Operational Mathematics of Quantum Operations: Extending the Iteration Count of CPTP Maps and their Inverses to the Complex Domain

This paper systematically transplants the core methodology of Operational Mathematics—the extension of the repetition count of fundamental operations from natural numbers to integers, rationals, reals, and ultimately complex numbers—onto a new class of binary operations: the quantum operation ΦEn(ρ,z) (a completely positive trace-preserving map, i.e. a quantum channel) and its inverse ΦE−1n (ρ,z) (the operational principal branch of the inverse channel). Acomplete set of seven independent axioms is established; integer-order, fractional-order,real-order, and complex-order quantum iterations are rigorously defined; and the existence of iterative roots at each level is proved by means of Schröder’s equation, Abel’s equation,and a suitably adapted Kneser construction. Uniqueness theorems under natural regularity conditions are provided.The singularity structure of complex-order quantum iterations is analysed in depth, revealing a fundamentally novel phenomenon: the branch points are of mixed superoperator valued algebraic (square-root type, from the degeneracy of Kraus operators) and logarithmic type (from the zeros of contractive eigenvalues of the channel). Three families of branch points are identified and their overlaps are discussed. The union of branch cuts accumulates densely on the negative real axis, forming a natural boundary. The local monodromy group contains both Z2 and Z factors.A fundamental structural discovery is rigorously proved: the quantum operational hierarchy collapses completely for all levels n ≥ 2, leaving only the base channel at level n = 1 and the collapsed iteration semigroup at level n = 2.Fractional calculus and the fractional calculus of variations with quantum kernels are shown to be special cases of the quantum operational framework, thereby unifying discrete quantum hyperoperations with continuous analysis.Acategorical duality between the mathematics of numbers and the mathematics of quantum operations is established, yielding a field isomorphism between the quantum hyperfield and the complex numbers. A functorial relationship reflecting complementarity of quantum channels is constructed.The connection between quantum iteration values and the arithmetic of the channel spectrum is explored, with particular emphasis on transcendence of special values and the Quantum Riemann Hypothesis, which is proved unconditionally via a Hilbert–Pólya self adjoint operator construction applied to the corrected quantum zeta function (defined using backward iterates, with a careful resolution of non-commutativity issues). The corrected quantum zeta function satisfies an exact functional equation and an Euler product over quantum prime periodic orbits. A conditional reduction of the classical Riemann Hypothesis to the compactification of the quantum iteration generator is established.The paper is self-contained, and every essential statement is accompanied by a detailed proof.