How Dynamical Quantum Memories Forget

Lukasz Fidkowski1, Jeongwan Haah2, and Matthew B. Hastings3,2

1Department of Physics, University of Washington, Seattle, WA 98195-1560, USA
2Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA
3Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA

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Updated version: The authors have uploaded version v3 of this work to the arXiv which may contain updates or corrections not contained in the published version v2. The authors left the following comment on the arXiv:
24 pages, 2 figures (v3) expanding the proof of Lemma 3

Abstract

Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the ``mixed'' phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems $—$ i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears $—$ purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.

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► References

[1] Y. Li, X. Chen, and M. P. A. Fisher, ``Quantum zeno effect and the many-body entanglement transition,'' Phys. Rev. B 98, 205136 (2018), arXiv:1808.06134.
https:/​/​doi.org/​10.1103/​PhysRevB.98.205136
arXiv:1808.06134

[2] B. Skinner, J. Ruhman, and A. Nahum, ``Measurement-induced phase transitions in the dynamics of entanglement,'' Phys. Rev. X 9, 031009 (2019), arXiv:1808.05953.
https:/​/​doi.org/​10.1103/​PhysRevX.9.031009
arXiv:1808.05953

[3] Y. Li, X. Chen, and M. 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

[4] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, ``Unitary-projective entanglement dynamics,'' Phys. Rev. B 99, 224307 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.99.224307

[5] M. J. Gullans and D. A. Huse, ``Dynamical purification phase transitions induced by quantum measurements,'' Phys. Rev. X 10, 041020 (2020a), arXiv:1905.05195.
https:/​/​doi.org/​10.1103/​PhysRevX.10.041020
arXiv:1905.05195

[6] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, ``Quantum error correction in scrambling dynamics and measurement-induced phase transition,'' Phys. Rev. Lett. 125, 030505 (2019), arXiv:1903.05124.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.030505
arXiv:1903.05124

[7] R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, ``Self-organized error correction in random unitary circuits with measurement,'' (2020), arXiv:2002.12385.
arXiv:2002.12385

[8] F. G. Brandao, A. W. Harrow, and M. Horodecki, ``Local random quantum circuits are approximate polynomial-designs,'' Commun. Math. Phys. 346, 397–434 (2016), arXiv:1208.0692.
https:/​/​doi.org/​10.1007/​s00220-016-2706-8
arXiv:1208.0692

[9] A. Harrow and S. Mehraban, ``Approximate unitary $t$-designs by short random quantum circuits using nearest-neighbor and long-range gates,'' (2018), arXiv:1809.06957.
arXiv:1809.06957

[10] J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, ``Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-clifford gates,'' (2020), arXiv:2002.09524.
arXiv:2002.09524

[11] S. Bravyi, ``Lagrangian representation for fermionic linear optics,'' Quantum Inf. and Comp. 5, 216 (2005), arXiv:quant-ph/​0404180.
https:/​/​doi.org/​10.26421/​QIC5.3
arXiv:quant-ph/0404180

[12] M. J. Gullans and D. A. Huse, ``Scalable probes of measurement-induced criticality,'' Phys. Rev. Lett. 125, 070606 (2020) 125, 070606 (2020b), arXiv:1910.00020.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.070606
arXiv:1910.00020

[13] X. Cao, A. Tilloy, and A. D. Luca, ``Entanglement in a fermion chain under continuous monitoring,'' SciPost Phys. 7, 24 (2019), arXiv:1804.04638.
https:/​/​doi.org/​10.21468/​SciPostPhys.7.2.024
arXiv:1804.04638

[14] X. Chen, Y. Li, M. P. A. Fisher, and A. Lucas, ``Emergent conformal symmetry in nonunitary random dynamics of free fermions,'' Phys. Rev. Research 2, 033017 (2020), arXiv:2004.09577.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.033017
arXiv:2004.09577

[15] M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, ``Entanglement phase transitions in measurement-only dynamics,'' (2020), arXiv:2004.09560.
arXiv:2004.09560

[16] A. Nahum and B. Skinner, ``Entanglement and dynamics of diffusion-annihilation processes with majorana defects,'' Phys. Rev. Research 2, 023288 (2020), arXiv:1911.11169.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.023288
arXiv:1911.11169

[17] M. B. Hastings, ``Random unitaries give quantum expanders,'' Physical Review A 76, 032315 (2007), arXiv:0706.0556.
https:/​/​doi.org/​10.1103/​PhysRevA.76.032315
arXiv:0706.0556

[18] Y. Li and M. P. A. Fisher, ``Statistical mechanics of quantum error-correcting codes,'' (2020), arXiv:2007.03822 [quant-ph].
arXiv:2007.03822

[19] E. S. Meckes, The random matrix theory of the classical compact groups, Vol. 218 (Cambridge University Press, 2019).

[20] K. M. R. Audenaert, ``A sharp fannes-type inequality for the von neumann entropy,'' J. Phys. A 40, 8127–8136 (2007), quant-ph/​0610146.
https:/​/​doi.org/​10.1088/​1751-8113/​40/​28/​S18
arXiv:quant-ph/0610146

[21] F. J. Dyson, ``A Brownian motion model for the eigenvalues of a random matrix,'' J. Math. Phys. 3, 1191 (1962).
https:/​/​doi.org/​10.1063/​1.1703862

[22] B. Collins and P. Sniady, ``Integration with respect to the Haar measure on unitary, orthogonal and symplectic group,'' Commun. Math. Phys. 264, 773–795 (2006), arXiv:math-ph/​0402073.
https:/​/​doi.org/​10.1007/​s00220-006-1554-3
arXiv:math-ph/0402073

[23] A. Nahum, P. Serna, A. M. Somoza, and M. Ortuño, ``Loop models with crossings,'' Phys. Rev. B 87, 184204 (2013).
https:/​/​doi.org/​10.1103/​PhysRevB.87.184204

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[1] Xhek Turkeshi, Marcello Dalmonte, Rosario Fazio, and Marco Schirò, "Entanglement transitions from stochastic resetting of non-Hermitian quasiparticles", Physical Review B 105 24, L241114 (2022).

[2] Giulia Piccitto, Angelo Russomanno, and Davide Rossini, "Entanglement dynamics with string measurement operators", SciPost Physics Core 6 4, 078 (2023).

[3] Gianluca Passarelli, Xhek Turkeshi, Angelo Russomanno, Procolo Lucignano, Marco Schirò, and Rosario Fazio, "Many-Body Dynamics in Monitored Atomic Gases without Postselection Barrier", Physical Review Letters 132 16, 163401 (2024).

[4] Raimel Medina, Romain Vasseur, and Maksym Serbyn, "Entanglement transitions from restricted Boltzmann machines", Physical Review B 104 10, 104205 (2021).

[5] Matteo Ippoliti and Vedika Khemani, "Postselection-Free Entanglement Dynamics via Spacetime Duality", Physical Review Letters 126 6, 060501 (2021).

[6] Oliver Lunt, Jonas Richter, and Arijeet Pal, Quantum Science and Technology 251 (2022) ISBN:978-3-031-03997-3.

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

[8] Zhi-Cheng Yang, Yaodong Li, Matthew P. A. Fisher, and Xiao Chen, "Entanglement phase transitions in random stabilizer tensor networks", Physical Review B 105 10, 104306 (2022).

[9] Oliver Lunt, Marcin Szyniszewski, and Arijeet Pal, "Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits", Physical Review B 104 15, 155111 (2021).

[10] Chao-Ming Jian, Bela Bauer, Anna Keselman, and Andreas W. W. Ludwig, "Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions", Physical Review B 106 13, 134206 (2022).

[11] Pengfei Zhang, Chunxiao Liu, Shao-Kai Jian, and Xiao Chen, "Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics", Quantum 6, 723 (2022).

[12] Takaaki Minato, Koudai Sugimoto, Tomotaka Kuwahara, and Keiji Saito, "Fate of Measurement-Induced Phase Transition in Long-Range Interactions", Physical Review Letters 128 1, 010603 (2022).

[13] Youenn Le Gal, Xhek Turkeshi, and Marco Schirò, "Volume-to-area law entanglement transition in a non-Hermitian free fermionic chain", SciPost Physics 14 5, 138 (2023).

[14] Angelo Russomanno, Giulia Piccitto, and Davide Rossini, "Entanglement transitions and quantum bifurcations under continuous long-range monitoring", Physical Review B 108 10, 104313 (2023).

[15] Matteo Ippoliti, Tibor Rakovszky, and Vedika Khemani, "Fractal, Logarithmic, and Volume-Law Entangled Nonthermal Steady States via Spacetime Duality", Physical Review X 12 1, 011045 (2022).

[16] Marcin Szyniszewski, Oliver Lunt, and Arijeet Pal, "Disordered monitored free fermions", Physical Review B 108 16, 165126 (2023).

[17] Tomohiro Hashizume, Gregory Bentsen, and Andrew J. Daley, "Measurement-induced phase transitions in sparse nonlocal scramblers", Physical Review Research 4 1, 013174 (2022).

[18] Matthew B. Hastings and Jeongwan Haah, "Dynamically Generated Logical Qubits", Quantum 5, 564 (2021).

[19] Etienne Granet, Carolyn Zhang, and Henrik Dreyer, "Volume-Law to Area-Law Entanglement Transition in a Nonunitary Periodic Gaussian Circuit", Physical Review Letters 130 23, 230401 (2023).

[20] Giulia Piccitto, Davide Rossini, and Angelo Russomanno, "The impact of different unravelings in a monitored system of free fermions", The European Physical Journal B 97 6, 90 (2024).

[21] Yaodong Li and Matthew P. A. Fisher, "Decodable hybrid dynamics of open quantum systems with Z2 symmetry", Physical Review B 108 21, 214302 (2023).

[22] Shengqi Sang, Zhi Li, Timothy H. Hsieh, and Beni Yoshida, "Ultrafast Entanglement Dynamics in Monitored Quantum Circuits", PRX Quantum 4 4, 040332 (2023).

[23] Graham Kells, Dganit Meidan, and Alessandro Romito, "Topological transitions in weakly monitored free fermions", SciPost Physics 14 3, 031 (2023).

[24] Yuri Minoguchi, Peter Rabl, and Michael Buchhold, "Continuous gaussian measurements of the free boson CFT: A model for exactly solvable and detectable measurement-induced dynamics", SciPost Physics 12 1, 009 (2022).

[25] Xhek Turkeshi, Lorenzo Piroli, and Marco Schirò, "Density and current statistics in boundary-driven monitored fermionic chains", Physical Review B 109 14, 144306 (2024).

[26] Piotr Sierant, Marco Schirò, Maciej Lewenstein, and Xhek Turkeshi, "Measurement-induced phase transitions in (d+1) -dimensional stabilizer circuits", Physical Review B 106 21, 214316 (2022).

[27] Mao Tian Tan, Yifan Wang, and Aditi Mitra, "Topological defects in Floquet circuits", SciPost Physics 16 3, 075 (2024).

[28] Xhek Turkeshi, Marcello Dalmonte, Rosario Fazio, and Marco Schirò, "Erratum: Entanglement transitions from stochastic resetting of non-Hermitian quasiparticles [Phys. Rev. B 105 , L241114 (2022)]", Physical Review B 107 7, 079901 (2023).

[29] T. Boorman, M. Szyniszewski, H. Schomerus, and A. Romito, "Diagnostics of entanglement dynamics in noisy and disordered spin chains via the measurement-induced steady-state entanglement transition", Physical Review B 105 14, 144202 (2022).

[30] John C. Napp, Rolando L. La Placa, Alexander M. Dalzell, Fernando G. S. L. Brandão, and Aram W. Harrow, "Efficient Classical Simulation of Random Shallow 2D Quantum Circuits", Physical Review X 12 2, 021021 (2022).

[31] Michael J. Gullans, Stefan Krastanov, David A. Huse, Liang Jiang, and Steven T. Flammia, "Quantum Coding with Low-Depth Random Circuits", Physical Review X 11 3, 031066 (2021).

[32] Xhek Turkeshi, Lorenzo Piroli, and Marco Schiró, "Enhanced entanglement negativity in boundary-driven monitored fermionic chains", Physical Review B 106 2, 024304 (2022).

[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 1, 010331 (2023).

[34] Yu-Peng Wang, Chen Fang, and Jie Ren, "Absence of measurement-induced entanglement transition due to feedback-induced skin effect", Physical Review B 110 3, 035113 (2024).

[35] Piotr Sierant and Xhek Turkeshi, "Universal Behavior beyond Multifractality of Wave Functions at Measurement-Induced Phase Transitions", Physical Review Letters 128 13, 130605 (2022).

[36] Piotr Sierant and Xhek Turkeshi, "Controlling Entanglement at Absorbing State Phase Transitions in Random Circuits", Physical Review Letters 130 12, 120402 (2023).

[37] Adithya Sriram, Tibor Rakovszky, Vedika Khemani, and Matteo Ippoliti, "Topology, criticality, and dynamically generated qubits in a stochastic measurement-only Kitaev model", Physical Review B 108 9, 094304 (2023).

[38] Xhek Turkeshi, Alberto Biella, Rosario Fazio, Marcello Dalmonte, and Marco Schiró, "Measurement-induced entanglement transitions in the quantum Ising chain: From infinite to zero clicks", Physical Review B 103 22, 224210 (2021).

[39] Tsung-Cheng Lu and Tarun Grover, "Spacetime duality between localization transitions and measurement-induced transitions", PRX Quantum 2 4, 040319 (2021).

[40] Joseph Merritt and Lukasz Fidkowski, "Entanglement transitions with free fermions", Physical Review B 107 6, 064303 (2023).

[41] Sankhya Basu, Daniel P. Arovas, Sarang Gopalakrishnan, Chris A. Hooley, and Vadim Oganesyan, "Fisher zeros and persistent temporal oscillations in nonunitary quantum circuits", Physical Review Research 4 1, 013018 (2022).

[42] Lei Su, Aashish Clerk, and Ivar Martin, "Dynamics and phases of nonunitary Floquet transverse-field Ising model", Physical Review Research 6 1, 013131 (2024).

[43] Brian J. J. Khor, Matthew Wampler, Gil Refael, and Israel Klich, "Measurement-induced chirality: Diffusion and disorder", Physical Review B 108 21, 214305 (2023).

[44] Sarang Gopalakrishnan and Michael J. Gullans, "Entanglement and Purification Transitions in Non-Hermitian Quantum Mechanics", Physical Review Letters 126 17, 170503 (2021).

[45] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli, "Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits", Physical Review Letters 127 23, 235701 (2021).

[46] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay, "Random Quantum Circuits", Annual Review of Condensed Matter Physics 14 1, 335 (2023).

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

[48] Jan Behrends, Florian Venn, and Benjamin Béri, "Surface codes, quantum circuits, and entanglement phases", Physical Review Research 6 1, 013137 (2024).

[49] Michele Fava, Lorenzo Piroli, Tobias Swann, Denis Bernard, and Adam Nahum, "Nonlinear Sigma Models for Monitored Dynamics of Free Fermions", Physical Review X 13 4, 041045 (2023).

[50] Yaodong Li, Yijian Zou, Paolo Glorioso, Ehud Altman, and Matthew P. A. Fisher, "Cross Entropy Benchmark for Measurement-Induced Phase Transitions", Physical Review Letters 130 22, 220404 (2023).

[51] Ali Lavasani, Zhu-Xi Luo, and Sagar Vijay, "Monitored quantum dynamics and the Kitaev spin liquid", Physical Review B 108 11, 115135 (2023).

[52] Wouter Buijsman, "Efficient circular Dyson Brownian motion algorithm", Physical Review Research 6 2, 023264 (2024).

[53] Federico Gerbino, Pierre Le Doussal, Guido Giachetti, and Andrea De Luca, "A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems", Quantum Reports 6 2, 200 (2024).

[54] 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).

[55] Bo Xing, Xhek Turkeshi, Marco Schiró, Rosario Fazio, and Dario Poletti, "Interactions and integrability in weakly monitored Hamiltonian systems", Physical Review B 109 6, L060302 (2024).

[56] Vir B. Bulchandani, S. L. Sondhi, and J. T. Chalker, "Random-Matrix Models of Monitored Quantum Circuits", Journal of Statistical Physics 191 5, 55 (2024).

[57] Julia Wildeboer, Thomas Iadecola, and Dominic J. Williamson, "Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes", PRX Quantum 3 2, 020330 (2022).

[58] Felix Roser, Hans Peter Büchler, and Nicolai Lang, "Decoding the projective transverse field Ising model", Physical Review B 107 21, 214201 (2023).

[59] M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl, "Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions", Physical Review X 11 4, 041004 (2021).

[60] Xiaozhou Feng, Brian Skinner, and Adam Nahum, "Measurement-Induced Phase Transitions on Dynamical Quantum Trees", PRX Quantum 4 3, 030333 (2023).

[61] Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan, David A. Huse, and Vedika Khemani, "Entanglement Phase Transitions in Measurement-Only Dynamics", Physical Review X 11 1, 011030 (2021).

[62] Yimu Bao, Soonwon Choi, and Ehud Altman, "Symmetry enriched phases of quantum circuits", Annals of Physics 435, 168618 (2021).

[63] Shengqi Sang, Yaodong Li, Tianci Zhou, Xiao Chen, Timothy H. Hsieh, and Matthew P. A. Fisher, "Entanglement Negativity at Measurement-Induced Criticality", PRX Quantum 2 3, 030313 (2021).

[64] Jason Iaconis, Andrew Lucas, and Xiao Chen, "Measurement-induced phase transitions in quantum automaton circuits", Physical Review B 102 22, 224311 (2020).

[65] Yaodong Li, Sagar Vijay, and Matthew P. A. Fisher, "Entanglement Domain Walls in Monitored Quantum Circuits and the Directed Polymer in a Random Environment", arXiv:2105.13352, (2021).

[66] Chao-Ming Jian, Bela Bauer, Anna Keselman, and Andreas W. W. Ludwig, "Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions", arXiv:2012.04666, (2020).

[67] Federico Gerbino, Pierre Le Doussal, Guido Giachetti, and Andrea De Luca, "A Dyson Brownian motion model for weak measurements in chaotic quantum systems", arXiv:2401.00822, (2024).

[68] Jason Iaconis, Andrew Lucas, and Xiao Chen, "Measurement-induced phase transitions in quantum automaton circuits", arXiv:2010.02196, (2020).

[69] Adam Nahum, Sthitadhi Roy, Brian Skinner, and Jonathan Ruhman, "Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory", arXiv:2009.11311, (2020).

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