Double categories for adaptive quantum computation

Abstract: Quantum computation can be formulated through various models, each highlighting distinct structural and resource-theoretic aspects of quantum computational power. This paper develops a unified categorical framework that encompasses these models and their interrelations using the language of double categories. We introduce double port graphs, a bidirectional generalization of port graphs, to represent the quantum (horizontal) and classical (vertical) flows of information within computational architectures. Quantum op- erations are described as adaptive instruments, organized into a one-object double category whose horizontal and vertical directions correspond to quantum channels and stochastic maps, respectively. Within this setting, we capture prominent adaptive quantum compu- tation models, including measurement-based and magic-state models. To analyze compu- tational power, we extend the theory of contextuality to an adaptive setting through the notion of simplicial instruments, which generalize simplicial distributions to double cat- egorical form. This construction yields a quantitative characterization of computational power in terms of contextual fraction, leading to a categorical formulation of the result that non-contextual resources can compute only affine Boolean functions. The frame- work thus offers a new perspective on the interplay between adaptivity, contextuality, and computational power in quantum computational models.