# Topological aspects of quantum cellular automata in one dimension

### APA

Gong, Z. (2021). Topological aspects of quantum cellular automata in one dimension. Perimeter Institute. https://pirsa.org/21110002

### MLA

Gong, Zongping. Topological aspects of quantum cellular automata in one dimension. Perimeter Institute, Nov. 03, 2021, https://pirsa.org/21110002

### BibTex

@misc{ pirsa_21110002, doi = {}, url = {https://pirsa.org/21110002}, author = {Gong, Zongping}, keywords = {Quantum Information}, language = {en}, title = {Topological aspects of quantum cellular automata in one dimension}, publisher = {Perimeter Institute}, year = {2021}, month = {nov}, note = {PIRSA:21110002 see, \url{https://pirsa.org}} }

## Abstract

Quantum cellular automata (QCA) are unitary transformations that preserve locality. In one dimension, QCA are known to be fully characterized by a topological chiral index that takes on arbitrary rational numbers [1]. QCA with nonzero indices are anomalous, in the sense that they are not finite-depth quantum circuits of local unitaries, yet they can appear as the edge dynamics of two-dimensional chiral Floquet topological phases [2].

In this seminar, I will focus on the topological aspects of one-dimensional QCA. First, I will talk about how the topological classification of QCA will be enriched by finite unitary symmetries [3]. On top of the cohomology character that applies equally to topological states, I will introduce a new class of topological numbers termed symmetry-protected indices. The latter, which include the chiral index as a special case, are genuinely dynamical topological invariants without state counterparts [4].

In the second part, I will show that the chiral index lower bounds the operator entanglement of QCA [5]. This rigorous bound enforces a linear growth of operator entanglement in the Floquet dynamics governed by nontrivial QCA, ruling out the possibility of many-body localization. In fact, this result gives a rigorous proof to a conjecture in Ref. [2]. Finally, I will present a generalized entanglement membrane theory that captures the large-scale (hydrodynamic) behaviors of typical (chaotic) QCA [6].

References:

[1] D. Gross, V. Nesme, H. Vogts, and R. F. Werner, Commun. Math. Phys. 310, 419 (2012).

[2] H. C. Po, L. Fidkowski, T. Morimoto, A. C. Potter, and A. Vishwanath, Phys. Rev. X 6, 041070 (2016).

[3] Z. Gong, C. Sünderhauf, N. Schuch, and J. I. Cirac, Phys. Rev. Lett. 124, 100402 (2020).

[4] Z. Gong and T. Guaita, arXiv:2106.05044.

[5] Z. Gong, L. Piroli, and J. I. Cirac, Phys. Rev. Lett. 126, 160601 (2021).

[6] Z. Gong, A. Nahum, and L. Piroli, arXiv:2109.07408.