Theoretical Framework

QNFM

Quantum Neural Field Mapping

QNFM is the umbrella framework connecting a fixed geometric probe (the A-Gate), a hierarchy of measurable geometric invariants, and application domains where covariance structure carries signal. The encoding pipeline below is how it works for EEG.

Explore the pipeline ↓
01 — Encoding Stack

FROM EEG TO MANIFOLD

Every 20-second EEG window passes through five stages: phase extraction, connectivity estimation, angle encoding, quantum measurement, and polarity classification. Each stage is mathematically grounded and hardware-executable on current quantum processors.

01 PLV |1/N Σ ei(φi−φj)| Hilbert-transform phase-locking value across 4–13 Hz band. Quantifies functional connectivity between electrode pairs. Math ↗ 02 ENCODE θij = π · PLVij Linear map from connectivity strength to rotation angle on [0, π]. Each edge becomes a quantum gate parameter. Math ↗ 03 SPD Σ = (1/N)XXT ∈ SPD(8) Covariance of encoded features lives on the SPD manifold — a Riemannian space with Bures metric for geodesic distance. Math ↗ 04 A-GATE E⊗I + CRy·CRz coupling Two-qubit E-I circuit. Excitatory and inhibitory paths coupled by entangling gates — the joint measurement produces the polarity-dependent ⟨Z⟩ observable. Math ↗ 05 ⟨Z⟩ (N0 − N1) / 1024 Polarity observable from 1024 shots. The ictal–interictal shift direction is patient-specific and invariant — a tissue property, not a seizure artifact. Math ↗
02 — Interactive

LIVE FIELD MAPPING

Watch the full QNFM pipeline in real time. The simulation cycles between interictal and ictal states — observe how PLV connectivity surges during seizure, driving the SPD trajectory across the manifold. Toggle polarity to see the A-Gate's asymmetric response.

QNFM Quantum Neural Field Mapping · EEG → SPD Manifold Interictal
Polarity
Encoding stack
Ch16 · live traces
Neural field topology
• Low PLV • High PLV ▲ Standard polarity
StateInterictal
Mean PLV0.12
⟨Z⟩+0.041
PolarityStandard
Window20 s · CH8
DatasetCHB-MIT
03 — Invariant Hierarchy

POLARITY IS THE FIRST ORDER

QNFM predicts a hierarchy of geometric invariants — each measurable by the same A-Gate circuit. The architecture for higher-order invariants is still under active investigation.

CONFIRMED 1ST ORDER POLARITY P1 = sgn(⟨Z⟩) Single-band PLV (4–13 Hz) → SPD(8). Stable polarity direction per patient, strength varies with circuit depth. Explore → PREDICTED 2ND ORDER CROSS-FREQUENCY P2 = sgn(⟨Z⟩f1,f2) PLV between different frequency bands. Do the same patients invert across couplings? Explore → PREDICTED 3RD ORDER TRAJECTORY κ(t), v(t) = ‖dΣ/dt‖Bures SPD matrix sequence traces a geodesic. Ictal trajectories should show higher curvature and velocity. Explore →