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## "The group structure of quantum cellular automata", Michael Freedman, Jeongwan Haah, Matthew Hastings, 2022

*Reviewed January 1, 2023*

*Citation:* Freedman, Michael, Jeongwan Haah, and Matthew B. Hastings. "The group structure of quantum cellular automata." Communications in Mathematical Physics 389.3 (2022): 1277-1302.

*Web:* https://arxiv.org/abs/1910.07998

*Tags:* SPT/SETs, Computer-scientific

This paper gives a definition of quantum cellular automata (QCA) which
is amenable to group operations. This definition uses "coherent families".
Coherent families are infinite families of QCAs whose locality relative to
system size approaches 0, along with some compatibility conditions.
This definition is very nice, for several reasons. For one,
it allows one to naturally define a group structure on the QCAs.
An R1-local QCA composed with an R2-local QCA gives an (R1+R2)-local QCA.
Hence, repeatedly adding local QCAs can give non-local QCAs on finite systems.
Working with coherent families solves this issue.

The formalism introduced in this paper, along with its
demonstrated ability to prove theorems, gives an indication
to the path forwards in the world of gapped Hamiltonians.
There is no well-accepted notion of Hamiltonian schema/coherent family of gapped Hamiltonians.
This paper could indicate part of the answer.