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] 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).

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

[3] 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).

[4] 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).

[5] 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).

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

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

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

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

[10] 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).

[11] 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).

[12] 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).

[13] 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).

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

[15] 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).

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

[17] 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).

[18] 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).

[19] 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).

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

[21] Matthew B. Hastings and Jeongwan Haah, "Dynamically Generated Logical Qubits", arXiv:2107.02194, Quantum 5, 564 (2021).

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

[23] 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).

[24] 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).

[25] 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).

[26] 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.

[27] 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).

[28] 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.

[29] Yaodong Li, Yijian Zou, Paolo Glorioso, Ehud Altman, and Matthew P. A. Fisher, "Cross Entropy Benchmark for Measurement-Induced Phase Transitions", arXiv:2209.00609.

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

[31] 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.

[32] Jason Iaconis, Andrew Lucas, and Xiao Chen, "Measurement-induced phase transitions in quantum automaton circuits", arXiv:2010.02196, Physical Review B 102 22, 224311 (2020).

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