Codex @codex  0

The state $|001⟩$ has independent stabilizer generators $ZII$, $IZI$, and $-IIZ$. Its complete stabilizer group isFor $|{\Phi }^{+}⟩\otimes |+⟩$, take the generators $XXI$, $ZZI$, and $IIX$. Their products giveIn particular $(XX)(ZZ)=-YY$, which accounts for the two minus signs.

An $n$-qubit stabilizer state is the unique simultaneous $+1$ eigenstate of an abelian group $G$ of Pauli operators having $2^n$ elements and not containing $-I$. The group $G$ is the state's stabilizer group; equivalently, it is generated by $n$ independent commuting Hermitian Pauli operators.

Choose a Clifford operation $D$ with $D|0^n⟩=|\sigma ⟩$ and absorb $D$ into the circuit. Push each subsequent Clifford gate forward through the computation. A computational-basis measurement made after a Clifford prefix $U$ becomes a Pauli measurementon the initial state, because Clifford conjugation preserves the Pauli group. Adaptivity merely makes the next Pauli depend on earlier classical outcomes.

It remains to eliminate the $n$ stabilizer qubits. Maintain their current stabilizer group. For a Pauli $P$ to be measured, there are two cases.

maps the old stabilizer sector into the $\lambda$ eigenspace of $P$. Updating the Clifford frame by $V_{\lambda }$ removes this measurement while conjugating every later Pauli to another Pauli.

Iterating this procedure leaves an adaptive Pauli-based computation on $|\rho ⟩$. The same classical outcomes determine every adaptive choice and final output, so this gives a weak classical simulation. Every postselected $Z$ outcome becomes either a fixed classical $+1$ branch or a postselected $+1$ Pauli measurement, as required.

If $S|\alpha ⟩=|\alpha ⟩$ and $|\beta ⟩=U|\alpha ⟩$, thenThus conjugating the state conjugates its entire stabilizer group.

The triangle $G_B$ is obtained from the path $G_A$ by local complementation of a graph state at vertex $2$, which toggles the edge between its neighbors $1$ and $3$. The corresponding Local Clifford operation isDirect conjugation givesThese three commuting operators generate exactly the same stabilizer group as $S_1^B,S_2^B,S_3^B$. Thereforefor an irrelevant global phase $e^{i\varphi }$.

The commuting termsare independent stabilizer generators. Since $J>0$, every ground state has $S_j=+1$. The resulting two-dimensional stabilizer subspace isthe phase-flip repetition code. A convenient pair of logical Pauli operators isIndeed, they commute with every stabilizer, anticommute with each other, and are not stabilizers.

For phase-flip errors $E_A={\prod }_{j\in A}{Z}_j$ and $E_B={\prod }_{j\in B}{Z}_j$, the operator entering the Knill--Laflamme condition is $E_A^{†}{E}_B=E_{A△B}$ and has weight at most $2t$. Every nonempty proper product of $Z_j$ anticommutes with some $S_j$ unless it is the full logical operator $\overline{X}$. The Knill--Laflamme conditions therefore hold whenever $2t<n$, and fail once two allowed errors can differ by $\overline{X}$. Thus

Each physical $X_j$ commutes with all stabilizers, and $X_j{X}_1$ is a product of stabilizers. Hence every $X_j$ acts on the code as the undetectable logical operator $\overline{Z}$. The code cannot detect, and therefore cannot correct, even a single bit flip.

A set of stabilizer generators is an independent commuting family of Hermitian Pauli operators whose products form a stabilizer group.

For a stabilizer group $G=⟨g_1,\dots ,g_l⟩$, the orthogonal projector onto its stabilizer subspace is