Noise redistribution by stabilizer codes: an organizing view of concatenated three-qubit codes

Abstract

We present an organizing framework for concatenated quantum error correction based on the observation that the syndrome projection of a stabilizer code reshapes the noise channel in a code-determined manner. For any stabilizer code, one level of QEC maps the physical noise to a block-diagonal logical channel whose diagonal Pauli weights are fixed by the code geometry. Coherences between different syndrome classes vanish, and no physical Pauli twirl is required to reach this block-diagonal form. Concatenation can then be viewed as the iteration of the same redistribution step: the dominant logical error after one level selects the code used at the next. We instantiate the framework with the two smallest stabilizer codes, the bit-flip and phase-flip codes, and apply it to amplitude damping. The redistribution rules give the concatenation protocol $C_2(\hatσ_y) C_1(\hatσ_x)$, whose ideal-recovery fidelity scales as $f_l=1-\mathcal O(γ^2)$. Solving the crossover condition $F_l>F_{l-1}$ in the ideal-gate limit yields a critical damping parameter $γ_c\approx0.30$; in terms of the initial channel fidelity, $f=1-1.72755γ^2$. Under the same ideal-recovery assumption, we compare the overall gate accuracy rate with that of the five-qubit code. The scope of the framework, in particular for coherent and correlated noise, is stated explicitly.