Geometric Complexity of Quantum Channels via Unitary Dilations

Nielsen's geometric approach to quantum circuit complexity provides a Riemannian framework for quantifying the cost of implementing unitary (closed--system) dynamics. For open dynamics, however, the reduced evolution is described by quantum channels and admits many inequivalent Stinespring realizations, so any meaningful complexity notion must specify which microscopic resources are counted as accessible and which transformations are regarded as gauge. We introduce and analyze a geometric complexity functional for families of quantum channels based on unitary dilations. We distinguish an implementation-dependent complexity, defined relative to explicit dilation data, from an intrinsic channel complexity obtained by minimizing over a physically motivated class of admissible dilations (e.g. bounded environment dimension, energy or norm constraints, and penalty structures). The functional has a subtractive form: it compares the geometric cost of the total unitary realization with a canonical surrogate term that removes purely environmental contributions. We justify this subtraction from concise postulates, including closed-system consistency, environment-only neutrality, and invariance under dilation gauge transformations that leave the channel unchanged. This leads to a companion quantity, noise complexity, quantifying the loss of geometric complexity relative to a prescribed ideal closed evolution. We establish a coherence-based lower bound for unitary geometric complexity, derive structural properties such as linear time scaling under time-homogeneous dilations, and obtain dissipator--controlled bounds in the Markovian (GKSL/Lindblad) regime under a standard dilation construction. Finally, we illustrate the framework on canonical benchmark noise models, including dephasing, amplitude damping, and depolarizing (Pauli) channels.