Spectroscopic Search for Topological Protection in Open Quantum Hardware: The Dissipative Mixed Hodge Module Approach

TL;DR

This work addresses the problem of certifying topological protection in open quantum systems, where Exceptional Points compromise standard spectral decompositions. It introduces Dissipative Mixed Hodge Modules (DMHM) and two spectroscopic protocols, Weight Filtered Spectroscopy (WFS) and Hodge Filtered Spectroscopy (HFS), built from the Hodge Filtration $F^\bullet$ and Weight Filtration $W_\bullet$ to separate coherence orders and decay topology, respectively. Theoretical results include Theorems 1 and 2 establishing the canonical filtrations and their physical implications, plus a practical dissipative insulation metric $F_{iso}$ and demonstrations on molecular polaritons and Non-Hermitian Aharonov-Bohm rings. The framework enables robust hardware design by distinguishing true topological insulation from spectral coincidences, enabling dissipation-aware engineering of next-generation quantum devices and motivating future exploration of singular geometry via Floquet-DMHM spectroscopy.

Abstract

Standard spectroscopic protocols model the dynamics of open quantum systems as a superposition of isolated, exponentially decaying eigenmodes. This paradigm fails fundamentally at Exceptional Points, where the eigenbasis collapses and the response becomes dominated by non-diagonalizable Jordan blocks. We resolve this ambiguity by introducing a geometric framework based on \textit{Dissipative Mixed Hodge Modules} (DMHM). By replacing the scalar linewidth with a topological \textit{Weight Filtration}, we derive ``Weight Filtered Spectroscopy'' (WFS)--a protocol that spatially separates decay channels based on the nilpotency rank of the Liouvillian. We demonstrate that WFS acts as a dissipative x-ray, quantifying dissipative leakage in molecular polaritons and certifying topological isolation in Non-Hermitian Aharonov-Bohm rings. This establishes that topological protection persists as an algebraic invariant even when the spectral gap is closed.

Figures (8)

• Figure 1: WFS as a "Dissipative X-Ray". (a) Standard spectroscopy shows a single conflated linewidth, obscuring the internal dynamics. (b) WFS resolves the hidden weight structure. The intensity of the off-diagonal cross-peak directly quantifies the mixing between the "dirty" bath ($\lambda_X$) and the "clean" channel ($\lambda_C$), serving as a figure of merit for hardware insulation.
• Figure 2: Geometric Tomography of the Weight Filtration.(a) The HWH Protocol ($\tilde{S}(\omega_1, s_2, \omega_3)$). Standard 2DES (blue projection, $s_2=0$) conflates the spectrum into a single linewidth. Our HWH protocol "unfolds" this ambiguity along the topological Weight axis ($s_2$). We observe two distinct weight-subspaces separated by the filtration: (i) The Cold Polaritons ($s_2 \approx 12$ meV), representing the protected quantum channel, and (ii) The Hot Vibron ($s_2 \approx 60$ meV), representing the decoherence sink. The separation visibly demonstrates the "lifting" of the sheaf structure. (b) The WWC Protocol ($\tilde{S}(s_1, s_2)$). A "dissipative x-ray" mapping the correlation between decay rates. Diagonal peaks (White Arrows) represent isolated evolution in the Polariton ($\lambda_C=12$) and Vibron ($\lambda_V=60$) subspaces. The Dissipative Leakage cross-peak at $(\lambda_C, \lambda_V)$ (Red Arrow) quantifies the violation of weight strictness. Its non-zero intensity $|\tilde{S}| > 0$ proves that the system is not dissipatively insulated; the bath is topologically connected to the qubit. Parameters: $\gamma_X=5.0$, $\gamma_C=0.1$, $g=20.0$ meV [See Supplemental Material Sec. S6].
• Figure 3: Topological Resolution: The AB Ring. (a) WFS Map of the Non-Hermitian Aharonov-Bohm Ring. Unlike the polariton case, the protected edge mode (Weight 12) and bulk bath (Weight 60) show zero cross-correlation, indicating perfect algebraic decoupling. (b) Parameter Independence. The leakage remains exponentially suppressed for the topological candidate (green) compared to the quadratic failure of standard hybrid systems (red), proving the robustness of the solution.
• Figure 4: Probing the Singularity. A preview of the Floquet-DMHM mapping. By driving the system around the Exceptional Point, the Monodromy operator $M$ mixes the weight filtrations Gritsev2017Bukov2015Eckardt2017. WFS can resolve this mixing, allowing for the future measurement of the "Singular Quantum Geometric Tensor" defined on the vanishing cohomology.
• Figure 5: The Topological Sieve Protocol. (a) In the standard framework, the spectrum is a messy "cloud" of overlapping peaks. (b) The Hodge Projector $\mathcal{P}_p$ separates response functions by coherence order ($p=\pm 1$). (c) The Weight Projector $\mathcal{P}_\lambda$ acts as a secondary filter, sieving the signal based on decay topology. The resulting "Refined Weight Spectrum" reveals the protected mode in isolation, free from the background of leakage modes.

Theorems & Definitions (3)

• Definition 1: Dissipative Mixed Hodge Module
• proof
• proof