IAS Special Physics Seminar

Abstract: Informally, a quantum cellular automaton (QCA) describes some discrete-time local dynamics of a quantum system.  Formally, a QCA is a *-automoprhism of the algebra of local operators.  Local quantum circuits provide one example of QCA, but the nontrivial QCA  are those that cannot be written as a local circuit.  I will discuss the classification of nontrivial QCA in various spatial dimensions, and relate it to the classification of invertible symmetry protected topological (SPT) phases of matter.  In one dimension, nontrivial QCA are classified by a rational “shift” index.  In two dimensions, all QCA are trivial up to shifts.  In dimensions three and higher, we give a general construction of a family of nontrivial QCA based on multiplication by a discrete Chern-Simons functional in an appropriate basis for the Hilbert space.  We conjecture a correspondence between unoriented cobordism (which classifies such SPTs) and this family of QCA.  This is joint work with L. Fidkowski and J. Haah.