Solvable model of deep thermalization with distinct design times

Matteo Ippoliti1 and Wen Wei Ho1,2

1Department of Physics, Stanford University, Stanford, CA 94305, USA
2Department of Physics, National University of Singapore, Singapore 117542

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We study the emergence over time of a universal, uniform distribution of quantum states supported on a finite subsystem, induced by projectively measuring the rest of the system. Dubbed $\textit{deep thermalization}$, this phenomenon represents a form of equilibration in quantum many-body systems stronger than regular thermalization, which only constrains the ensemble-averaged values of observables. While there exist quantum circuit models of dynamics in one dimension where this phenomenon can be shown to arise exactly, these are special in that deep thermalization occurs at precisely the same time as regular thermalization. Here, we present an exactly-solvable model of chaotic dynamics where the two processes can be shown to occur over different time scales. The model is composed of a finite subsystem coupled to an infinite random-matrix bath through a small constriction, and highlights the role of locality and imperfect thermalization in constraining the formation of such universal wavefunction distributions. We test our analytical predictions against exact numerical simulations, finding excellent agreement.

A projective measurement on a part of a quantum system yields a random state in the rest of the system—but how random? Recent works suggest the emergence of a universal, maximally-random distribution of states at late times in generic quantum dynamics, a condition strictly stronger than conventional thermalization and potentially useful for quantum information processing tasks. Here we introduce a simple model of quantum dynamics where the emergence of this random distribution can be studied analytically thanks to random matrix methods.

► BibTeX data

► References

[1] Ehud Altman, Kenneth R. Brown, Giuseppe Carleo, Lincoln D. Carr, Eugene Demler, Cheng Chin, et al. ``Quantum Simulators: Architectures and Opportunities''. PRX Quantum 2, 017003 (2021).

[2] Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``Probing many-body dynamics on a 51-atom quantum simulator''. Nature 551, 579–584 (2017).

[3] Soonwon Choi, Joonhee Choi, Renate Landig, Georg Kucsko, Hengyun Zhou, Junichi Isoya, Fedor Jelezko, Shinobu Onoda, Hitoshi Sumiya, Vedika Khemani, Curt von Keyserlingk, Norman Y. Yao, Eugene Demler, and Mikhail D. Lukin. ``Observation of discrete time-crystalline order in a disordered dipolar many-body system''. Nature 543, 221–225 (2017).

[4] Xiao Mi, Matteo Ippoliti, Chris Quintana, Ami Greene, Zijun Chen, Jonathan Gross, et al. ``Time-crystalline eigenstate order on a quantum processor''. Nature 601, 531–536 (2022).

[5] J. Randall, C. E. Bradley, F. V. van der Gronden, A. Galicia, M. H. Abobeih, M. Markham, D. J. Twitchen, F. Machado, N. Y. Yao, and T. H. Taminiau. ``Many-body–localized discrete time crystal with a programmable spin-based quantum simulator''. Science 374, 1474–1478 (2021).

[6] Philipp T. Dumitrescu, Justin G. Bohnet, John P. Gaebler, Aaron Hankin, David Hayes, Ajesh Kumar, Brian Neyenhuis, Romain Vasseur, and Andrew C. Potter. ``Dynamical topological phase realized in a trapped-ion quantum simulator''. Nature 607, 463–467 (2022).

[7] Brian Skinner, Jonathan Ruhman, and Adam Nahum. ``Measurement-Induced Phase Transitions in the Dynamics of Entanglement''. Physical Review X 9, 031009 (2019).

[8] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. ``Quantum Zeno effect and the many-body entanglement transition''. Physical Review B 98, 205136 (2018).

[9] Soonwon Choi, Yimu Bao, Xiao-Liang Qi, and Ehud Altman. ``Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition''. Physical Review Letters 125, 030505 (2020).

[10] Michael J. Gullans and David A. Huse. ``Dynamical Purification Phase Transition Induced by Quantum Measurements''. Physical Review X 10, 041020 (2020).

[11] Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan, David A. Huse, and Vedika Khemani. ``Entanglement Phase Transitions in Measurement-Only Dynamics''. Physical Review X 11, 011030 (2021).

[12] Andrew C. Potter and Romain Vasseur. ``Entanglement dynamics in hybrid quantum circuits'' (2021). arXiv:2111.08018.

[13] Matthew P. A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random Quantum Circuits'' (2022). arXiv:2207.14280.

[14] Crystal Noel, Pradeep Niroula, Daiwei Zhu, Andrew Risinger, Laird Egan, Debopriyo Biswas, Marko Cetina, Alexey V. Gorshkov, Michael J. Gullans, David A. Huse, and Christopher Monroe. ``Measurement-induced quantum phases realized in a trapped-ion quantum computer''. Nature Physics 18, 760–764 (2022).

[15] Jin Ming Koh, Shi-Ning Sun, Mario Motta, and Austin J. Minnich. ``Experimental Realization of a Measurement-Induced Entanglement Phase Transition on a Superconducting Quantum Processor'' (2022). arXiv:2203.04338.

[16] Scott Aaronson. ``Shadow Tomography of Quantum States'' (2018). arXiv:1711.01053.

[17] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. Nature Physics 16, 1050–1057 (2020).

[18] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoı̂t Vermersch, and Peter Zoller. ``The randomized measurement toolbox'' (2022). arXiv:2203.11374.

[19] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019).

[20] Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, et al. ``Strong Quantum Computational Advantage Using a Superconducting Quantum Processor''. Physical Review Letters 127, 180501 (2021).

[21] Scott Aaronson and Lijie Chen. ``Complexity-theoretic foundations of quantum supremacy experiments''. In Proceedings of the 32nd Computational Complexity Conference. Pages 1–67. CCC '17Dagstuhl, DEU (2017).

[22] Yiqing Zhou, E. Miles Stoudenmire, and Xavier Waintal. ``What Limits the Simulation of Quantum Computers?''. Physical Review X 10, 041038 (2020).

[23] Xun Gao, Marcin Kalinowski, Chi-Ning Chou, Mikhail D. Lukin, Boaz Barak, and Soonwon Choi. ``Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage'' (2021). arXiv:2112.01657.

[24] Abhinav Deshpande, Pradeep Niroula, Oles Shtanko, Alexey V. Gorshkov, Bill Fefferman, and Michael J. Gullans. ``Tight bounds on the convergence of noisy random circuits to the uniform distribution'' (2021). arXiv:2112.00716.

[25] Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G. S. L. Brandão. ``Random quantum circuits transform local noise into global white noise'' (2021). arXiv:2111.14907.

[26] Joonhee Choi, Adam L. Shaw, Ivaylo S. Madjarov, Xin Xie, Jacob P. Covey, Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Anant Kale, Hannes Pichler, Fernando G. S. L. Brandão, Soonwon Choi, and Manuel Endres. ``Emergent Randomness and Benchmarking from Many-Body Quantum Chaos'' (2021). arXiv:2103.03535.

[27] Daniel K. Mark, Joonhee Choi, Adam L. Shaw, Manuel Endres, and Soonwon Choi. ``Benchmarking Quantum Simulators using Quantum Chaos'' (2022). arXiv:2205.12211.

[28] Mark Srednicki. ``Chaos and quantum thermalization''. Physical Review E 50, 888–901 (1994).

[29] Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. ``Thermalization and its mechanism for generic isolated quantum systems''. Nature 452, 854–858 (2008).

[30] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. ``Colloquium: Many-body localization, thermalization, and entanglement''. Reviews of Modern Physics 91, 021001 (2019).

[31] Álvaro M. Alhambra. ``Quantum many-body systems in thermal equilibrium'' (2022). arXiv:2204.08349.

[32] Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Felipe Hernandez, Joonhee Choi, Adam L. Shaw, Manuel Endres, and Soonwon Choi. ``Emergent quantum state designs from individual many-body wavefunctions'' (2021). arXiv:2103.03536.

[33] Wen Wei Ho and Soonwon Choi. ``Exact Emergent Quantum State Designs from Quantum Chaotic Dynamics''. Physical Review Letters 128, 060601 (2022).

[34] Matteo Ippoliti and Wen Wei Ho. ``Dynamical purification and the emergence of quantum state designs from the projected ensemble'' (2022). arXiv:2204.13657.

[35] Andris Ambainis and Joseph Emerson. ``Quantum t-designs: t-wise Independence in the Quantum World''. In Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07). Pages 129–140. (2007).

[36] D. Gross, K. Audenaert, and J. Eisert. ``Evenly distributed unitaries: On the structure of unitary designs''. Journal of Mathematical Physics 48, 052104 (2007).

[37] Richard A. Low. ``Pseudo-randomness and Learning in Quantum Computation'' (2010). arXiv:1006.5227.

[38] M. Akila, D. Waltner, B. Gutkin, and T. Guhr. ``Particle-time duality in the kicked Ising spin chain''. Journal of Physics A: Mathematical and Theoretical 49, 375101 (2016).

[39] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos''. Physical Review Letters 121, 264101 (2018).

[40] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions''. Phys. Rev. Lett. 123, 210601 (2019).

[41] Sarang Gopalakrishnan and Austen Lamacraft. ``Unitary circuits of finite depth and infinite width from quantum channels''. Physical Review B 100, 064309 (2019).

[42] Pieter W. Claeys and Austen Lamacraft. ``Maximum velocity quantum circuits''. Physical Review Research 2, 033032 (2020).

[43] Pieter W. Claeys and Austen Lamacraft. ``Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics''. Quantum 6, 738 (2022).

[44] Richard Jozsa, Daniel Robb, and William K. Wootters. ``Lower bound for accessible information in quantum mechanics''. Physical Review A 49, 668–677 (1994).

[45] Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka, and Nino Zanghì. ``On the Distribution of the Wave Function for Systems in Thermal Equilibrium''. Journal of Statistical Physics 125, 1193–1221 (2006).

[46] Sheldon Goldstein, Joel L. Lebowitz, Christian Mastrodonato, Roderich Tumulka, and Nino Zanghì. ``Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment''. Communications in Mathematical Physics 342, 965–988 (2016).

[47] Lorenzo Piroli, Bruno Bertini, J. Ignacio Cirac, and Tomaž Prosen. ``Exact dynamics in dual-unitary quantum circuits''. Physical Review B 101, 094304 (2020).

[48] Fabio Anza and James P. Crutchfield. ``Beyond density matrices: Geometric quantum states''. Physical Review A 103, 062218 (2021).

[49] Fabio Anza and James P. Crutchfield. ``Quantum Information Dimension and Geometric Entropy''. PRX Quantum 3, 020355 (2022).

[50] Joseph M. Renes, Robin Blume-Kohout, A. J. Scott, and Carlton M. Caves. ``Symmetric informationally complete quantum measurements''. Journal of Mathematical Physics 45, 2171–2180 (2004).

[51] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland. ``Randomized benchmarking of quantum gates''. Physical Review A 77, 012307 (2008).

[52] Daniel A. Roberts and Beni Yoshida. ``Chaos and complexity by design''. Journal of High Energy Physics 2017, 121 (2017).

[53] M. C. Bañuls, M. B. Hastings, F. Verstraete, and J. I. Cirac. ``Matrix Product States for Dynamical Simulation of Infinite Chains''. Phys. Rev. Lett. 102, 240603 (2009).

[54] M. B. Hastings and R. Mahajan. ``Connecting entanglement in time and space: Improving the folding algorithm''. Phys. Rev. A 91, 032306 (2015).

[55] Alessio Lerose, Michael Sonner, and Dmitry A. Abanin. ``Influence Matrix Approach to Many-Body Floquet Dynamics''. Physical Review X 11, 021040 (2021).

[56] Michael Sonner, Alessio Lerose, and Dmitry A. Abanin. ``Influence functional of many-body systems: Temporal entanglement and matrix-product state representation''. Annals of Physics 435, 168677 (2021).

[57] Giacomo Giudice, Giuliano Giudici, Michael Sonner, Julian Thoenniss, Alessio Lerose, Dmitry A. Abanin, and Lorenzo Piroli. ``Temporal Entanglement, Quasiparticles, and the Role of Interactions''. Physical Review Letters 128, 220401 (2022).

[58] Matteo Ippoliti and Vedika Khemani. ``Postselection-Free Entanglement Dynamics via Spacetime Duality''. Physical Review Letters 126, 060501 (2021).

[59] Matteo Ippoliti, Tibor Rakovszky, and Vedika Khemani. ``Fractal, Logarithmic, and Volume-Law Entangled Nonthermal Steady States via Spacetime Duality''. Physical Review X 12, 011045 (2022).

[60] Tsung-Cheng Lu and Tarun Grover. ``Spacetime duality between localization transitions and measurement-induced transitions''. PRX Quantum 2, 040319 (2021).

[61] David T. Stephen, Wen Wei Ho, Tzu-Chieh Wei, Robert Raussendorf, and Ruben Verresen. ``Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual-unitary circuits'' (2022). arXiv:2209.06191.

[62] Felix A. Pollock, Cesar Rodriguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi. ``Non-Markovian quantum processes: Complete framework and efficient characterization''. Physical Review A 97, 012127 (2018).

[63] Georg Köstenberger. ``Weingarten Calculus'' (2021). arXiv:2101.00921.

[64] Hyungwon Kim and David A. Huse. ``Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System''. Physical Review Letters 111, 127205 (2013).

[65] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. ``Quantum Entanglement Growth under Random Unitary Dynamics''. Physical Review X 7, 031016 (2017).

[66] Cheryne Jonay, David A. Huse, and Adam Nahum. ``Coarse-grained dynamics of operator and state entanglement'' (2018). arXiv:1803.00089.

[67] Tianci Zhou and Adam Nahum. ``Emergent statistical mechanics of entanglement in random unitary circuits''. Phys. Rev. B 99, 174205 (2019).

[68] Tianci Zhou and Adam Nahum. ``Entanglement Membrane in Chaotic Many-Body Systems''. Phys. Rev. X 10, 031066 (2020).

[69] Yaodong Li and Matthew P. A. Fisher. ``Statistical mechanics of quantum error correcting codes''. Physical Review B 103, 104306 (2021).

[70] Jac Bensa and Marko Znidaric. ``Fastest Local Entanglement Scrambler, Multistage Thermalization, and a Non-Hermitian Phantom''. Physical Review X 11, 031019 (2021).

[71] Jac Bensa and Marko Znidaric. ``Two-step phantom relaxation of out-of-time-ordered correlations in random circuits''. Physical Review Research 4, 013228 (2022).

[72] Henrik Wilming and Ingo Roth. ``High-temperature thermalization implies the emergence of quantum state designs'' (2022). arXiv:2202.01669.

[73] Tibor Rakovszky, Frank Pollmann, and C. W. von Keyserlingk. ``Diffusive Hydrodynamics of Out-of-Time-Ordered Correlators with Charge Conservation''. Phys. Rev. X 8, 031058 (2018).

[74] Vedika Khemani, Ashvin Vishwanath, and David A. Huse. ``Operator Spreading and the Emergence of Dissipative Hydrodynamics under Unitary Evolution with Conservation Laws''. Phys. Rev. X 8, 031057 (2018).

[75] Nicholas Hunter-Jones. ``Operator growth in random quantum circuits with symmetry'' (2018). arXiv:1812.08219.

[76] 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'' (2021). arXiv:2107.10279.

[77] Sarang Gopalakrishnan. ``Operator growth and eigenstate entanglement in an interacting integrable Floquet system''. Physical Review B 98, 060302 (2018).

[78] Katja Klobas, Bruno Bertini, and Lorenzo Piroli. ``Exact Thermalization Dynamics in the ``Rule 54'' Quantum Cellular Automaton''. Physical Review Letters 126, 160602 (2021).

[79] Berislav Buča, Katja Klobas, and Tomaž Prosen. ``Rule 54: exactly solvable model of nonequilibrium statistical mechanics''. Journal of Statistical Mechanics: Theory and Experiment 2021, 074001 (2021).

[80] Hansveer Singh, Romain Vasseur, and Sarang Gopalakrishnan. ``The Fredkin staircase: An integrable system with a finite-frequency Drude peak'' (2022). arXiv:2205.08542.

[81] Maxime Lucas, Lorenzo Piroli, Jacopo De Nardis, and Andrea De Luca. ``Generalized Deep Thermalization for Free Fermions'' (2022). arXiv:2207.13628.

[82] C. W. von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and S. L. Sondhi. ``Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems without Conservation Laws''. Physical Review X 8, 021013 (2018).

Cited by

[1] Hugo Lóio, Andrea De Luca, Jacopo De Nardis, and Xhek Turkeshi, "Purification timescales in monitored fermions", Physical Review B 108 2, L020306 (2023).

[2] Matteo Ippoliti and Wen Wei Ho, "Dynamical Purification and the Emergence of Quantum State Designs from the Projected Ensemble", PRX Quantum 4 3, 030322 (2023).

[3] Anastasiia Tiutiakina, Andrea De Luca, and Jacopo De Nardis, "Frame potential of Brownian SYK model of Majorana and Dirac fermions", Journal of High Energy Physics 2024 1, 115 (2024).

[4] Matteo Ippoliti and Vedika Khemani, "Learnability Transitions in Monitored Quantum Dynamics via Eavesdropper’s Classical Shadows", PRX Quantum 5 2, 020304 (2024).

[5] Minh C. Tran, Daniel K. Mark, Wen Wei Ho, and Soonwon Choi, "Measuring Arbitrary Physical Properties in Analog Quantum Simulation", Physical Review X 13 1, 011049 (2023).

[6] Pieter W. Claeys, "Universality in quantum snapshots", Quantum Views 7, 71 (2023).

[7] Saúl Pilatowsky-Cameo, Ceren B. Dag, Wen Wei Ho, and Soonwon Choi, "Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives", Physical Review Letters 131 25, 250401 (2023).

[8] Heitor P. Casagrande, Bo Xing, Marcello Dalmonte, Alex Rodriguez, Vinitha Balachandran, and Dario Poletti, "Complexity of spin configuration dynamics due to unitary evolution and periodic projective measurements", Physical Review E 108 4, 044128 (2023).

[9] Maxime Lucas, Lorenzo Piroli, Jacopo De Nardis, and Andrea De Luca, "Generalized deep thermalization for free fermions", Physical Review A 107 3, 032215 (2023).

[10] Matteo Ippoliti, "Classical shadows based on locally-entangled measurements", Quantum 8, 1293 (2024).

[11] Felix Fritzsch and Tomaž Prosen, "Boundary Chaos: Spectral Form Factor", arXiv:2312.12452, (2023).

[12] Shiyong Guo, Martin Sasieta, and Brian Swingle, "Complexity is not Enough for Randomness", arXiv:2405.17546, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-21 18:26:05) and SAO/NASA ADS (last updated successfully 2024-06-21 18:26:05). The list may be incomplete as not all publishers provide suitable and complete citation data.

1 thought on “Solvable model of deep thermalization with distinct design times

  1. Pingback: Perspective in Quantum Views by Pieter W. Claeys "Universality in quantum snapshots"