Magic Flow in Quantum Circuits: Island Graphs, Orbit-Valence Magic Current, and the 6-7 Magic Diode

This paper develops the island graph picture of quantum circuit computation. At any time step, a quantum circuit partitions into entanglement islands — maximal sets of mutually entangled qubits forming totally isotropic subspaces of the symplectic polar space W(2k-1,2). Each island carries a Fano orbit valence label {p_L} — its magic valence. Magic flows between islands via non-Clifford gates, typed by the XOR-Fano composition rule.

The central claim: entanglement determines the topology of the island graph (which qubits are correlated), while magic determines the computational content (what the computation can achieve beyond classical simulation). These are independent.

Within the island graph, magic current obeys a diode law: primary-orbit magic (L ∈ {1,2,3}) flows naturally toward secondary-orbit states (L ∈ {4,5,6}) via the 6-7 broken-Fano topology, with a measurable asymmetry ratio D decreasing monotonically from 6/7 (stabiliser state) to approximately 0.650 (triple-CS state). Wrong-valence magic — primary-orbit magic used in a secondary-orbit context — scores below the classical bound (0.472 < 0.500), making magic valence mismatch an experimentally detectable computational error.

The ORBIT opcode serves as a magic ammeter: seven Pauli measurements on a TriQ identify the orbit valence label at any node, providing real-time monitoring of magic flow throughout the circuit at zero additional syndrome cost.

The paper connects the island graph picture to the biological FMO complex: the bacteriochlorophyll network is a natural 6-7 topology whose directed energy transport is magic flow at biological temperatures, with the same diode asymmetry ratio measured experimentally.

Keywords

Magic Flow, Island Graph, Magic Current, Orbit Valence, Fano Orbit, Magic Valence, ORBIT Opcode, Magic Ammeter, 6-7 Topology, Magic Diode, XOR-Fano Rule, Entanglement, Quantum Circuit, W(5,2), Symplectic Polar Space, Wigner Negativity, TriQ, SQU, Origami ISA, FMO Complex, Bacteriochlorophyll, Biological Quantum Coherence, Fault-Tolerant Quantum Computation, Valence Mismatch, CS Gate, Non-Clifford Gates, Adelic Simplicial Architecture