On complexity of unitaries on many-qubit systems

If a Hilbert space is built from qubits, we may define complexity of a global unitary U by the minimal number of one- and two-qubit unitaries that comprise the global U. I will discuss three questions that can be posed using this complexity. (i) What quantum error correcting codes are there on D-dimensional arrays of qubits modulo an equivalence relation given by low complexity unitaries? (ii) What locality-preserving automorphisms of local operator algebras are there modulo those that are implemented by low complexity unitaries? (iii) What is the complexity of typical unitary consisting of random 2-qubit unitaries as a function of the number of constituents?