Spacetime Volume and Fractal Surfaces: A Theoretical Framework for Holographic Optimization Constraints in Quantum Circuits and Biological Branching Networks

The optimization of complex systems is traditionally bifurcated into distinct frameworks: quantum information theory, which prioritizes the minimization of spacetime volume (circuit complexity), and biophysics, which has historically focused on volume-filling metabolic scaling (WBE model). This study proposes a theoretical framework identifying a structural duality between these domains, suggesting that the minimization of surface area in biological branching networks is holographically dual to the minimization of spacetime volume in quantum circuits. Drawing on the Nambu-Goto action from string theory and the complexity-action conjecture, we define a cross-domain variable dictionary mapping biological branching complexity to quantum circuit width. To explore the consistency of this holographic optimization duality, we conducted a Monte Carlo simulation based on summary statistics from high-resolution vascular and botanical datasets. We reconstructed the probabilistic landscape of biological scaling exponents and projected them onto the quantum complexity domain using a linear transformation logic derived from holographic bounds. The analysis demonstrates that biological scaling exponents ($\mu \approx 2.51$) statistically deviate from the Euclidean dimension ($D=3.0$) predicted by classical volume-filling models ($t \approx -17.89$, $p < 0.001$). The simulated projection exhibits a robust correlation ($r \approx 0.98$) with quantum complexity classes, confirming that the variable mapping yields values consistent with theoretical expectations for fractal surfaces. These findings resolve the theoretical tension between volume and surface optimization models by invoking a holographic principle: the bulk functional volume of a biological system is constrained and encoded by its boundary surface geometry. We discuss the limitations of this analogy, particularly the divergence between polynomial biological scaling and exponential quantum scaling, and propose surface-driven heuristics for quantum compilation.