Finite Depth Quantum Circuits and Symmetry Protected Topological Phases

Suppose you start with a bunch of uncorrelated qubits. This is called the “trivial product state”, state P. Now, you have a desired end state, state Q. There are 4 cases. You can either use a finite depth quantum circuit (FDQC) to change P to Q. Or, you can’t. You can also choose to use matrices (a quantum gate can represent a matrix) that commute with the symmetry group matrices of Q (called “enforcing symmetry”). Case 1: Using FDQC (constraint) and enforcing symmetry (constraint), you can stil