Raw EEG is a real-valued time series. Before it can enter a quantum circuit, it must be converted to a rotation angle. PLV is the bridge: it extracts instantaneous phase via the Hilbert transform, measures how tightly two channels are synchronized, and maps that synchrony to a number in [0, 1] that becomes a gate parameter.
A real-valued EEG signal x(t) is extended to a complex analytic signal z(t) = x(t) + iH[x(t)], where H is the Hilbert transform. The instantaneous phase is the argument of this complex signal.
The Hilbert transform shifts every frequency component by 90°, creating a complex envelope whose argument is instantaneous phase.
For channels i and j, the phase difference Δφij(t) = φi(t) − φj(t). If two signals are synchronized, Δφ is approximately constant. If independent, Δφ wanders uniformly around the circle.
Left: unit circle with two rotating phasors and their difference arc. Right: Δφ(t) — near-constant. Synchronized signals maintain a stable phase relationship.
PLV is the magnitude of the mean unit phasor of the phase difference distribution over N time samples. Each sample contributes a unit vector eiΔφ(t) on the circle. Their centroid’s length is the PLV.
PLV = 1 means perfect phase locking. PLV = 0 means fully random phase relationship. During seizure, widespread synchronization drives PLV upward across electrode pairs.
QNFM maps PLV to a quantum rotation angle via θ = π · PLV. This maps [0, 1] → [0, π], placing the encoding in the natural range of an Rx rotation gate. No nonlinearity is introduced between the EEG measurement and the gate parameter — the A-Gate circuit is the first nonlinear transformation in the pipeline.
The encoding is deliberately linear. No nonlinearity is introduced between the EEG measurement and the quantum gate parameter — the A-Gate circuit is the first nonlinear transformation in the pipeline.
The A-Gate encodes amplitude (a) and synchrony (c) on separate qubits. The coupling layer mixes them — neither qubit alone determines the outcome. ⟨Z⟩ is a joint measurement: the polarity signal lives in the interaction, not in either path separately.