  Classifying phases of local quantum systems is a general problem that
includes special cases such as free fermions, commuting projectors, and others.
An important distinction in this classification should be made between
classifying periodic and aperiodic systems. A related distinction is that
between homotopy invariants (invariants which remain constant so long as
certain general properties such as locality, gap, and others hold) and locally
computable invariants (properties of the system that cannot change from one
region to another without producing a gapless edge between them). We attack
this problem using a technique inspired by Kirby's "torus trick" in topology.
We use this trick to reproduce results for free fermions (in particular, using
the trick to reduce the aperiodic classification to the simpler problem of
periodic classification). We also show that a similar trick works for
interacting phases which are nontrivial but lack anyons; these results include
symmetry protected phases. A key part of this work is an attempt to classify
quantum cellular automata (QCA).
