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

Pieter W. Claeys1,2 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

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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.

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.

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► References

[1] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).

[2] H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).

[3] F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, Vol. 54 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010).

[4] M. Akila, D. Waltner, B. Gutkin, and T. Guhr, J. Phys. A: Math. Theor. 49, 375101 (2016).

[5] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. Lett. 121, 264101 (2018).

[6] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. X 9, 021033 (2019a).

[7] S. Gopalakrishnan and A. Lamacraft, Phys. Rev. B 100, 064309 (2019).

[8] B. Bertini, P. Kos, and T. Prosen, Phys. Rev. Lett. 123, 210601 (2019b).

[9] S. A. Rather, S. Aravinda, and A. Lakshminarayan, Phys. Rev. Lett. 125, 070501 (2020).

[10] B. Gutkin, P. Braun, M. Akila, D. Waltner, and T. Guhr, Phys. Rev. B 102, 174307 (2020).

[11] S. Aravinda, S. A. Rather, and A. Lakshminarayan, Phys. Rev. Research 3, 043034 (2021).

[12] P. W. Claeys and A. Lamacraft, Phys. Rev. Lett. 126, 100603 (2021).

[13] T. Prosen, Chaos 31, 093101 (2021).

[14] S. Singh and I. Nechita, arXiv:2112.11123 (2021).

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

[16] P. W. Claeys and A. Lamacraft, Phys. Rev. Research 2, 033032 (2020).

[17] B. Bertini and L. Piroli, Phys. Rev. B 102, 064305 (2020).

[18] R. Suzuki, K. Mitarai, and K. Fujii, Quantum 6, 631 (2022).

[19] L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, Phys. Rev. B 101, 094304 (2020).

[20] B. Jonnadula, P. Mandayam, K. Życzkowski, and A. Lakshminarayan, Phys. Rev. Research 2, 043126 (2020).

[21] I. Reid and B. Bertini, Phys. Rev. B 104, 014301 (2021).

[22] P. Kos, B. Bertini, and T. Prosen, Phys. Rev. X 11, 011022 (2021a).

[23] A. Lerose, M. Sonner, and D. A. Abanin, Phys. Rev. X 11, 021040 (2021).

[24] G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D. A. Abanin, and L. Piroli, Phys. Rev. Lett. 128, 220401 (2022).

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

[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).

[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).

[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.

[29] B. Bertini, P. Kos, and T. Prosen, Commun. Math. Phys. 387, 597 (2021).

[30] P. Kos, B. Bertini, and T. Prosen, Phys. Rev. Lett. 126, 190601 (2021b).

[31] F. Fritzsch and T. Prosen, Phys. Rev. E 103, 062133 (2021).

[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).

[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).

[34] W. W. Ho and S. Choi, Phys. Rev. Lett. 128, 060601 (2022).

[35] D. Gross, K. Audenaert, and J. Eisert, J. Math. Phys. 48, 052104 (2007).

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

[37] D. A. Roberts and B. Yoshida, J. High Energ. Phys. 2017, 121 (2017).

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

[39] D. J. Reutter and J. Vicary, Higher Structures 3, 109 (2019).

[40] A. Chandran and C. R. Laumann, Phys. Rev. B 92, 024301 (2015).

[41] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Phys. Rev. X 7, 031016 (2017).

[42] V. Khemani, A. Vishwanath, and D. A. Huse, Phys. Rev. X 8, 031057 (2018).

[43] C. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. Sondhi, Phys. Rev. X 8, 021013 (2018).

[44] A. Nahum, S. Vijay, and J. Haah, Phys. Rev. X 8, 021014 (2018).

[45] A. Chan, A. De Luca, and J. Chalker, Phys. Rev. X 8, 041019 (2018).

[46] T. Rakovszky, F. Pollmann, and C. von Keyserlingk, Phys. Rev. X 8, 031058 (2018).

[47] T. Rakovszky, F. Pollmann, and C. von Keyserlingk, Phys. Rev. Lett. 122, 250602 (2019).

[48] T. Zhou and A. Nahum, Phys. Rev. X 10, 031066 (2020).

[49] S. Garratt and J. Chalker, Phys. Rev. X 11, 021051 (2021).

[50] J. Bensa and M. Žnidarič, Phys. Rev. X 11, 031019 (2021).

[51] R. Orús, Ann. Phys. 349, 117 (2014).

[52] B. Bertini, P. Kos, and T. Prosen, SciPost Phys. 8, 067 (2020a).

[53] D. Weingarten, J. Math. Phys. 19, 999 (1978).

[54] B. Collins, Int. Math. Res. Not. 2003, 953 (2003).

[55] B. Collins and P. Śniady, Commun. Math. Phys. 264, 773 (2006).

[56] B. Bertini, P. Kos, and T. Prosen, SciPost Phy. 8, 068 (2020b).

[57] Z. Webb, QIC 16, 1379 (2016).

[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).

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

[60] R. F. Werner, J. Phys. A: Math. Gen. 34, 7081 (2001).

[61] J. Hauschild and F. Pollmann, SciPost Phys. Lect. Notes , 005 (2018).

[62] Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B 98, 205136 (2018).

[63] B. Skinner, J. Ruhman, and A. Nahum, Phys. Rev. X 9, 031009 (2019).

[64] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Phys. Rev. B 99, 224307 (2019).

[65] M. J. Gullans and D. A. Huse, Phys. Rev. X 10, 041020 (2020).

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

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

[11] 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", Physical Review Letters 132 25, 250601 (2024).

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

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

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

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

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

[17] Max McGinley, "Postselection-Free Learning of Measurement-Induced Quantum Dynamics", PRX Quantum 5 2, 020347 (2024).

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

[19] Márton Mestyán, Balázs Pozsgay, and Ian M. Wanless, "Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states", SciPost Physics 16 1, 010 (2024).

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

[21] Xie-Hang Yu, Zhiyuan Wang, and Pavel Kos, "Hierarchical generalization of dual unitarity", Quantum 8, 1260 (2024).

[22] 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).

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

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