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Deep learning and quantum variational architectures share a profound mathematical bottleneck: as parameters scale or cross phase boundaries, the underlying loss landscape ceases to behave like a smooth manifold. Instead, it fractures into a singular algebraic variety where the metric tensor collapses ($\det(G) = 0$), causing conventional natural gradients (and standard AdamW) to stall or deflect chaotically. The standard optimization reflex relies on heuristic, coordinate-dependent smoothing (like Tikhonov damping), which dilutes geometric structures and breaks gauge invariance.
This project implements a fundamentally different approach. Instead of treating metric degeneracies as numerical errors to be smoothed away, we exploit the algebraic structure of these singular fibers natively. Drawing from regular holonomic $\mathcal{D}_X$-modules and Mixed Hodge structures, our Singular Natural Gradient Descent (SNGD) optimization engine maps parameter spaces to the Brieskorn lattice. This allows updates to compute strictly covariant paths, letting gradients natively tunnel directly through the topological barriers and barren plateaus where classical and Quantum Natural Gradient (QNG) algorithms fail.
Having already established this framework within open-source computational pipelines to resolve non-Hermitian root bifurcations in molecular potential surfaces, this grant funds a focused 6-month project to translate this algebraic engine directly into Quantum Native Machine Learning (QNML). Crucially, it also funds the launch of an advanced QNML training course in Nepal—cultivating rare, highly specialized technical talent in an LMIC where quantum infrastructure and AI resources are historically scarce. This dual approach establishes an immediate operational runway and seeds a localized tech ecosystem while a major multi-year institutional grant proposal is under review.
The goal is to deliver a functional, high-performance JAX-native optimization engine that routes around manifold fracturing, paired with an advanced curriculum to seed uptake in the quantum computing ecosystem.
This is a milestone program:
M1: Mathematical Engine & Local Invariant Tracking via Dissipative Mixed Hodge Modules (done, open-source via QuMorpheus, more details in track recod and validation section).
M2: JAX-native Framework Integration, QNML benchmarking, and advanced course launch (this 6-month grant).
M3 & M4: Multi-year scaling (via Simons Foundation), physical quantum hardware validation, and full institutional integration at the Bhaweshwar Das Center of Advanced Study (BDCAS) under NepAl Applied Mathematics and Informatics Institute for research (NAAMII).
M2 is an accelerated 6-month program to build the software and educational runway. Deliverables:
D2.1 (~5 weeks) - Core Engine & JAX Compliance: Build the core JAX compliance layer. Construct a high-performance FastAlgebraicSNGDEngine capable of dynamically tracking metric degeneration during runtime optimization without numerical approximations, benchmarked on a simulated Transverse-Field Ising Model (TFIM).
D2.2 (~7 weeks) - Monodromy-Driven Regularization: Implement anisotropic, monodromy-driven updates. Run structural sweeps to verify that the SNGD engine preserves gauge invariance and prevents parity sign-flips across non-stoquastic boundaries where standard Tikhonov regularizations break down.
D2.3 (~8 weeks) - The QNML Payoff: Apply the SNGD engine to a variational quantum-classical network experiencing severe parameter space fracturing. Demonstrate that covariant tunneling maintains stable convergence rates, establishing a direct speedup over standard QNG algorithms under identical noise profiles.
D2.4 (~4 weeks) - Course Delivery & Open-Access Dissemination: Launch and manage an advanced specialized course curriculum detailing singularity resolutions in quantum learning landscapes. Consolidate codebase documentation and publish a comprehensive arXiv technical report to engage directly with geometric deep learning groups and quantum hardware validation teams.
Funding Goal ($44,548): Secures the full 6-month runway (D2.1–D2.4).
Stipend (85% FTE for 6 months): $31,458
Hardware & Compute: $5,040 (Dedicated workstation setup for local JAX/Ubuntu parallelization scaling)
Course Management & Curriculum Development: $4,000
Contingency Buffer (10%): $4,050 (To safeguard against hardware price fluctuations or local overhead discrepancies)
Minimum ($12,000): Deceptively high leverage. This minimum threshold guarantees the delivery and management of the specialized advanced course ($4,000), covers the core hardware/compute setup ($5,040), and funds the initial codebase mapping ($2,960)—deferring full-time scaling but ensuring the educational infrastructure is launched.
The project will be executed independently by the Founding Director and Principal Investigator (me) of the newly established Bhaweshwar Das Center for Advanced Study (BDCAS) under the umbrella of NAAMII. The Center operates through two newly formed divisions—Algebraic Quantum Dynamics & Topological Informatics and Quantum Native Machine Learning—providing the exact institutional framework needed to bridge pure mathematical physics with deployable software engineering.
Track Record & Validation:
Mathematical Foundations & Preprints: The core theoretical mechanics of this proposal are grounded in an extensive portfolio of independent research. During a recent career break taken to serve as a primary caregiver in Nepal for my grandmother who had Alzheimer's, I maintained a high-volume research output, authoring multiple foundational preprints (available on arXiv, links in the Metric and relevant publications section) detailing the mathematical resolution of algebraic singularities and fractured parameter manifolds.
Deployable Software: I successfully translated these pure mathematical foundations into the open-source QuMorpheus: Topological Resolution framework. This pipeline utilizes symbolic engines to map non-linear singular systems and continuous toroidal loops in multi-variable chemical potential energy surfaces using machine-precision integer invariants. It is published open source under copyleft.
Peer Visibility: These mechanics for resolving the $X_9$ singularity were recently presented directly to the field at the 6th International Quantum Matter Conference & Expo (QUANTUMatter 2026) in Barcelona (abstract , session link).
Metrics and relevant publications: [Google Scholar / Mathematical Foundations and Topological Framework / Computational Quantum Physics / Computational Chemistry and Photochemistry / Quantum Hardware & Interaction Metrology ] demonstrating a sustained research trajectory in theoretical quantum physics and computational chemistry. These recent works has garnered massive readership traction in the niche field according to adsabs.harvard.com
Technical Risk: Computing local algebraic varieties or Gröbner bases could introduce steep runtime overhead during JAX's just-in-time compilation loops as network depth increases.
Mitigation: We do not compute global varieties. The engine uses a localized spectral partitioner that isolates the active singular mask of the metric tensor dynamically, triggering the algebraic engine only when the system approaches a critical metric collapse threshold ($\det(G) < \epsilon$).
Outreach & Uptake Risk: Advanced algebraic optimization methods risk being viewed as overly abstract by industry ML practitioners who default to empirical heuristic tuning.
Mitigation: The concurrent launch of the specialized advanced course directly builds an active user base. Furthermore, benchmarking the engine explicitly against standard QNG and AdamW under identical conditions provides legible, performance-driven proof of utility.
None for this specific operational branch; core framework development has been self-directed and self-funded.
Parallel Applications: I am currently structuring an application for the Simons Foundation Targeted Grant in MPS to secure long-term scaling (covering 1/12 of my calendar PI time, a full-year postdoc, and two graduate students). Because the Simons Foundation explicitly accommodates career interruptions for caregiving, my recent independent output of foundational arXiv preprints positions the Center well for their theoretical mandate.
However, that multi-month institutional review process creates a critical gap. This Manifund request acts as a fast-acting 6-month operational bridge. It allows me to immediately pivot the mathematical foundations built during my caregiving period into deployable JAX software and launch the Center's initial course deliverable while the larger Simons grant is under review.