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

Donghoon Kim1, Yusuke Kimura1, Hugo Mackay1,2, Yosuke Mitsuhashi1, Hideaki Nishikawa1,3, Carla Rubiliani1,4, Cheng Shang1, Ayumi Ukai1,5, and Tomotaka Kuwahara1,6

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Phys. Rev. X 16, 041035 – Published 9 November, 2026

DOI: https://doi.org/10.1103/8fcr-5tf4

Abstract

How the detailed entanglement structure emerges from quantum dynamics remains a fundamental challenge motivated by recent advances in quantum simulators and information processing. As a central milestone, the small-incremental-entangling (SIE) theorem bounds the entanglement-entropy growth rate but does not control the entanglement spectrum itself. In this work, we define the spectral-entangling strength, which quantifies how strongly an operator can reshape the distribution of Schmidt coefficients across a bipartition. We then prove a spectral-SIE theorem: for Rényi index α≥1/2, the growth rate of Rényi-entanglement entropies admits a universal bound. Remarkably, our bound at α=1/2 is both qualitatively and quantitatively optimal; below this threshold (α<1/2), no universal speed limit on entanglement growth can exist. This result yields a sharp 1/s2 threshold in the tail of the ordered squared Schmidt coefficients, or equivalently in the entanglement-spectrum weights (with s the Schmidt index). It enables rigorous truncation-based-error control and establishes a quantitative link between entanglement-spectrum structure and computational complexity. As a practical highlight, for one-dimensional power-law interactions 1/rη with η>2, this implies matrix-product-state approximations with bond dimension polynomial in (n/ε) for ground states, real-time evolved states, and Gibbs states, thereby closing the quasipolynomial gap, where n is the system size, and ε denotes the target approximation error in the relevant state or operator norm. By controlling Rényi entanglement, we further obtain a rigorous a priori truncation-error bound for the time-dependent density-matrix renormalization-group scheme analyzed here. Overall, we extend the SIE paradigm from bounding entanglement entropies to constraining the entanglement spectrum itself.

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How fast and in what form quantum entanglement grows shapes our understanding and classical simulation of many-body quantum systems. To capture this behavior, we introduce the spectral-entangling strength and establish a spectral theorem that places universal limits on the speed and structural evolution of entanglement growth across arbitrary bipartite systems. These theoretical constraints yield algorithms with polynomial complexity for approximating ground states, real-time dynamics, and Gibbs states in long-range interacting systems. Consequently, our findings define clear boundaries for the reliability of tensor-network simulations and clarify when analog quantum platforms can be faithfully modeled classically. Ultimately, this framework explains why many complex quantum materials remain computationally tractable and guides the development of more efficient simulation techniques.

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Phys. Rev. X 16, 041035– Published 9 November, 2026
  • Received 23 October 2025
  • Revised 23 June 2026
  • Accepted 13 August 2026
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