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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Cited by

[1] Matteo Ippoliti and Wen Wei Ho, "Dynamical purification and the emergence of quantum state designs from the projected ensemble", arXiv:2204.13657.

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

[3] Pavel Kos and Georgios Styliaris, "Circuits of space-time quantum channels", arXiv:2206.12155.

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