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  • Open Access

Lieb-Robinson Bounds with Exponential-in-Volume Tails

Ben T. McDonough1, Chao Yin1, Andrew Lucas1,*, and Carolyn Zhang2

  • *Contact author: andrew.j.lucas@colorado.edu
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PRX Quantum 6, 040322 – Published 3 November, 2025

DOI: https://doi.org/10.1103/n242-m1l2

Abstract

Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance r after a time t decays as exp(vt−r), where v is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on rd sites in d spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as exp(−rd). We confirm this intuition, showing that, for r>vt, the volume-filling operator is suppressed by exp(−(r−vt)d/(vt)d−1). This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance ε for any finite time t: as ε becomes sufficiently small, only ε−O(td−1) resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.

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Popular Summary

Lieb-Robinson bounds show that, up to small error, information propagates with finite velocity in quantum many-body systems with spatially local interactions. They underlie many of the most important mathematically rigorous results about many-body quantum lattice models in condensed matter physics and provide sharp constraints on the capability of locally interacting models to perform quantum information processing. Characterizing the nature of the error, i.e. the leakage outside of the light cone defined by the Lieb-Robinson velocity, has been an important challenge for analyzing problems involving Hamiltonian evolution.

In our manuscript, we develop new Lieb-Robinson-type bounds which imply novel, strong constraints on the size and shape of the tail of a growing operator. We apply these bounds to resolve the simulation complexity problem by providing the first classical algorithm that scales optimally for both accuracy and time, surpassing all previous methods by at least a super-polynomial speedup. The broad scope of our results is illustrated through other applications. Subject to certain reasonable assumptions, we can prove a number of conjectured bounds about states which are close to fixed points of spontaneous symmetry-breaking phases for finite symmetries, such as the Z2 ferromagnet state. Looking forward, the weak tails in Lieb-Robinson bounds are known to hinder the applicability and generalizability of many important results in the literature: the stability of a gapped phase of matter to perturbations, the lifetime of a false vacuum, and entanglement area laws in gapped ground states. We hope that our improved Lieb-Robinson bounds may help the community revisit many of these important open questions.

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Outline

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PRX Quantum 6, 040322– Published 3 November, 2025
  • Received 21 February 2025
  • Revised 30 June 2025
  • Accepted 20 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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