Entanglement dynamics in U(1) symmetric hybrid quantum automaton circuits
Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
| Published: | 2023-12-06, volume 7, page 1200 |
| Eprint: | arXiv:2305.18141v2 |
| Doi: | https://doi.org/10.22331/q-2023-12-06-1200 |
| Citation: | Quantum 7, 1200 (2023). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We study the entanglement dynamics of quantum automaton (QA) circuits in the presence of U(1) symmetry. We find that the second Rényi entropy grows diffusively with a logarithmic correction as $\sqrt{t\ln{t}}$, saturating the bound established by Huang [1]. Thanks to the special feature of QA circuits, we understand the entanglement dynamics in terms of a classical bit string model. Specifically, we argue that the diffusive dynamics stems from the rare slow modes containing extensively long domains of spin 0s or 1s. Additionally, we investigate the entanglement dynamics of monitored QA circuits by introducing a composite measurement that preserves both the U(1) symmetry and properties of QA circuits. We find that as the measurement rate increases, there is a transition from a volume-law phase where the second Rényi entropy persists the diffusive growth (up to a logarithmic correction) to a critical phase where it grows logarithmically in time. This interesting phenomenon distinguishes QA circuits from non-automaton circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an area-law phase transition exists, and any non-zero rate of projective measurements in the volume-law phase leads to a ballistic growth of the Rényi entropy.
Featured image: The entanglement dynamics of quantum automaton circuits can be mapped to a classical bit string model. More specifically, the second Renyi entropy can be approximated by the dynamics of particles representing the difference of a bit string pair whose spins perform simple symmetric exclusion random walks under the circuit dynamics. Under U(1) symmetry, there are two distinct modes of particle dynamics, namely, the fast mode and the slow mode. In the example illustrated in the cartoon, although both modes have the same particle configuration, the bit string configuration of the slow mode consists of extensively long domains of spin 0s or 1s, resulting in a diffusive (with a logarithmic correction) expansion of the region occupied by particles.
Popular summary
In this work, we use random circuit models to study U(1)-symmetric quantum systems. Specifically, we focus on quantum automaton (QA) circuits, one of the few circuit models that allow an analytic understanding of the entanglement dynamics, and demonstrate that the second Renyi entropy scales as $\sqrt{t\ln{t}}$, saturating the bound mentioned above. By mapping the second Renyi entropy to the quantity of a classical particle model, we show that this diffusive dynamics is the consequence of the emergence of rare slow modes under U(1) symmetry.
In addition, we introduce measurements into QA circuits and examine the monitored entanglement dynamics. Interestingly, as we manipulate the measurement rate, we observe a phase transition from a volume-law phase where the second Renyi entropy persists the diffusive growth, to a critical phase where it grows logarithmically. This is different from non-automaton U(1)-symmetric hybrid quantum circuits where a volume-law to area-law entanglement phase transition exists, and any non-zero rate of measurements below the critical point induces a linear growth of the Renyi entropy.
► BibTeX data
► References
[1] Yichen Huang. ``Dynamics of rényi entanglement entropy in diffusive qudit systems''. IOP SciNotes 1, 035205 (2020).
https://doi.org/10.1088/2633-1357/abd1e2
[2] Hyungwon Kim and David A. Huse. ``Ballistic spreading of entanglement in a diffusive nonintegrable system''. Phys. Rev. Lett. 111, 127205 (2013).
https://doi.org/10.1103/PhysRevLett.111.127205
[3] Elliott H. Lieb and Derek W. Robinson. ``The finite group velocity of quantum spin systems''. Communications in Mathematical Physics 28, 251–257 (1972).
https://doi.org/10.1007/BF01645779
[4] Pasquale Calabrese and John Cardy. ``Evolution of entanglement entropy in one-dimensional systems''. Journal of Statistical Mechanics: Theory and Experiment 2005, P04010 (2005).
https://doi.org/10.1088/1742-5468/2005/04/P04010
[5] Christian K. Burrell and Tobias J. Osborne. ``Bounds on the speed of information propagation in disordered quantum spin chains''. Phys. Rev. Lett. 99, 167201 (2007).
https://doi.org/10.1103/PhysRevLett.99.167201
[6] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. ``Quantum entanglement growth under random unitary dynamics''. Phys. Rev. X 7, 031016 (2017).
https://doi.org/10.1103/PhysRevX.7.031016
[7] Winton Brown and Omar Fawzi. ``Scrambling speed of random quantum circuits'' (2013). arXiv:1210.6644.
arXiv:1210.6644
[8] Tibor Rakovszky, Frank Pollmann, and C. W. von Keyserlingk. ``Sub-ballistic growth of rényi entropies due to diffusion''. Phys. Rev. Lett. 122, 250602 (2019).
https://doi.org/10.1103/PhysRevLett.122.250602
[9] Marko Žnidarič. ``Entanglement growth in diffusive systems''. Communications Physics 3, 100 (2020).
https://doi.org/10.1038/s42005-020-0366-7
[10] Tianci Zhou and Andreas W. W. Ludwig. ``Diffusive scaling of rényi entanglement entropy''. Phys. Rev. Res. 2, 033020 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033020
[11] Yiqiu Han and Xiao Chen. ``Measurement-induced criticality in ${\mathbb{z}}_{2}$-symmetric quantum automaton circuits''. Phys. Rev. B 105, 064306 (2022).
https://doi.org/10.1103/PhysRevB.105.064306
[12] Yiqiu Han and Xiao Chen. ``Entanglement structure in the volume-law phase of hybrid quantum automaton circuits''. Phys. Rev. B 107, 014306 (2023).
https://doi.org/10.1103/PhysRevB.107.014306
[13] Jason Iaconis, Andrew Lucas, and Xiao Chen. ``Measurement-induced phase transitions in quantum automaton circuits''. Phys. Rev. B 102, 224311 (2020).
https://doi.org/10.1103/PhysRevB.102.224311
[14] Brian Skinner, Jonathan Ruhman, and Adam Nahum. ``Measurement-induced phase transitions in the dynamics of entanglement''. Phys. Rev. X 9, 031009 (2019).
https://doi.org/10.1103/PhysRevX.9.031009
[15] Amos Chan, Rahul M. Nandkishore, Michael Pretko, and Graeme Smith. ``Unitary-projective entanglement dynamics''. Phys. Rev. B 99, 224307 (2019).
https://doi.org/10.1103/PhysRevB.99.224307
[16] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. ``Quantum zeno effect and the many-body entanglement transition''. Phys. Rev. B 98, 205136 (2018).
https://doi.org/10.1103/PhysRevB.98.205136
[17] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. ``Measurement-driven entanglement transition in hybrid quantum circuits''. Phys. Rev. B 100, 134306 (2019).
https://doi.org/10.1103/PhysRevB.100.134306
[18] Michael J. Gullans and David A. Huse. ``Dynamical purification phase transition induced by quantum measurements''. Phys. Rev. X 10, 041020 (2020).
https://doi.org/10.1103/PhysRevX.10.041020
[19] Yimu Bao, Soonwon Choi, and Ehud Altman. ``Theory of the phase transition in random unitary circuits with measurements''. Phys. Rev. B 101, 104301 (2020).
https://doi.org/10.1103/PhysRevB.101.104301
[20] Chao-Ming Jian, Yi-Zhuang You, Romain Vasseur, and Andreas W. W. Ludwig. ``Measurement-induced criticality in random quantum circuits''. Phys. Rev. B 101, 104302 (2020).
https://doi.org/10.1103/PhysRevB.101.104302
[21] Xiao Chen, Yaodong Li, Matthew P. A. Fisher, and Andrew Lucas. ``Emergent conformal symmetry in nonunitary random dynamics of free fermions''. Phys. Rev. Res. 2, 033017 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033017
[22] O. Alberton, M. Buchhold, and S. Diehl. ``Entanglement transition in a monitored free-fermion chain: From extended criticality to area law''. Physical Review Letters 126 (2021).
https://doi.org/10.1103/physrevlett.126.170602
[23] Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan, David A. Huse, and Vedika Khemani. ``Entanglement phase transitions in measurement-only dynamics''. Phys. Rev. X 11, 011030 (2021).
https://doi.org/10.1103/PhysRevX.11.011030
[24] Shengqi Sang and Timothy H. Hsieh. ``Measurement-protected quantum phases''. Phys. Rev. Res. 3, 023200 (2021).
https://doi.org/10.1103/PhysRevResearch.3.023200
[25] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. ``Measurement-induced topological entanglement transitions in symmetric random quantum circuits''. Nature Physics 17, 342–347 (2021).
https://doi.org/10.1038/s41567-020-01112-z
[26] Utkarsh Agrawal, Aidan Zabalo, Kun Chen, Justin H. Wilson, Andrew C. Potter, J. H. Pixley, Sarang Gopalakrishnan, and Romain Vasseur. ``Entanglement and charge-sharpening transitions in u(1) symmetric monitored quantum circuits''. Phys. Rev. X 12, 041002 (2022).
https://doi.org/10.1103/PhysRevX.12.041002
[27] Matthew B. Hastings, Iván González, Ann B. Kallin, and Roger G. Melko. ``Measuring renyi entanglement entropy in quantum monte carlo simulations''. Phys. Rev. Lett. 104, 157201 (2010).
https://doi.org/10.1103/PhysRevLett.104.157201
[28] Zhi-Cheng Yang. ``Distinction between transport and rényi entropy growth in kinetically constrained models''. Phys. Rev. B 106, L220303 (2022).
https://doi.org/10.1103/PhysRevB.106.L220303
[29] Richard Arratia. ``The motion of a tagged particle in the simple symmetric exclusion system on $z$''. The Annals of Probability 11, 362 – 373 (1983).
https://doi.org/10.1214/aop/1176993602
[30] Soonwon Choi, Yimu Bao, Xiao-Liang Qi, and Ehud Altman. ``Quantum error correction in scrambling dynamics and measurement-induced phase transition''. Phys. Rev. Lett. 125, 030505 (2020).
https://doi.org/10.1103/PhysRevLett.125.030505
[31] Ruihua Fan, Sagar Vijay, Ashvin Vishwanath, and Yi-Zhuang You. ``Self-organized error correction in random unitary circuits with measurement''. Phys. Rev. B 103, 174309 (2021).
https://doi.org/10.1103/PhysRevB.103.174309
[32] Yaodong Li and Matthew P. A. Fisher. ``Statistical mechanics of quantum error correcting codes''. Phys. Rev. B 103, 104306 (2021).
https://doi.org/10.1103/PhysRevB.103.104306
[33] Yaodong Li, Sagar Vijay, and Matthew P.A. Fisher. ``Entanglement domain walls in monitored quantum circuits and the directed polymer in a random environment''. PRX Quantum 4, 010331 (2023).
https://doi.org/10.1103/PRXQuantum.4.010331
[34] Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. ``Measuring entanglement entropy in a quantum many-body system''. Nature 528, 77–83 (2015).
https://doi.org/10.1038/nature15750
[35] Scott Aaronson and Daniel Gottesman. ``Improved simulation of stabilizer circuits''. Phys. Rev. A 70, 052328 (2004).
https://doi.org/10.1103/PhysRevA.70.052328
[36] Hansveer Singh, Brayden A. Ware, Romain Vasseur, and Aaron J. Friedman. ``Subdiffusion and many-body quantum chaos with kinetic constraints''. Phys. Rev. Lett. 127, 230602 (2021).
https://doi.org/10.1103/PhysRevLett.127.230602
Cited by
[1] Shuo Liu, Ming-Rui Li, Shi-Xin Zhang, Shao-Kai Jian, and Hong Yao, "Noise-induced phase transitions in hybrid quantum circuits", Physical Review B 110 6, 064323 (2024).
[2] Jimin Li, Bruno Bertini, Juan P. Garrahan, and Robert L. Jack, "Exact large deviations and emergent long-range correlations in sequential quantum East circuits", Physical Review B 113 10, L100305 (2026).
[3] Shuo Liu, Ming-Rui Li, Shi-Xin Zhang, and Shao-Kai Jian, "Entanglement Structure and Information Protection in Noisy Hybrid Quantum Circuits", Physical Review Letters 132 24, 240402 (2024).
[4] Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio, Lev Vidmar, and Jakub Zakrzewski, "Many-body localization in the age of classical computing* ", Reports on Progress in Physics 88 2, 026502 (2025).
[5] Bruno Bertini, Katja Klobas, Pavel Kos, and Daniel Malz, "Random permutation circuits beyond qubits are quantum chaotic", Physical Review B 113 10, L100302 (2026).
[6] Chun-Yue Zhang, Zi-Xiang Li, and Shi-Xin Zhang, "Entanglement Growth from Entangled States: A Unified Perspective on Entanglement Generation and Transport", Physical Review Letters 137 2, 020404 (2026).
[7] Shuo Liu, Shao-Kai Jian, and Shi-Xin Zhang, "Noisy monitored quantum circuits", Journal of Physics: Condensed Matter 38 20, 203001 (2026).
The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 13:54:08) and SAO/NASA ADS (last updated successfully 2026-08-09 13:54:09). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.