Intrinsic locality dimension of quantum codes

Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes that is independent of any background geometry and naturally incorporates flexible architectures and accommodates noninteger values, drawing on mathematical machinery from fractal geometry and geometric measure theory. Important scenarios include topological codes and algebraic codes such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--K\"{o}nig bounds for regular topological codes, respectively. Furthermore, we discuss implications on thermal properties, presenting a conditional no-go result for self-correcting quantum memories in dimension $3-\epsilon$ for any $\epsilon>0$. Our theory lays a versatile and unifying mathematical foundation for studying the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.