Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

Pieter W. Claeys; Austen Lamacraft

doi:10.22331/q-2022-06-15-738

Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

Pieter W. Claeys^1,2 and Austen Lamacraft²

¹Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
²TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

Published:	2022-06-15, volume 6, page 738
Eprint:	arXiv:2202.12306v2
Doi:	https://doi.org/10.22331/q-2022-06-15-738
Citation:	Quantum 6, 738 (2022).

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

Abstract

Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a $\textit{projected ensemble}$. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a $\textit{quantum state design}$. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.

Featured image: Schematic illustration of a quantum circuit for which measurements on the observed subsystem B return random states on the unobserved subsystem A.

Popular summary

Recent demonstrations of quantum supremacy have been based on preparing random quantum states. In these experiments randomness was introduced by choosing experimental parameters using ordinary (pseudo-)random number generators. Recently, an alternative approach was suggested: by measuring a part of a large quantum system, the uncertainty inherent in the quantum measurement process itself could be used to generate a random quantum state in the unobserved part of the system.

For this approach to work the state must have a high degree of entanglement between the two subsystems. On the other hand, feasible experimental realisations must be local: formed by operations on neighbouring qubits, for example. In this paper we show that a recently introduced family of quantum circuits made from dual-unitary gates provides precisely the necessary ingredients to build arbitrarily random quantum states by the method of partial measurements. Besides potential applications to benchmarking of quantum computers, our results provide a detailed view of the quantum chaotic properties of the wavefunctions of an extended system.

► BibTeX data

@article{Claeys2022emergentquantum,
  doi = {10.22331/q-2022-06-15-738},
  url = {https://doi.org/10.22331/q-2022-06-15-738},
  title = {Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics},
  author = {Claeys, Pieter W. and Lamacraft, Austen},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {738},
  month = jun,
  year = {2022}
}

► References

[1] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).
https://doi.org/10.1080/00018732.2016.1198134

[2] H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).
https://doi.org/10.1017/CBO9780511524622

[3] F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, Vol. 54 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010).
https://doi.org/10.1007/978-3-642-05428-0

[4] M. Akila, D. Waltner, B. Gutkin, and T. Guhr, J. Phys. A: Math. Theor. 49, 375101 (2016).
https://doi.org/10.1088/1751-8113/49/37/375101

[5] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. Lett. 121, 264101 (2018).
https://doi.org/10.1103/PhysRevLett.121.264101

[6] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. X 9, 021033 (2019a).
https://doi.org/10.1103/PhysRevX.9.021033

[7] S. Gopalakrishnan and A. Lamacraft, Phys. Rev. B 100, 064309 (2019).
https://doi.org/10.1103/PhysRevB.100.064309

[8] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. Lett. 123, 210601 (2019b).
https://doi.org/10.1103/PhysRevLett.123.210601

[9] S. A. Rather, S. Aravinda, and A. Lakshminarayan, Phys. Rev. Lett. 125, 070501 (2020).
https://doi.org/10.1103/PhysRevLett.125.070501

[10] B. Gutkin, P. Braun, M. Akila, D. Waltner, and T. Guhr, Phys. Rev. B 102, 174307 (2020).
https://doi.org/10.1103/PhysRevB.102.174307

[11] S. Aravinda, S. A. Rather, and A. Lakshminarayan, Phys. Rev. Research 3, 043034 (2021).
https://doi.org/10.1103/PhysRevResearch.3.043034

[12] P. W. Claeys and A. Lamacraft, Phys. Rev. Lett. 126, 100603 (2021).
https://doi.org/10.1103/PhysRevLett.126.100603

[13] T. Prosen, Chaos 31, 093101 (2021).
https://doi.org/10.1063/5.0056970

[14] S. Singh and I. Nechita, arXiv:2112.11123 (2021).
https://doi.org/10.1088/1751-8121/ac7017
arXiv:2112.11123v1

[15] M. Borsi and B. Pozsgay, arXiv:2201.07768 (2022).
arXiv:2201.07768

[16] P. W. Claeys and A. Lamacraft, Phys. Rev. Research 2, 033032 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033032

[17] B. Bertini and L. Piroli, Phys. Rev. B 102, 064305 (2020).
https://doi.org/10.1103/PhysRevB.102.064305

[18] R. Suzuki, K. Mitarai, and K. Fujii, Quantum 6, 631 (2022).
https://doi.org/10.22331/q-2022-01-24-631

[19] L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, Phys. Rev. B 101, 094304 (2020).
https://doi.org/10.1103/PhysRevB.101.094304

[20] B. Jonnadula, P. Mandayam, K. Życzkowski, and A. Lakshminarayan, Phys. Rev. Research 2, 043126 (2020).
https://doi.org/10.1103/PhysRevResearch.2.043126

[21] I. Reid and B. Bertini, Phys. Rev. B 104, 014301 (2021).
https://doi.org/10.1103/PhysRevB.104.014301

[22] P. Kos, B. Bertini, and T. Prosen, Phys. Rev. X 11, 011022 (2021a).
https://doi.org/10.1103/PhysRevX.11.011022

[23] A. Lerose, M. Sonner, and D. A. Abanin, Phys. Rev. X 11, 021040 (2021).
https://doi.org/10.1103/PhysRevX.11.021040

[24] G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D. A. Abanin, and L. Piroli, Phys. Rev. Lett. 128, 220401 (2022).
https://doi.org/10.1103/PhysRevLett.128.220401

[25] A. Lerose, M. Sonner, and D. A. Abanin, arXiv:2201.04150 (2022).
arXiv:2201.04150

[26] A. Zabalo, M. Gullans, J. Wilson, R. Vasseur, A. Ludwig, S. Gopalakrishnan, D. A. Huse, and J. Pixley, Phys. Rev. Lett. 128, 050602 (2022).
https://doi.org/10.1103/PhysRevLett.128.050602

[27] E. Chertkov, J. Bohnet, D. Francois, J. Gaebler, D. Gresh, A. Hankin, K. Lee, R. Tobey, D. Hayes, B. Neyenhuis, R. Stutz, A. C. Potter, and M. Foss-Feig, arXiv:2105.09324 (2021).
arXiv:2105.09324

[28] X. Mi, P. Roushan, C. Quintana, S. Mandrà, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, J. Basso, A. Bengtsson, S. Boixo, A. Bourassa, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. R. Derk, A. Dunsworth, D. Eppens, C. Erickson, E. Farhi, A. G. Fowler, B. Foxen, C. Gidney, M. Giustina, J. A. Gross, M. P. Harrigan, S. D. Harrington, J. Hilton, A. Ho, S. Hong, T. Huang, W. J. Huggins, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, C. Jones, D. Kafri, J. Kelly, S. Kim, A. Kitaev, P. V. Klimov, A. N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, E. Lucero, O. Martin, J. R. McClean, T. McCourt, M. McEwen, A. Megrant, K. C. Miao, M. Mohseni, S. Montazeri, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley, M. Newman, M. Y. Niu, T. E. O’Brien, A. Opremcak, E. Ostby, B. Pato, A. Petukhov, N. Redd, N. C. Rubin, D. Sank, K. J. Satzinger, V. Shvarts, D. Strain, M. Szalay, M. D. Trevithick, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, I. Aleiner, K. Kechedzhi, V. Smelyanskiy, and Y. Chen, Science (2021), 10.1126/science.abg5029.
https://doi.org/10.1126/science.abg5029

[29] B. Bertini, P. Kos, and T. Prosen, Commun. Math. Phys. 387, 597 (2021).
https://doi.org/10.1007/s00220-021-04139-2

[30] P. Kos, B. Bertini, and T. Prosen, Phys. Rev. Lett. 126, 190601 (2021b).
https://doi.org/10.1103/PhysRevLett.126.190601

[31] F. Fritzsch and T. Prosen, Phys. Rev. E 103, 062133 (2021).
https://doi.org/10.1103/PhysRevE.103.062133

[32] J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernandez, J. Choi, A. L. Shaw, M. Endres, and S. Choi, arXiv:2103.03536 (2021).
arXiv:2103.03536

[33] J. Choi, A. L. Shaw, I. S. Madjarov, X. Xie, J. P. Covey, J. S. Cotler, D. K. Mark, H.-Y. Huang, A. Kale, H. Pichler, F. G. S. L. Brandão, S. Choi, and M. Endres, arXiv:2103.03535 (2021).
arXiv:2103.03535

[34] W. W. Ho and S. Choi, Phys. Rev. Lett. 128, 060601 (2022).
https://doi.org/10.1103/PhysRevLett.128.060601

[35] D. Gross, K. Audenaert, and J. Eisert, J. Math. Phys. 48, 052104 (2007).
https://doi.org/10.1063/1.2716992

[36] A. Ambainis and J. Emerson, in Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07) (2007) pp. 129–140, iSSN: 1093-0159.
https://doi.org/10.1109/CCC.2007.26

[37] D. A. Roberts and B. Yoshida, J. High Energ. Phys. 2017, 121 (2017).
https://doi.org/10.1007/JHEP04(2017)121

[38] H. Wilming and I. Roth, arXiv:2202.01669 (2022).
arXiv:2202.01669

[39] D. J. Reutter and J. Vicary, Higher Structures 3, 109 (2019).
https://doi.org/10.48550/arXiv.1609.07775

[40] A. Chandran and C. R. Laumann, Phys. Rev. B 92, 024301 (2015).
https://doi.org/10.1103/PhysRevB.92.024301

[41] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Phys. Rev. X 7, 031016 (2017).
https://doi.org/10.1103/PhysRevX.7.031016

[42] V. Khemani, A. Vishwanath, and D. A. Huse, Phys. Rev. X 8, 031057 (2018).
https://doi.org/10.1103/PhysRevX.8.031057

[43] C. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. Sondhi, Phys. Rev. X 8, 021013 (2018).
https://doi.org/10.1103/PhysRevX.8.021013

[44] A. Nahum, S. Vijay, and J. Haah, Phys. Rev. X 8, 021014 (2018).
https://doi.org/10.1103/PhysRevX.8.021014

[45] A. Chan, A. De Luca, and J. Chalker, Phys. Rev. X 8, 041019 (2018).
https://doi.org/10.1103/PhysRevX.8.041019

[46] T. Rakovszky, F. Pollmann, and C. von Keyserlingk, Phys. Rev. X 8, 031058 (2018).
https://doi.org/10.1103/PhysRevX.8.031058

[47] T. Rakovszky, F. Pollmann, and C. von Keyserlingk, Phys. Rev. Lett. 122, 250602 (2019).
https://doi.org/10.1103/PhysRevLett.122.250602

[48] T. Zhou and A. Nahum, Phys. Rev. X 10, 031066 (2020).
https://doi.org/10.1103/PhysRevX.10.031066

[49] S. Garratt and J. Chalker, Phys. Rev. X 11, 021051 (2021).
https://doi.org/10.1103/PhysRevX.11.021051

[50] J. Bensa and M. Žnidarič, Phys. Rev. X 11, 031019 (2021).
https://doi.org/10.1103/PhysRevX.11.031019

[51] R. Orús, Ann. Phys. 349, 117 (2014).
https://doi.org/10.1016/j.aop.2014.06.013

[52] B. Bertini, P. Kos, and T. Prosen, SciPost Phys. 8, 067 (2020a).
https://doi.org/10.21468/SciPostPhys.8.4.067

[53] D. Weingarten, J. Math. Phys. 19, 999 (1978).
https://doi.org/10.1063/1.523807

[54] B. Collins, Int. Math. Res. Not. 2003, 953 (2003).
https://doi.org/10.1155/S107379280320917X

[55] B. Collins and P. Śniady, Commun. Math. Phys. 264, 773 (2006).
https://doi.org/10.1007/s00220-006-1554-3

[56] B. Bertini, P. Kos, and T. Prosen, SciPost Phy. 8, 068 (2020b).
https://doi.org/10.21468/SciPostPhys.8.4.068

[57] Z. Webb, QIC 16, 1379 (2016).
https://doi.org/10.26421/QIC16.15-16-8

[58] E. Knill, Non-binary unitary error bases and quantum codes, Tech. Rep. LA-UR-96-2717 (Los Alamos National Lab. (LANL), Los Alamos, NM (United States), 1996).
https://doi.org/10.2172/373768

[59] P. Shor, in Proceedings of 37th Conference on Foundations of Computer Science (1996) pp. 56–65, iSSN: 0272-5428.
https://doi.org/10.1109/SFCS.1996.548464

[60] R. F. Werner, J. Phys. A: Math. Gen. 34, 7081 (2001).
https://doi.org/10.1088/0305-4470/34/35/332

[61] J. Hauschild and F. Pollmann, SciPost Phys. Lect. Notes , 005 (2018).
https://doi.org/10.21468/SciPostPhysLectNotes.5

[62] Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B 98, 205136 (2018).
https://doi.org/10.1103/PhysRevB.98.205136

[63] B. Skinner, J. Ruhman, and A. Nahum, Phys. Rev. X 9, 031009 (2019).
https://doi.org/10.1103/PhysRevX.9.031009

[64] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Phys. Rev. B 99, 224307 (2019).
https://doi.org/10.1103/PhysRevB.99.224307

[65] M. J. Gullans and D. A. Huse, Phys. Rev. X 10, 041020 (2020).
https://doi.org/10.1103/PhysRevX.10.041020

[66] M. Ippoliti and W. W. Ho, arXiv:2204.13657 (2022).
arXiv:2204.13657

Cited by

[1] 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).

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

[3] Pieter W. Claeys, Marius Henry, Jamie Vicary, and Austen Lamacraft, "Exact dynamics in dual-unitary quantum circuits with projective measurements", Physical Review Research 4 4, 043212 (2022).

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

[5] Grace M. Sommers, David A. Huse, and Michael J. Gullans, "Crystalline Quantum Circuits", PRX Quantum 4 3, 030313 (2023).

[6] 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).

[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] Max McGinley and Michele Fava, "Shadow Tomography from Emergent State Designs in Analog Quantum Simulators", Physical Review Letters 131 16, 160601 (2023).

[9] Pavel Kos and Georgios Styliaris, "Circuits of space and time quantum channels", Quantum 7, 1020 (2023).

[10] Berislav Buča, "Unified Theory of Local Quantum Many-Body Dynamics: Eigenoperator Thermalization Theorems", Physical Review X 13 3, 031013 (2023).

[11] Michael A. Rampp, Roderich Moessner, and Pieter W. Claeys, "From Dual Unitarity to Generic Quantum Operator Spreading", Physical Review Letters 130 13, 130402 (2023).

[12] Alessandro Foligno and Bruno Bertini, "Growth of entanglement of generic states under dual-unitary dynamics", Physical Review B 107 17, 174311 (2023).

[13] Matteo Ippoliti and Wen Wei Ho, "Solvable model of deep thermalization with distinct design times", Quantum 6, 886 (2022).

[14] Kai Klocke and Michael Buchhold, "Majorana Loop Models for Measurement-Only Quantum Circuits", Physical Review X 13 4, 041028 (2023).

[15] Márton Mestyán, Balázs Pozsgay, and Ian M. Wanless, "Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states", arXiv:2210.13017, (2022).

[16] Suhail Ahmad Rather, S. Aravinda, and Arul Lakshminarayan, "Construction and Local Equivalence of Dual-Unitary Operators: From Dynamical Maps to Quantum Combinatorial Designs", PRX Quantum 3 4, 040331 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2023-11-29 17:18:38) and SAO/NASA ADS (last updated successfully 2023-11-29 17:18:39). 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.