Transitions in Entanglement Complexity in Random Circuits

Sarah True; Alioscia Hamma

doi:10.22331/q-2022-09-22-818

Transitions in Entanglement Complexity in Random Circuits

Sarah True¹ and Alioscia Hamma^1,2,3

¹Physics Department, University of Massachusetts Boston, 02125, USA
²Dipartimento di Fisica `Ettore Pancini', Università degli Studi di Napoli Federico II, Via Cintia 80126, Napoli, Italy
³INFN, Sezione di Napoli, Italy

Published:	2022-09-22, volume 6, page 818
Eprint:	arXiv:2202.02648v4
Doi:	https://doi.org/10.22331/q-2022-09-22-818
Citation:	Quantum 6, 818 (2022).

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Abstract

Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple pattern of entanglement to a universal, complex pattern can be driven by doping a random Clifford circuit with $T$ gates. This work shows that quantum complexity and complex entanglement stem from the conjunction of entanglement and non-stabilizer resources, also known as magic.

Featured image: Color map representation of a 16-qubit state after $10N^2$ random Clifford gates, then $n_T$ random $T$ gates $($Top row: $n_T{=}5$. Bottom row: $n_T{=}20)$, then another $10N^2$ random Clifford gates are applied, starting with the initial state $\left|\psi_0\right>{=}\left|0\right>^{\otimes N}$. The image is arranged by partitioning the state into two equal subsystems and expanding one along each axis. A pixel at some position $(x, y)$ is mapped from the magnitude of the amplitude for the computational basis state $\left|\sigma_{xy}\right>{=}\left|\sigma_x\right>{\otimes}\left|\sigma_y\right>$ $($for example, the pixel at $(0, 0)$ corresponds to the state $\left|\sigma_{00}\right>{=}\left|0\right>^{\otimes N/2}{\otimes}\left|0\right>^{\otimes N/2})$. The action of a $T$ gate, being a phase gate, does not change the color map representation at all. However, after being spread around by the second Clifford circuit, the color map representation becomes more mixed for the circuit with the larger number of inserted $T$ gates, showing the interplay between the non-Clifford resources and the operator spreading through Clifford-driven entanglement in the onset of quantum chaos.

► BibTeX data

@article{True2022transitionsin,
  doi = {10.22331/q-2022-09-22-818},
  url = {https://doi.org/10.22331/q-2022-09-22-818},
  title = {Transitions in {E}ntanglement {C}omplexity in {R}andom {C}ircuits},
  author = {True, Sarah and Hamma, Alioscia},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {818},
  month = sep,
  year = {2022}
}

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[7] J. Odavić, G. Torre, N. Mijić, D. Davidović, F. Franchini, and S. M. Giampaolo, "Random unitaries, Robustness, and Complexity of Entanglement", arXiv:2210.13495, (2022).

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