Spectral Small-Incremental-Entangling: Breaking Quasi-Polynomial Complexity Barriers in Long-Range Interacting Systems
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arXiv
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| Format: | Preprint |
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2025
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| author | Kim, Donghoon Kimura, Yusuke Mackay, Hugo Mitsuhashi, Yosuke Nishikawa, Hideaki Rubiliani, Carla Shang, Cheng Ukai, Ayumi Kuwahara, Tomotaka |
| author_facet | Kim, Donghoon Kimura, Yusuke Mackay, Hugo Mitsuhashi, Yosuke Nishikawa, Hideaki Rubiliani, Carla Shang, Cheng Ukai, Ayumi Kuwahara, Tomotaka |
| contents | 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 $α\ge 1/2$, revealing a robust $1/s^2$ decay threshold in the entanglement spectrum. Remarkably, our bound at $α=1/2$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12014 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Small-Incremental-Entangling: Breaking Quasi-Polynomial Complexity Barriers in Long-Range Interacting Systems Kim, Donghoon Kimura, Yusuke Mackay, Hugo Mitsuhashi, Yosuke Nishikawa, Hideaki Rubiliani, Carla Shang, Cheng Ukai, Ayumi Kuwahara, Tomotaka Quantum Physics Disordered Systems and Neural Networks Statistical Mechanics Mathematical Physics 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 $α\ge 1/2$, revealing a robust $1/s^2$ decay threshold in the entanglement spectrum. Remarkably, our bound at $α=1/2$ 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. |
| title | Spectral Small-Incremental-Entangling: Breaking Quasi-Polynomial Complexity Barriers in Long-Range Interacting Systems |
| topic | Quantum Physics Disordered Systems and Neural Networks Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2509.12014 |