Eigenstate Correlations in Dual-Unitary Quantum Circuits: Partial Spectral Form Factor

Felix Fritzsch; Maximilian F. I. Kieler; Arnd Bäcker

doi:10.22331/q-2025-04-17-1709

Eigenstate Correlations in Dual-Unitary Quantum Circuits: Partial Spectral Form Factor

Felix Fritzsch¹, Maximilian F. I. Kieler², and Arnd Bäcker²

¹Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
²Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany

Published:	2025-04-17, volume 9, page 1709
Editor:	Álvaro Alhambra
Eprint:	arXiv:2407.19929v4
Doi:	https://doi.org/10.22331/q-2025-04-17-1709
Citation:	Quantum 9, 1709 (2025).

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Abstract

While the notion of quantum chaos is tied to random matrix spectral correlations, also eigenstate properties in chaotic systems are often assumed to be described by random matrix theory. Analytic insights into eigenstate correlations can be obtained by the recently introduced partial spectral form factor. Here, we study the partial spectral form factor in chaotic dual-unitary quantum circuits in the thermodynamic limit. We compute the latter for a finite subsystem in a brickwork circuit coupled to an infinite complement. For initial times, shorter than the subsystem's size, spatial locality and (dual) unitarity implies a constant partial spectral form factor, clearly deviating from the linear ramp of the random matrix prediction. In contrast, for larger times we prove, that the partial spectral form factor follows the random matrix result up to exponentially suppressed corrections. We supplement our exact analytical results by semi-analytic computations performed in the thermodynamic limit as well as with numerics for finite-size systems.

Featured image: Partial spectral form factor $K_A(t)$ for a subsystem $A$ of dimension $D_A$ in a dual-unitary quantum circuit. The red line represents analytical result for dual unitary circuits in the thermodynamic limit. They exhibit an initial plateau caused by spatially local interactions, a subsequent transient regime, and a linear ramp $\sim t$ at late times in accordance with the corresponding random matrix result (black line). The inset depicts a tensor network representation of the partial spectral form factor for times in the transient regime.

Popular summary

In many-body quantum systems deterministic interactions between the particles often lead to dynamics which appear completely random. Thus at macroscopic scales a statistical description is very efficient. However, an important conceptual challenge is to explain this randomness in terms of the highly structured, local interactions at the microscopic level. Such a situation can be realized by local quantum circuits, which therefore provide simple models to understand the emergence of macroscopic randomness from local interactions.
This paper establishes this connection for dual-unitary quantum circuits, which obey an additional symmetry between space and time. As an observable the partial spectral form factor is used, which simultaneously captures the correlations among energy levels and the associated stationary eigenstates, revealing aspects of quantum entanglement. It is shown that at short times, spatial locality represents an obstruction to randomness, whereas at late times fully random behavior emerges. Therefore this result explains the macroscopic statistical description of many body quantum dynamics at late times. Moreover, it also provides a potential benchmark for experimental tests of quantum chaos and complexity.

► BibTeX data

@article{Fritzsch2025eigenstate,
  doi = {10.22331/q-2025-04-17-1709},
  url = {https://doi.org/10.22331/q-2025-04-17-1709},
  title = {Eigenstate {C}orrelations in {D}ual-{U}nitary {Q}uantum {C}ircuits: {P}artial {S}pectral {F}orm {F}actor},
  author = {Fritzsch, Felix and Kieler, Maximilian F. I. and B{\"{a}}cker, Arnd},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1709},
  month = apr,
  year = {2025}
}

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