TITLE:
Quantum Circuit Complexity as a Physical Observable
AUTHORS:
Logan Nye
KEYWORDS:
Quantum Circuit Complexity, Physical Observables, Operator Theory, Quantum Gravity, Quantum Measurement
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.1,
January
13,
2025
ABSTRACT: This work proposes quantum circuit complexity—the minimal number of elementary operations needed to implement a quantum transformation—be established as a legitimate physical observable. We prove that circuit complexity satisfies all requirements for physical observables, including self-adjointness, gauge invariance, and a consistent measurement theory with well-defined uncertainty relations. We develop complete protocols for measuring complexity in quantum systems and demonstrate its connections to gauge theory and quantum gravity. Our results suggest that computational requirements may constitute physical laws as fundamental as energy conservation. This framework grants insights into the relationship between quantum information, gravity, and the emergence of spacetime geometry while offering practical methods for experimental verification. Our results indicate that the physical universe may be governed by both energetic and computational constraints, with profound implications for our understanding of fundamental physics.