Figure 1

Embedding a single QMBS (A) and an exponential number of QMBS (B) in a DU circuit. The top row (a),(d) shows the bipartite entanglement of eigenstates $|{\phi }_{\alpha }⟩$ of the Floquet unitary $\widehat{\mathbb{U}}$ in each case. The second row (b),(e) shows the growth of bipartite entanglement, starting from different separable states. The gray line is the average entanglement growth for 100 random product states of the form $|\psi (0)⟩=|i_0,i_1,\dots ,i_{N-1}⟩$, $i_n\in \{0,1,\dots ,d-1\}$, while the surrounding gray region shows the 5–95 percentile range. The dashed black line in (a), (b), (d), and (e) is the Page entropy, $S_{\text{Page}}=[N\ln (d)-1]/2$. The dotted line in (b) and (d) shows the maximum possible rate of entanglement growth, saturating at the maximum value $S_{\max }=N\ln (d)/2$. “Solvable” initial states have this maximal entanglement growth in the thermodynamic limit [23, 25, 43]. The bottom row (c),(f) shows the overlap with the initial state $F(t)=|⟨\psi (0)|\psi (t)⟩|$. The blue line in (b), (c), (e), and (f) shows the results for $|{\mathrm{\Psi }}_1⟩=|0{⟩}^{N-1}\otimes [(|0⟩+|d-1⟩)/\sqrt{2}]$, and the green line for $|{\mathrm{\Psi }}_2⟩=|0,0,d-1,d-1{⟩}^{\otimes N/4-1}\otimes |0,0,d-1⟩\otimes (|d-1⟩+|1⟩)/\sqrt{2}$. [Parameters: $N=8$ and $d=3$.]