleneck by utilizing the system’s intrinsic tendency to minimize free energy.

   from 1.2 The Promise

4. Secondly, we derive a 2D Hamiltonian explicitly designed to induce ultrametric relaxation dynamics, utilizing strain engineering to create hierarchical potential barriers.

   from 1.3 Research Objectives and

What would falsify it

1. We predict a logarithmic decay characteristic of ultrametric relaxation.

   from 5.7 Experimental Roadmap

Where the claim stops

1. The primary limitation is the restriction to abelian models, precluding universal computation.

   from 7.3 Limitations of the Study

2. This active control model assumes that the quantum state must be forcibly held against the entropic gradient of the environment.

   from 1.1 The Active

3. However, this assumption ignores the potential for engineering ground states that are intrinsically protected by their geometry.

   from 1.1 The Active

Where to read

1. Addressing24 words · 1 min
2. Abstract162 words · 1 min
3. Keywords13 words · 1 min
4. 1.0 Introduction916 words · 4 min
5. 2.0 Theoretical903 words · 4 min
6. 3.0 Structural766 words · 3 min
7. 4.0646 words · 3 min
8. 5.0666 words · 3 min
9. 6.0396 words · 2 min
10. 7.0 Conclusion & Future332 words · 1 min
11. Appendices2,220 words · 10 min