A Fundamental Energy-Complexity Uncertainty Relation

We establish quantum circuit complexity as a fundamental physical observable and prove that it satisfies an uncertainty relation with energy, analogous to Heisenberg’s canonical uncertainty principle. Through rigorous operator theory, we demonstrate that the complexity operator meets all mathematical requirements for a legitimate quantum observable, including self-adjointness, gauge invariance, and proper spectral decomposition. This enables us to derive a fundamental bound that constrains how quickly complexity can increase in physical systems given available energy resources. We provide complete mathematical proofs of these results and demonstrate their far-reaching implications across quantum computation, black hole physics, and computational complexity theory. In particular, we show that this uncertainty relation imposes fundamental speed limits on quantum circuits, explains maximal complexity growth in black holes, and suggests that physical constraints may enforce an effective separation between complexity classes independent of their mathematical relationships. We outline explicit experimental protocols for testing these predictions using current quantum computing platforms and discuss the profound implications for our understanding of the relationship between computational complexity and fundamental physics. Our results indicate that computational requirements may be as basic to physics as energy conservation, suggesting a deep connection between the structure of physical law and fundamental limits on computation.