Random Permutation Circuits Beyond Qubits are Quantum Chaotic

Abstract

Random permutation circuits were recently introduced as minimal models for local many-body dynamics that can be interpreted both as classical and quantum. Standard dynamical complexity indicators such as damage spreading and out-of-time-order correlators (OTOCs), show that these systems exhibit sensitivity to initial conditions in the classical setting and operator scrambling in the quantum setting. Here, we address their quantum chaoticity - a stricter property - by studying the time evolution of local operator entanglement (LOE). We show that the behaviour of LOE in random permutation circuits depends on the dimension of the local configuration space q. When q = 2, i.e. the circuits act on qubits, random permutations are Clifford and the LOE of any local operator is bounded by a constant, indicating that they are not truly chaotic. On the other hand, when the dimension of the local configuration space exceeds two, the LOE grows linearly in time. We prove this in the limit of large q and present numerical evidence that a three-dimensional local configuration space is sufficient for a linear growth of LOE. Our findings highlight that quantum chaos can be produced by essentially classical dynamics. Moreover, we show that LOE can be defined also in the classical realm and put it forward as a universal indicator chaos, both quantum and classical.

Figures (2)

• Figure 1: Schematic illustration of the dominant contribution to local operator purity in cases with $\ell>0$ and $\ell<0$. The large rectangle represents (up to normalisation) the partition sum from Eq. \ref{['eq:puritydiagram']} after taking into account the unitarity and locality of the interaction, and the symbols outside the border represent the boundary conditions given by the partition states. Interpreting purity as a partition sum of a two-dimensional statistical spin model, the leading contribution comes from the bulk configuration that minimises the lengths of domain walls (red lines) between different partition states jonay2018coarsezhou2020entanglement. Note that for $\ell=0$, both the configurations above give the same contribution, which explains the prefactor two in Eq. \ref{['eq:puritymain']}.
• Figure 2: Averaged Rényi-$2$ local operator entanglement $\overline{S}_{\mathcal{O},2}(t)$ for $q=3$ for an off-diagonal operator $\tau$ (blue), a diagonal operator $\mathcal{D}$ (green), and $\tau$ under the dynamics with phases (yellow), and their linear fits. Note that for the off-diagonal operator $\overline{S}_{\mathcal{O},2}(t)$ grows approximately twice as fast as for the diagonal operator, ($v_{\mathrm{OE},\tau}=0.50$ vs $v_{\mathrm{OE},\mathcal{D}}=0.25$), and dynamics with phases is on top of the data without phases. The inset shows the numerically estimated velocities for various $q$. Note that for $q \gg 1$ and the diagonal operator the velocity approaches $1/2$, agreeing with the analytical results (dashed line). We also observe that at $q=2$, only dynamics with phases shows non-zero velocity, as it is not Clifford. The off-diagonal operator $\tau$ is the one-site cyclic shift of basis states, $\tau\ket{s}=\ket{s+1\pmod q}$, while the diagonal operator $\mathcal{D}$ is taken to be $\mathrm{diag}(+1,-1,+1,\ldots)$, and for odd $q$ we need to further normalise it as $\tr[\mathcal{D}]=0$ and $\tr[\mathcal{D}^2]=q$.