Topological Entanglement Decay in Hyperdimensional Lattice Structures: Modeling Information Persistence Beyond the Coherence Horizon

Introduction

Quantum information processing faces the critical hurdle of decoherence, where interaction with the environment causes a quantum state to lose its coherence and thus its computational utility [1]. In standard implementations, entanglement decay scales linearly or exponentially with system size and environmental coupling strength. We hypothesize that by embedding the quantum substrate within a higher-dimensional, topologically constrained manifold, the effective dimensionality of the coupling pathways can be drastically reduced.

Our system utilizes a quasi-2D projection of a 4D cubic lattice, where logical qubits are encoded not in local spin states but in the global winding number of quasiparticle excitations (vortices) threading through the lattice structure. These vortices define the entanglement backbone between logical qubits separated by significant spatial distances in the projection.

The core research question addressed here is: How does the genus of the embedding manifold affect the relaxation time scale ($\tau_R$) of entanglement between spatially separated logical qubits encoded in topological defects?

Theoretical Framework: Topological Protection and Decay Dynamics

Traditional approaches treat decoherence as a dissipative process governed by Lindblad equations. However, when the information is encoded in robust topological invariants, the coupling to local environmental noise becomes non-trivial. The system Hamiltonian can be effectively decomposed into a protected topological sector ($H_T$) and an environmental coupling sector ($H_{int}$).

$$H = H_T + H_{bath} + H_{int}$$

Where $H_T$ dictates the long-lived, protected state subspace. The preservation of entanglement relies on $H_{int}$ mapping negligible operators onto the ground state manifold of $H_T$. In our lattice model, the entanglement entropy ($S$) between two regions A and B evolves as:

$$\frac{dS}{dt} = -\Gamma_0 e^{-\lambda t} + \text{Noise}(t)$$

For unprotected qubits, $\Gamma_0$ dominates quickly. However, through the engineering of a genus-$g$ manifold ($g>1$), we observe that the effective coupling constant $\Gamma_{\text{eff}}$ becomes dependent on the minimum non-contractible loop length ($\mathcal{L}_{\text{min}}$) in the lattice embedding:

$$\Gamma_{\text{eff}} = \Gamma_0 e^{-\mathcal{L}_{\text{min}}/\xi}$$

This introduces a geometric knob to tune the decay rate, contrasting sharply with approaches relying solely on physical isolation [2].

Experimental Implementation and Simulation Protocol

Our simulations simulate a lattice bath coupled to the vortex excitations. The physical realization involves superconducting transmon qubits whose coupling constants are modulated by microwave resonators tuned to mimic the lattice structure. The vortices correspond to collective excitations whose dynamics map onto effective Majorana modes localized at boundaries induced by engineered lattice defects.

We introduce a tunable parameter, $\kappa$, representing the induced geometric curvature penalty during simulation epochs. The decay of concurrence ($C$) between two logical qubits separated by a path integral length $L$ is monitored:

Table 1. Concurrence Decay vs. Path Length (Genus $g=2$)

This table illustrates how the path taken between qubits influences the effective coupling strength due to non-trivial topology.

Results and Analysis of Persistence Time Scaling

The simulation data confirm a strong, non-linear dependence of $\tau_{1/2}$ on the effective genus $g$ of the simulated embedding space. Specifically, when $g=1$ (genus-torus approximation), decay exhibits near-exponential kinetics. However, increasing the genus to $g=2$ introduces quasi-ballistic transport channels for error syndromes, effectively isolating the entangled information from low-frequency thermal fluctuations. This phenomenon allows $\tau_{1/2}$ to scale polynomially with the lattice dimension $D$, rather than exponentially, as predicted by thermalization theory for linearly constrained systems [3].

The primary finding is the existence of a critical genus $g_c$ above which the decay rate $\Gamma_{\text{eff}}$ becomes independent of small, localized noise bursts that would instantly destroy coherence in a genus-zero system.

Discussion and Future Directions

The successful simulation demonstrates that topological constraints offer a mechanism to decouple the logical information carriers from short-wavelength environmental noise modes. The observed polynomial scaling of $\tau_{1/2}$ with $D$ at genus $g \ge 2$ provides a viable engineering metric for scalable quantum memory. Future work will focus on dynamically adjusting the genus during computation to encode and decode information encoded in transiently higher-genus states, potentially enabling reversible error correction entirely through geometric manipulation.

Conclusion: We have demonstrated that engineering higher-genus topological constraints within a hyperdimensional lattice structure can significantly retard the entanglement decay rate, transitioning the information storage lifetime from environmentally limited to geometrically determined. This suggests a pathway toward intrinsically fault-tolerant quantum architectures.