The Pauli spectrum is defined in terms of a basis of $L$-qubit Pauli operators. Such a basis is formed by all Hermitian Pauli strings $P$ with phase $+1$.

This notion of magic is relevant to active quantum error correction and monitored quantum dynamics, where a key question related to the complexity of classically simulating the dynamics is which measurements reduce the amount of magic in a state [56].

Focusing on contiguous systems also naturally leads to interesting connections between magic length scales for projectively destroyable magic resource and the linear and contiguous code distances of QECC [95, 96].

Here, by generators, we mean both the logical and stabilizer generators of, for example, $\rho$ from Eq. (2).

We can relabel the logicals $\overline{X},\overline{Y}$, and $\overline{Z}$ as needed, provided they follow the $\mathfrak{su}(2)$ anticommutation rules; the prefactors $a_{\pm }$ depend on the non-Clifford gate used to produce $|\psi ⟩$, but for suitably chosen $\overline{Z}$ and $\overline{Y}$, they satisfy $|a_{\pm }|\ne 0$, ensuring that ${\tilde{\mathcal{M}}}_2({\rho }_A)>0$ and $\mathcal{R}({\rho }_A)>1$ for a subsystem $A$ where at least one logical can be reduced. The specific values of $|a_{\pm }|$ set the values of “full” [case ($i$)] and “partial” [cases ($iv),(v$)] NS. See also Supplemental Material [81].

For intermediate times, the data from Fig. 3 is consistent with a growth rate of FLEOM given by ${\partial }_{t}W\sim 4v_{\mathrm{E}}/3$. We attribute the distinct intermediate-time growth of $W(t)$ to a combination of effects due to boundary conditions and shrinking of large logicals, as detailed in [81].