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

• URL: http://arxiv.org/abs/2509.12014v2
• Date: Thu, 23 Oct 2025 04:59:24 GMT
• Title: Spectral Small-Incremental-Entangling: Breaking Quasi-Polynomial Complexity Barriers in Long-Range Interacting Systems
• Authors: Donghoon Kim, Yusuke Kimura, Hugo Mackay, Yosuke Mitsuhashi, Hideaki Nishikawa, Carla Rubiliani, Cheng Shang, Ayumi Ukai, Tomotaka Kuwahara,
• Abstract summary: We introduce the concept of Spectral-Entangling strength, which captures the structural entangling power of an operator.<n>We establish a spectral SIE theorem: a universal speed limit for R'enyi entanglement growth at $alpha ge 1/2$.<n>Our results extend the SIE theorem to the spectral domain and establish a unified framework that unveils the detailed and universal structure underlying quantum complexity.
• Score: 2.911917667184046
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: How the detailed structure of quantum complexity emerges from quantum dynamics remains a fundamental challenge highlighted by advances in quantum simulators and information processing. The celebrated Small-Incremental-Entangling (SIE) theorem provides a universal constraint on the rate of entanglement generation, yet it leaves open the problem of fully characterizing fine entanglement structures. Here we introduce the concept of Spectral-Entangling strength, which captures the structural entangling power of an operator, and establish a spectral SIE theorem: a universal speed limit for R'enyi entanglement growth at $\alpha \ge 1/2$, revealing a robust $1/s^2$ decay threshold in the entanglement spectrum. Remarkably, our bound at $\alpha=1/2$ is both qualitatively and quantitatively optimal, defining the universal threshold beyond which entanglement growth becomes unbounded. This exposes the detailed structure of Schmidt coefficients and enables rigorous truncation-based error control, linking entanglement structure to computational complexity. Building on this, we derive a generalized entanglement area law under an adiabatic-path condition, extending a central principle of quantum many-body physics to general interactions. As a concrete application, we show that one-dimensional long-range interacting systems admit polynomial bond-dimension approximations for ground, time-evolved, and thermal states, thereby closing the long-standing quasi-polynomial gap and demonstrating that such systems can be simulated efficiently with tensor-network methods. By explicitly controlling R'enyi entanglement, we obtain a rigorous, a priori error guarantee for the time-dependent density-matrix renormalization-group algorithm. Overall, our results extend the SIE theorem to the spectral domain and establish a unified framework that unveils the detailed and universal structure underlying quantum complexity.

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