Beyond Heredity, Axiomatic Foundations of Quantum Greedoids for Time Ordered Circuits

Greedoid theory generalizes matroids to feasible systems where the natural operation is extension along a trajectory rather than closure under all deletions. Quantum circuit families, especially variational ansatze defined by time ordered gate words, can violate the heredity property in a structural way. Feasibility can depend on interference and cancellation, so that a feasible circuit word need not have any feasible single deletion. This paper introduces an axiomatic framework for “quantum greedoids” on the free monoid of gate words, explicitly incorporating order and multiplicity. We define word accessibility and a suffix exchange axiom adapted to repeated gates, give minimal separations between heredity, accessibility, and exchange on one qubit, and identify stabilizer regimes where hereditary behaviour reappears. We then connect accessibility to pruning based training and to local information geometry diagnostics through the quantum Fisher information, framing greedy constructability as a structural correlate of navigability rather than a guarantee of optimization success.