Stabilizer formalism
A quantum state is “stabilized” by an operator if it’s a +1 eigenstate of the operator.
Examples:
-
+Z stabilizes
|0⟩; -
+Z does not stabilize
|1⟩; -
+X stabilizes
|0⟩ + |1⟩; -
−XX stabilizes
|00⟩ - |11⟩.
“The stabilizer group S is some Abelian subgroup of 𝒢. The code space T is the space of vectors stabilized by S: ∀ (M ∈ S) (|ψᵢ⟩ ∈ T), M |ψᵢ⟩ = |ψᵢ⟩.”
The stabilizer group of a tensor product of two states is the Cartesian product of their individual stabilizer groups.
For a code to encode k qubits in n, T has 2ᵏ dimensions and S has 2ⁿ⁻ᵏ elements.”
Consider an error E that anticommutes with a stabilizer M, i.e. ME = -EM. Then
⟨ψᵢ|E|ψⱼ⟩ = ⟨ψᵢ|ME|ψⱼ⟩ = - ⟨ψᵢ|E|ψⱼ⟩ = 0.
Meaning: The error E takes the basis codeword to a state orthogonal to any state stabilized by M.
Find the state stabilized by the given stabilizers;