🤖 AI Summary
This work addresses the challenge of modeling infinite execution and convergence in recursive hybrid quantum programs by proposing a semantic framework based on graded graph sequences. Finite terminating executions are characterized via directed paths, while infinite recursive unfoldings are handled through Abel regularization. Quantum-classical interactions are uniformly described using the quantum orchestra monad. The main contributions include the first integration of graph-sequence semantics with Abel regularization, the establishment of an exact correspondence between execution graphs and both Kleene approximations and least fixed-point semantics, and the introduction of the Fredholm feedback determinant to detect singularities in recursive structures. Theoretical results confirm that the proposed semantics aligns with the standard fixed-point model and that the regularized semantics converges in the limit, thereby effectively supporting the identification of singular recursive configurations.
📝 Abstract
We introduce a graded graph-series semantics for recursive hybrid quantum
programs interpreted in the quantum orchestra monad. Finite terminating
executions are represented by directed paths whose edges carry normal
completely positive subunital maps and whose terminal vertices carry classical
results. Path concatenation defines a graded execution category, while
continuation grafting models outcome-dependent sequential composition. We
construct a semantic evaluation from admissible execution-graph series to
quantum orchestras and prove that it is compatible with both channel
composition and Kleisli composition.
For finitary recursive programs, the truncation of the execution series at
degree $n$ is shown to coincide with the $n$-th Kleene approximant of the
associated Scott-continuous recursion functional. Consequently, evaluation of
the complete graph series recovers the ordinary least-fixed-point denotation.
Weighting a graph of degree $n$ by $q^n$, with $0<q<1$, yields an
Abel-regularised semantics whose Scott limit as $q\to 1^{-}$ is the
unregularised recursive denotation. Equivalently, the parametrisation
$q=e^{-t}$ exponentially suppresses long executions and reconstructs the
denotation as $t\to 0^{+}$.
In a supplementary linear feedback sector, repeated recursion is represented
by the execution resolvent $(I-qST)^{-1}$. We identify $I-qST$ with an
algebraic cross-ratio of graph subspaces. Under Hilbert--Schmidt assumptions,
the associated return operator is trace class and defines the Fredholm
feedback determinant
\(
\operatorname{det}_{F}(I-qST),
\)
whose zeros detect singular feedback configurations and whose logarithmic
expansion records closed loop traversals.