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Quantum complexity and generalized area law in fully connected models

Donghoon Kim, Tomotaka Kuwahara

TL;DR

This work establishes a generalized area law for ground states of fully connected quantum systems with finite gap and bounded boundary Schmidt rank, showing entanglement entropy across any bipartition grows only polylogarithmically with system size $n$. The authors introduce the mean-field renormalization group (MFRG), a hierarchical projection/renormalization scheme that coherently reduces the effective Hilbert-space dimension while preserving the ground-state properties, enabling an efficient MPS approximation with bond dimensions that are quasi-polynomial in $n$. Under 2-local or $k$-local all-to-all interactions with $d_H=\mathcal{O}(1)$ (or bounded $\sum_s|J_s|$), they prove $S_{AB}(\ket{\Omega}) \le 2(\log n)^{\alpha}+\log(d)+1$ with a computable exponent $\alpha=\mathcal{O}(1)$ dependent on the gap $\Delta$ and interaction strength. The results illuminate why certain infinite-dimensional fully connected models admit low-complexity ground states despite nonlocal connectivity, providing a rigorous pathway for classical simulations and a deeper understanding of entanglement in nonlocal quantum many-body systems. Numerical studies on the Lipkin-Meshkov-Glick model and all-to-all bilinear fermions corroborate the theoretical findings, showing sub-logarithmic growth or saturation of entanglement in gapped regimes.

Abstract

The area law for entanglement entropy fundamentally reflects the complexity of quantum many-body systems, demonstrating ground states of local Hamiltonians to be represented with low computational complexity. While this principle is well-established in one-dimensional systems, little is known beyond 1D cases, and attempts to generalize the area law on infinite-dimensional graphs have largely been disproven. In this work, for non-critical ground states of Hamiltonians on fully connected graphs, we establish a generalized area law up to a polylogarithmic factor in system size, by effectively reducing the boundary area to a constant scale for interactions between subsystems. This result implies an efficient approximation of the ground state by the matrix product state up to an approximation error of $1/\text{poly}(n)$. As the core technique, we develop the mean-field renormalization group approach, which rigorously guarantees efficiency by systematically grouping regions of the system and iteratively approximating each as a product state. This approach provides a rigorous pathway to efficiently simulate ground states of complex systems, advancing our understanding of infinite-dimensional quantum many-body systems and their entanglement structures.

Quantum complexity and generalized area law in fully connected models

TL;DR

This work establishes a generalized area law for ground states of fully connected quantum systems with finite gap and bounded boundary Schmidt rank, showing entanglement entropy across any bipartition grows only polylogarithmically with system size . The authors introduce the mean-field renormalization group (MFRG), a hierarchical projection/renormalization scheme that coherently reduces the effective Hilbert-space dimension while preserving the ground-state properties, enabling an efficient MPS approximation with bond dimensions that are quasi-polynomial in . Under 2-local or -local all-to-all interactions with (or bounded ), they prove with a computable exponent dependent on the gap and interaction strength. The results illuminate why certain infinite-dimensional fully connected models admit low-complexity ground states despite nonlocal connectivity, providing a rigorous pathway for classical simulations and a deeper understanding of entanglement in nonlocal quantum many-body systems. Numerical studies on the Lipkin-Meshkov-Glick model and all-to-all bilinear fermions corroborate the theoretical findings, showing sub-logarithmic growth or saturation of entanglement in gapped regimes.

Abstract

The area law for entanglement entropy fundamentally reflects the complexity of quantum many-body systems, demonstrating ground states of local Hamiltonians to be represented with low computational complexity. While this principle is well-established in one-dimensional systems, little is known beyond 1D cases, and attempts to generalize the area law on infinite-dimensional graphs have largely been disproven. In this work, for non-critical ground states of Hamiltonians on fully connected graphs, we establish a generalized area law up to a polylogarithmic factor in system size, by effectively reducing the boundary area to a constant scale for interactions between subsystems. This result implies an efficient approximation of the ground state by the matrix product state up to an approximation error of . As the core technique, we develop the mean-field renormalization group approach, which rigorously guarantees efficiency by systematically grouping regions of the system and iteratively approximating each as a product state. This approach provides a rigorous pathway to efficiently simulate ground states of complex systems, advancing our understanding of infinite-dimensional quantum many-body systems and their entanglement structures.

Paper Structure

This paper contains 27 sections, 16 theorems, 304 equations, 4 figures.

Table of Contents

  1. Setup and Assumptions
  2. Assumption for 2-local Hamiltonian
  3. Assumption for $k$-local Hamiltonian
  4. Preliminaries
  5. Trade-off between variance and spectral gap
  6. Probability distribution of $1$-local operator
  7. General hopping amplitude
  8. Small hopping amplitude
  9. Main Theorem and Overview of the Proof
  10. Main theorem
  11. Overview of the proof
  12. Local Robustness and Largest Schmidt Coefficient
  13. Robustness of the ground state for local unitary operators
  14. Largest local Schmidt coefficient
  15. Reduction of the block Hilbert space
  16. ...and 12 more sections

Key Result

Lemma 3

Let $A_L$ be a $k_{A}$-local and $1$-extensive operator as in Eq. supp_def:A_observable. For an arbitrary ground state $\ket{\Omega}$ of the $k$-local Hamiltonian $H$ with the property g_0_extensive_2, the variance $\mathrm{Var}_{\Omega} (A_L)$ and the spectral gap $\Delta$ satisfies the following t where we define In particular, $\gamma = 6k^{2}$ when $k_A=k$.

Figures (4)

  • Figure 1: Schematic picture of the MFRG flow. In each block (blue-shaded box), we perform dimension reduction by truncating the product bases where more than $z$ sites deviate from the mean-field states. This truncation reduces the Hilbert space dimension in each block to a polynomial form with respect to the block size. After applying the Hilbert space truncations, we construct new qudits and an effective Hamiltonian that now describes the interactions between these renormalized qudits. The same process is then iteratively repeated for this new Hamiltonian.
  • Figure S.1: Outline of the proof of the main theorem.
  • Figure S.2: The figures present the entanglement entropy and energy gaps for the Lipkin-Meshkov-Glick model in Eq. \ref{['All_LMG']} at $\gamma = 0.8$ and $\gamma = 0.9$ across various values of $h$. Panels (a-c) focus on $\gamma = 0.8$, detailing the entropy as a function of $n$, its logarithmic scaling, and the energy gap. Panels (d-f) display analogous measurements for $\gamma = 0.9$, showcasing consistent observations and trends.
  • Figure S.3: Entanglement entropy and energy gap for the all-to-all bilinear fermion model in Eq. \ref{['All_TB']}. Panels (a)-(c) depict the entanglement entropy between two halves of the system as a function of system size $n$ for $\mu = 0$, $0.04$, and $1$, respectively. Each graph shows the results for six distinct values of $\kappa$, ranging from 0 to 100. Panels (d)-(f) depict the energy gap for $\mu = 0$, $0.04$, and $1$, respectively, where the gap is defined as the difference between the energy of the first excited state and that of the ground state. All results are averaged over 1000 samples, with the values of $t_{i,j}$ in Eq. \ref{['All_TB']} randomly chosen between 0 and 1.

Theorems & Definitions (16)

  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 1
  • Proposition 1
  • Lemma 6
  • Proposition 2
  • Lemma 7
  • Lemma 8
  • Corollary 9
  • ...and 6 more