Monte Carlo Simulation of Operator Dynamics and Entanglement in Dual-Unitary Circuits

Menghan Song; Zhaoyi Zeng; Ting-Tung Wang; Yi-Zhuang You; Zi Yang Meng; Pengfei Zhang

doi:10.22331/q-2025-04-01-1681

Monte Carlo Simulation of Operator Dynamics and Entanglement in Dual-Unitary Circuits

Menghan Song¹, Zhaoyi Zeng², Ting-Tung Wang¹, Yi-Zhuang You³, Zi Yang Meng¹, and Pengfei Zhang^2,4,5,6

¹Department of Physics and HK Institute of Quantum Science & Technology, The University of Hong Kong, Pokfulam Road, Hong Kong
²Department of Physics, Fudan University, Shanghai, 200438, China
³Department of Physics, University of California, San Diego, California 92093, USA
⁴State Key Laboratory of Surface Physics, Fudan University, Shanghai, 200438, China
⁵Shanghai Qi Zhi Institute, AI Tower, Xuhui District, Shanghai 200232, China
⁶Hefei National Laboratory, Hefei 230088, China

Published:	2025-04-01, volume 9, page 1681
Editor:	Alessio Benavoli
Eprint:	arXiv:2410.00953v5
Doi:	https://doi.org/10.22331/q-2025-04-01-1681
Citation:	Quantum 9, 1681 (2025).

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Abstract

We investigate operator dynamics and entanglement growth in dual-unitary circuits, a class of locally scrambled quantum systems that enables efficient simulation beyond the exponential complexity of the Hilbert space. By mapping the operator evolution to a classical Markov process, we perform Monte Carlo simulations to access the time evolution of local operator density and entanglement with polynomial computational cost. Our results reveal that the operator density converges exponentially to a steady-state value, with analytical bounds that match our simulations. Additionally, we observe a volume-law scaling of operator entanglement across different subregions, and identify a critical transition from maximal to sub-maximal entanglement growth, governed by the circuit’s gate parameter. This transition, confirmed by both mean-field theory and Monte Carlo simulations, provides new insights into operator entanglement dynamics in quantum many-body systems. Our work offers a scalable computational framework for studying long-time operator evolution and entanglement, paving the way for deeper exploration of quantum information dynamics.

Popular summary

In this paper, we explore the dynamics of operator growth and entanglement in dual-unitary circuits, leveraging their locally scrambled nature to bypass the exponential complexity often encountered in quantum many-body systems. By mapping the operator dynamics to a classical Markov process, we develop an efficient Monte Carlo simulation framework that allows for the precise computation of local operator density and entanglement over time.

Our analysis focuses on the evolution of two distinct operators and examines how their density profiles develop near the light cone. We derive analytical expressions for the operator density, demonstrating that it converges exponentially to a steady-state value of $ \frac{3}{4} $, with bounds that are validated through simulations.

In addition, we investigate the entanglement dynamics in various subregions of the light cone, revealing a volume-law scaling for operator entanglement. A key result of our study is the identification of a transition from maximal to sub-maximal volume-law entanglement growth for left-moving operators near the left light cone, governed by the gate parameter $ \alpha $. This transition is corroborated by both mean-field theory and numerical simulations, offering new insights into the relationship between gate parameters and entanglement dynamics in quantum systems.

Our work establishes a scalable computational framework for studying long-time operator evolution and entanglement in quantum many-body systems, providing a novel approach to understanding operator spreading and entanglement growth in complex quantum circuits.

► BibTeX data

@article{Song2025montecarlo,
  doi = {10.22331/q-2025-04-01-1681},
  url = {https://doi.org/10.22331/q-2025-04-01-1681},
  title = {Monte {C}arlo {S}imulation of {O}perator {D}ynamics and {E}ntanglement in {D}ual-{U}nitary {C}ircuits},
  author = {Song, Menghan and Zeng, Zhaoyi and Wang, Ting-Tung and You, Yi-Zhuang and Meng, Zi Yang and Zhang, Pengfei},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1681},
  month = apr,
  year = {2025}
}

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Cited by

[1] Mao Tian Tan and Tomaž Prosen, "Logarithmic growth of operator entanglement in a clean nonintegrable circuit", Physical Review B 113 22, 224316 (2026).

[2] Bruno Bertini, Pieter W. Claeys, and Tomaž Prosen, "Exactly solvable quantum many-body dynamics from space-time duality", Reviews of Modern Physics 98 2, 025001 (2026).

[3] Ning Sun, Lei Feng, and Pengfei Zhang, "Post-Selection Probability and Fidelity of Bidirectional Teleportation", arXiv:2606.17251, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-19 14:05:23) and SAO/NASA ADS (last updated successfully 2026-08-19 14:05:24). The list may be incomplete as not all publishers provide suitable and complete citation data.

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