From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy

Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are “as random as a coin toss.” Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power $e_p\left(U\right)$ of the basic two-particle unitary building block, $U$, of the circuit that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally, we show, both analytically and numerically, how local averaging over random realizations of the single-particle unitaries $u_i$ and $v_i$ such that the building block is $U^{\prime }=(u_1\otimes u_2)U(v_1\otimes v_2)$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power $e_p\left(U\right)$. Finally, we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map, which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.

Figure 1

As the entangling power of the two-particle unitary $U$ is increased, by multiples of $1/(q^2-1)$, beyond $e_{p,k}^{*}$, this ensures that $q^2-k$ modes of the dual-unitary circuit are mixing. At least one mode is mixing if $e_p\left(U\right)>e_{p,q^2-1}^{*}=0$ and all the $q^2-1$ nontrivial modes are mixing if $e_p\left(U\right)>e_{p,1}=e_p^{*}$.

Figure 2

The real and imaginary parts of the nontrivial largest and smallest magnitude eigenvalues ${\lambda }_1$ and ${\lambda }_{q^2-1}$ of $M_{+}\left[U^{\prime }\right]$ for four dual-unitary $U$, the first three being of the form $D_{1}S$, where $D_1$ is a diagonal unitary and $S$ the swap and $q=3$. The $U^{\prime }$ are generated in all cases with ${10}^4$ local, random single-particle unitaries, while a solid circle of radius $\sqrt{1-e_p\left(U\right)}$ is shown in each figure.

Figure 3

The single-particle averaged spectral radius ${\mathbb{E}}_{u_i,v_i}\left(\right|{\lambda }_1\left|\right)$ vs the entangling power. The dual-unitaries are 200 realizations each of the $D_{1}S$ and dual-CUE ensembles, for all the local dimensions $q$. The number of single-particle unitaries used in the averaging is ${10}^4$. The analytical approximations shown are discussed in the text. The inset shows how well the analytical expression fits in the neighborhood of $e_p\left(U\right)=1$.

Figure 4

The maximal mixing rate ${\nu }_{+}$ and the average mixing rate ${\mu }_{+}$ for dual-unitaries with entangling power $e_p\left(U\right)$ close to $e_p\left(U\right)=0$ for $q=2,3$, and 4. ${10}^4$ local unitary realizations are used in the averages.

Figure 5

The real and imaginary parts of the nontrivial largest eigenvalue ${\lambda }_1$ of $M_{+}\left[U^{\prime }\right]$ for two dual-unitary $U_i$ that have the same entangling power, $e_p\left(U_i\right)=3/4$. The circle shown is of unit radius, while for $U_1$ the complex eigenvalues are bounded by a hypocycloid called a deltoid.

Figure 6

The maximal mixing rate ${\nu }_{+}\left[U\right]$ for $q=2$ as a function of the entangling power. Shown are the results of maximization over the restricted subset of local unitaries represented by $w$ (${\nu }_{+}^{\prime }\left[U\right]$ with its analytical evaluation in Eq. (71)) as well as over the unrestricted set comprising $u\in \mathrm{SU}\left(2\right)$. The analytical results for the restricted set $v$ in Eq. (74) is seen to fit exactly the data from the optimization over the unrestricted set. The inset shows the corresponding average mixing rates ${\mu }_{+}^{\prime }\left[U\right]$, for which we compare with the analytical expression in Eq. (72).

Figure 7

The distribution of entangling power $e_p\left(U\right)$ of the dual-unitary $S{D}_1$ ensemble, consisting of the swap gate multiplied by diagonal unitaries, for $q=2$ and $q=3$, the former is an arcsin law given in Eq. (78).

Figure 8

Distribution of $e_p\left(U\right)$ of dual-CUE ensemble generated by applying the $M_R$ map on the CUE of ensemble. It can be seen that for $q=3$, the $M_R$ map also produces a fraction of 2-unitaries whose $e_p\left(U\right)=1$. If $e_p\left(U\right)>e_p^{*}$, the dual-unitary circuit is guaranteed to be mixing. For comparison, the average entangling power of CUE ${\overline{e}}_p$ is also indicated.

Figure 9

The maximal mixing rate ${\nu }_{+}$ and the average mixing rate ${\mu }_{+}$ for $q=3$ and $K=1,3$ cases of dual-unitary operators of the form $D_{3}S$, and $D_{1}S$. In the former case the three blocks are random unitaries drawn from the CUE, while in the latter the diagonal entries are random phases drawn uniformly from the unit circle. The maximum possible entangling power $3/4$ is indicated by dashed vertical line. The inset shows the case of dual-CUE, again with $q=3$, which can explore a larger range of entangling power.

Figure 10

The largest and second (inset) largest eigenvalues $|{\lambda }_{1,2}|$ of the channel $M_{+}$ for all possible dual-unitary permutations for qutrits, $q=3$. The figure also illustrates the bounds of the entangling power, beyond which the corresponding modes are definitely mixing.

Figure 11

The real and imaginary parts of the nontrivial largest eigenvalue ${\lambda }_1$ of $M_{+}\left[U_i\right]$ for two dual-unitary $U_i$ with $e_p\left(U_i\right)=4/5$. A finite fraction of spectrum of $M_{+}\left[U_i\right]$ is deviated from the 4-hypocycloid. The matrix $U_2$ converges from iterating the map $\mathcal{M}$ for a large number times $n$ to a particular permutation matrix $P$.

Figure 12

The ordered eigenvalues $|{\lambda }_i|$ of the qubit channel $M_{+}\left[U^{\prime }\left(J\right)\right]$ as a function of $\theta$ for $J=\pi /16$, when the local single particle gates are of the restricted form $w$.

Figure 13

The ordered nontrivial eigenvalues $|{\lambda }_i|,i=1,2,3$ of $M_{+}\left[U^{\prime }\left(J\right)\right]$ as a function of $\varphi$ for $J=\pi /16$, when the locals are drawn from the restriction of local unitaries to the form $v$ in Eq. (73). The smallest value of $|{\lambda }_1|$ is ${(sin2J)}^{2/3}$ at $\varphi =\pi /2$ for all values of $J$.

Figure 14

Absolute value of the largest nontrivial eigenvalue; ${\lambda }_1$, of $M\left[U^{\prime }\right]=(u\otimes u^{*})M\left[J\right],u\in \mathrm{SU}\left(2\right)$, is plotted as a function of $\theta ,\varphi$ parametrizing single qubit gate $u$. For given value of $J$, $|{\lambda }_1|$ is bounded below by ${sin}^{\frac{2}{3}}\left(2J\right)$.

Figure 15

Function $f(J,\theta ,\varphi )$ is evaluated on the $(\theta ,\varphi )$ plane for four different values of $J$. Pairs of $(\theta ,\varphi )$ angles for which $f(J,\theta ,\varphi )$ diverges are the values at which the minimum of $|{\lambda }_1(J,\theta ,\varphi )|$ is reached.

Figure 16

Minimum value of $|{\lambda }_1|$ (obtained numerically by searching over $\theta ,\varphi \in [0,2\pi ]$ parametrizing single qubit gate) is plotted as a function of $J$. The minimum obtained perfectly fits the predicted theoretical value obtained using locals from a restricted set.