Unraveling the emergence of quantum state designs in systems with symmetry

Naga Dileep Varikuti; Soumik Bandyopadhyay

doi:10.22331/q-2024-08-29-1456

Unraveling the emergence of quantum state designs in systems with symmetry

Naga Dileep Varikuti^1,2 and Soumik Bandyopadhyay^3,4

¹Department of Physics, Indian Institute of Technology Madras, Chennai, India, 600036
²Center for Quantum Information, Communication and Computing (CQuICC), Indian Institute of Technology Madras, Chennai, India 600036
³Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento, Via Sommarive 14, Trento, I-38123, Italy
⁴INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Via Sommarive 14, Trento, I-38123, Italy

Published:	2024-08-29, volume 8, page 1456
Eprint:	arXiv:2402.08949v3
Doi:	https://doi.org/10.22331/q-2024-08-29-1456
Citation:	Quantum 8, 1456 (2024).

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Abstract

Quantum state designs, by enabling an efficient sampling of random quantum states, play a quintessential role in devising and benchmarking various quantum protocols with broad applications ranging from circuit designs to black hole physics. Symmetries, on the other hand, are expected to reduce the randomness of a state. Despite being ubiquitous, the effects of symmetry on quantum state designs remain an outstanding question. The recently introduced projected ensemble framework generates efficient approximate state $t$-designs by hinging on projective measurements and many-body quantum chaos. In this work, we examine the emergence of state designs from the random generator states exhibiting symmetries. Leveraging on translation symmetry, we analytically establish a sufficient condition for the measurement basis leading to the state $t$-designs. Then, by making use of the trace distance measure, we numerically investigate the convergence to the designs. Subsequently, we inspect the violation of the sufficient condition to identify bases that fail to converge. We further demonstrate the emergence of state designs in a physical system by studying the dynamics of a chaotic tilted field Ising chain with translation symmetry. We find faster convergence of the trace distance during the early time evolution in comparison to the cases when the symmetry is broken. To delineate the general applicability of our results, we extend our analysis to other symmetries. We expect our findings to pave the way for further exploration of deep thermalization and equilibration of closed and open quantum many-body systems.

Featured image: Schematic representation of the projected ensemble framework for a $Q$-symmetric state (an eigenstate of a symmetry operator $Q$), showcasing the interplay between measurement bases and symmetry. (a) Haar random state, wherein the spins are randomly aligned and entangled. An application of the subspace projector $\mathbf{Q}_k$ takes the Haar random state to an invariant subspace with charge $k$, which is depicted in (b). For simplicity, we are showing a $Z_2$-symmetric state (see Sec. 5.1) in this schematic. The labels $\sigma^x_j$ indicate that the state is invariant under spin-flip operations. The resultant state is subjected to projective measurements over the subsystem-$B$. These measurements result in the so-called projected ensembles supported over $A$ (see Sec. 2 ). (c)-(e) Distributions of the resultant projected ensembles for three different measurement bases. While in (c), the measurements in $\mathcal{B}_1$ yield a uniformly distributed projected ensemble, $\mathcal{B}_2$ and $\mathcal{B}_3$ result in ensembles of pure states localized near $|\pm\rangle$ states. We understand that the latter cases largely violate a sufficient condition we derive in this text for the emergence of state designs.

Popular summary

Symmetry plays a crucial role in quantum systems, acting as a guiding principle that unifies different physical laws and helps predict new phenomena. While symmetry enforces order within a system, quantum chaos tends to introduce randomness. The interplay between symmetry and chaos can give rise to novel phenomena.

Quantum state $t$-designs facilitate the efficient sampling of uniform random states, making them valuable for various quantum protocols such as randomized benchmarking, randomized measurements, quantum state tomography, and several quantum machine learning schemes. A recently proposed framework, the projected ensemble, leverages quantum chaos as a resource to construct approximate higher-order state designs through measurements, a process known as deep thermalization.

Despite the widespread occurrence of symmetries, their effects on the emergence of quantum state designs remain an open question. Our study uncovers the intricate relationship between symmetries and measurements in constructing these approximate state designs. We expect our findings to pave the way for further exploration of deep thermalization and the equilibration of closed and open quantum many-body systems.

► BibTeX data

@article{Varikuti2024unravelingemergence,
  doi = {10.22331/q-2024-08-29-1456},
  url = {https://doi.org/10.22331/q-2024-08-29-1456},
  title = {Unraveling the emergence of quantum state designs in systems with symmetry},
  author = {Varikuti, Naga Dileep and Bandyopadhyay, Soumik},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1456},
  month = aug,
  year = {2024}
}

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