Spectral Small-Incremental-Entangling: Breaking Quasi-Polynomial Complexity Barriers in Long-Range Interacting Systems

Abstract: A key challenge in quantum complexity is how entanglement structure emerges from dynamics, highlighted by advances in simulators and information processing. The Lieb--Robinson bound sets a locality-based speed limit on information propagation, while the Small-Incremental-Entangling (SIE) theorem gives a universal constraint on entanglement growth. Yet, SIE bounds only total entanglement, leaving open the fine entanglement structure. In this work, we introduce Spectral-Entangling Strength, measuring the structural entangling power of an operator, and prove a Spectral SIE theorem: a universal limit for R\'enyi entanglement growth at $\alpha \ge 1/2$, revealing a robust $1/s^2$ tail in the entanglement spectrum. At $\alpha=1/2$ the bound is qualitatively and quantitatively optimal, identifying the universal threshold beyond which growth is unbounded. This exposes the detailed structure of Schmidt coefficients, enabling rigorous truncation-based error control and linking entanglement to computational complexity. Our framework further establishes a generalized entanglement area law under adiabatic paths, extending a central principle of many-body physics to general interactions. Practically, we show that 1D long-range interacting systems admit polynomial bond-dimension approximations for ground, time-evolved, and thermal states. This closes the quasi-polynomial gap and proves such systems are simulable with polynomial complexity comparable to short-range models. By controlling R\'enyi entanglement, we also derive the first rigorous precision-guarantee bound for the time-dependent density-matrix-renormalization-group algorithm. Overall, our results extend SIE and provide a unified framework that reveals the detailed structure of quantum complexity.