Figure 1

(a) Diagrammatic representation of $|{\mathrm{\Psi }}_V⟩$, where the upward direction represents the progression of time. The steps of the decoding algorithm generating $|\mathrm{\Psi }{⟩}_V$ are (1) parallel application of $U_t$ (obtaining so $|{\mathrm{\Psi }}_t⟩$, diagrammatically shown in the purple box) and of the decrypter $V^{\prime }$ on the initial state $|RA⟩|B{B}^{\prime }⟩|A^{\prime }{R}^{\prime }⟩$; (2) application of ${\mathcal{D}}^{†}$ (obtaining $\mathcal{D}|\mathrm{\Psi }{⟩}_t$, as diagrammatically shown in the violet box) and ${\mathcal{D}}^T$. This process dumps the stabilizer entropy onto the $F$, $F^{\prime }$ subspaces. (3) The final steps involve the application of ${\mathcal{R}}^{*}$ and of a projective measurement on $D{D}^{\prime }$. Panel (b) provides an example of a doped random Clifford circuit $U_t$, whereas panel (c) displays the circuit ${\mathcal{D}}^{†}{U}_t$, where $\mathcal{D}$ is a diagonalizer for the circuit $U_t$ given in panel (b). Panel (d) illustrates the adjoint action of both circuits on the generators of the group ${\mathbb{P}}_E$. The circuit $U_t$ shown in panel (b) only preserves the generator $IIZII$, transforming it into another Pauli operator, whereas the adjoint action of ${\mathcal{D}}^{†}{U}_t$ preserves all generators, showing how the diagonalizer can move non-Cliffordness away from the subsystem of interest.