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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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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[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.

[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.

[3] Y. Li, X. Chen, and M. P. A. Fisher, ``Measurement-driven entanglement transition in hybrid quantum circuits,'' Phys. Rev. B 100, 134306 (2019).

[4] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, ``Unitary-projective entanglement dynamics,'' Phys. Rev. B 99, 224307 (2019).

[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.

[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.

[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.

[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.

[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.

[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.

[11] S. Bravyi, ``Lagrangian representation for fermionic linear optics,'' Quantum Inf. and Comp. 5, 216 (2005), 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.

[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.

[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.

[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.

[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.

[17] M. B. Hastings, ``Random unitaries give quantum expanders,'' Physical Review A 76, 032315 (2007), arXiv:0706.0556.

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

[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.

[21] F. J. Dyson, ``A Brownian motion model for the eigenvalues of a random matrix,'' J. Math. Phys. 3, 1191 (1962).

[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.

[23] A. Nahum, P. Serna, A. M. Somoza, and M. Ortuño, ``Loop models with crossings,'' Phys. Rev. B 87, 184204 (2013).

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

[3] 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).

[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, 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).

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

[8] 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).

[9] 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).

[10] Yimu Bao, Soonwon Choi, and Ehud Altman, "Symmetry enriched phases of quantum circuits", arXiv:2102.09164.

[11] 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.

[12] 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).

[13] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli, "Topological order and criticality in (2+1)D monitored random quantum circuits", arXiv:2011.06595.

[14] 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.

[15] 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.

[16] 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).

[17] Zhi-Cheng Yang, Yaodong Li, Matthew P. A. Fisher, and Xiao Chen, "Entanglement phase transitions in random stabilizer tensor networks", arXiv:2107.12376.

[18] Sankhya Basu, Daniel P. Arovas, Sarang Gopalakrishnan, Chris A. Hooley, and Vadim Oganesyan, "Fisher zeros and persistent temporal oscillations in non-unitary quantum circuits", arXiv:2103.10628.

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-22 08:34:23) and SAO/NASA ADS (last updated successfully 2021-10-22 08:34:24). The list may be incomplete as not all publishers provide suitable and complete citation data.