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 still change P to Q. We call Q “trivial”.

Case 2: Using FDQC and enforcing symmetry CANNOT change P to Q. BUT, when you stop using FDQC (you start using any depth), you CAN change to P to Q. We say that Q is a “Symmetry Protected Topological Phase”. P needs to undergo a phase transition to get to Q.

Case 3: Using FDQC but do not enforce symmetry, you can change P to Q. We again call Q “trivial”. The “topology wasn’t intrinsic”. It was held together by the symmetry constraint.

Case 4: Only by NOT using FDQC and NOT using symmetry, you can change P to Q. Q is “intrinsically topological.” “Real” topology.