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).

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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}
}

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

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