Stabilizer state

”Stabilizer states are states that a Stabilizer circuit can generate, starting from ​|00...0⟩.”

”Here’s an amazing fact, which we won’t give a proof of: The ​n​-qubit Stabilizer states are exactly the ​n​-qubit states that have a stabilizer group of size 2ⁿ. So the 1-qubit stabilizer states are those states with a 2-element stabilizer group, the 2-qubit stabilizer states are those states with a 4-element stabilizer group, and so on. This is a completely different characterization of Stabilizer states, a structural one. It makes no mention of Stabilizer circuits, but tells us something about the invariant that Stabilizer circuits are preserving.”

Another fact the notes don’t prove: “Given any n-qubit Stabilizer state, its stabilizer group is always generated by only ​n​ elements.”

“How many bits does it take to store such a generating set in your computer?

Well, there are ​n​ generators, and each one takes 2​n​+1 bits to specify: 2 bits for each of the ​n​ Pauli matrices, plus 1 additional bit for the ± sign. So the total number of bits is n​(2​n​+1) = 2​n²​ + ​n​ = O(​n²​).

Naïvely writing out the entire amplitude vector, or the entire stabilizer group, would have taken ~2ⁿ​ bits, so we’ve gotten an exponential savings. We’re already starting to see the power of the Stabilizer formalism.”

Stabilizer states and Graph states;