Two-Channel Admissibility Diagnostic Applied to Magic Spreading in Random Quantum Circuits

This work develops a geometric diagnostic framework for analyzing resource propagation and equilibration in complex systems, with particular emphasis on two-channel dynamics under conserved coherence constraints. Using a Thales-based partition geometry and an admissibility inequality, the framework distinguishes between linear (additive, local) transport channels and multiplicative (global, contractive) coherence channels, showing that these equilibrate on fundamentally different timescales.

Applied to random quantum circuits, the diagnostic clarifies the observed separation between entanglement growth, which saturates ballistically in system size, and nonstabilizerness (“magic”), which equilibrates exponentially and saturates logarithmically. The analysis explains why global quantum computational power can emerge before long-range entanglement is fully established, and why sparse non-Clifford resources lead to a crossover from logarithmic to linear saturation behavior.

The framework is explicitly diagnostic and phenomenological: it does not replace domain-specific dynamical equations or predict microscopic rates. Instead, it provides a map of admissible regimes, identifies which resources set global convergence, and offers falsifiable criteria for regime transitions. These insights are relevant to quantum information processing, quantum communication, and other systems where coherence, transport, and constraint satisfaction interact across scales.