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

TL;DR

This work introduces a spectral extension of the Small-Incremental-Entangling (SIE) framework by defining the Spectral-Entangling (SE) Strength and proving a Spectral SIE bound for Rényi entanglement with α ≥ 1/2. The key finding is a universal 1/s^2 decay in the entanglement spectrum at the critical α = 1/2, with a precise constant, and an unbounded growth threshold for α < 1/2. Leveraging SE strength, the authors derive a generalized entanglement area law under an adiabatic-path condition and show that 1D long-range interacting systems with η > 2 admit polynomial-bond-dimension MPS/MPO descriptions for ground, time-evolved, and Gibbs states, accompanied by rigorous error guarantees for t-DMRG. Collectively, these results unify structural entanglement properties with computational complexity, closing the quasi-polynomial gap for certain long-range systems and offering a rigorous framework for certifying tensor-network simulations.

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 $α\ge 1/2$, 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.

Figures (5)

• Figure 1: Overview of the main results. (a) Spectral Small-Incremental-Entangling (SIE): Introducing the spectral-entangling strength $\bar{J}$, we establish that Rényi entanglement entropies satisfy $\lvert dE_\alpha/dt \rvert \le c_\alpha \bar{J}(V_{AB})$ for $\alpha \ge 1/2$, with the constant $c_{1/2}=2$ being sharp, while for $\alpha < 1/2$ the growth rate can diverge. The optimal $\alpha=1/2$ case corresponds to an entanglement spectrum with Schmidt coefficients decaying as $1/s^{2}$, as illustrated in the panel. (b) Applications to 1D long-range interacting systems: When interactions decay faster than $r^{-2}$, ground states, real-time dynamics, and Gibbs states admit polynomial-bond-dimension MPS/MPO approximations. In addition, our results yield rigorous error guarantees for t-DMRG simulations.
• Figure 2: Plot of $c_\alpha$ with respect to $\alpha$. We have $c_{\alpha=1/2}=2$, $c_{\alpha=1}=4/e$, and $c_{\alpha=\infty}=2$.
• Figure 3: Schematic picture of the generalized area law. (a) The generalized area law states that the entanglement entropy is bounded in terms of the dimension of the interacting region. However, this bound is known to be violated for general ground states. (b) By assuming an adiabatic path for the boundary interaction, we establish the generalized area law under a condition formulated with the SE strength $\tilde{g}$. Roughly speaking, $\tilde{g}$ corresponds to the effective size of the boundary. For example, in systems defined on graphs, it becomes more evident that $\tilde{g}$ is proportional to the boundary size. If $A_0$ and $B_0$ consist of $m$ qubits and the interaction $V_{A_0B_0}$ is described by $\mathcal{O}(m)$ two-qubit interactions, then one obtains $\tilde{g} \propto m = \log(\mathcal{D}_{A_0})$ from the inequality \ref{['Trivial_Ineq_interaction_strength']}.
• Figure 4: Conceptual illustration of the monitoring-based analysis in t-DMRG algorithm. The exact evolution (top row, blue) is compared with the algorithmic trajectory: each pre-truncation state (middle row, green, dashed) is compressed to a rank-$D$ MPS (bottom row, red, dashed). Vertical arrows labelled "Compression" indicate the truncation step, during which the Rényi-$1/2$ proxy (sum of Schmidt coefficients) can only decrease. By tracking this proxy across steps, one can bound the entanglement growth and thus control the accumulated truncation error.
• Figure 5: Schematic picture of the Haah-Hastings-Kothari-Low approximation along the cut $A_0A_1 \sqcup B_0B_1$.

Theorems & Definitions (29)

• Definition 1: Spectral-Entangling (SE) Strength
• Lemma 1
• Corollary 2
• Corollary 3
• Lemma 4: Sub-additivity of the SE strength
• Lemma 5
• Theorem 1
• Corollary 6
• Proposition 7
• Proposition 8