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

TL;DR

This paper explores how dual-unitary circuits exhibit emergent quantum state designs and biunitarity, revealing their exact solvability and connection to random matrix behavior in chaotic quantum dynamics.

Contribution

It introduces a new construction for projected ensembles in dual-unitary circuits, extending previous results and highlighting the role of dual-unitarity and biunitary connections.

Findings

Dual-unitary circuits exhibit quantum state designs.

Exact solvability is demonstrated in chaotic dual-unitary models.

Connections to complex Hadamard matrices and unitary error bases are established.

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 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 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.