Probes of chaos over the Clifford group and approach to Haar values

Abstract

Chaotic behavior of quantum systems can be characterized by the adherence of the expectation values of given probes to moments of the Haar distribution. In this work, we analyze the behavior of several probes of chaos using a technique known as Isospectral Twirling [1]. This consists in fixing the spectrum of the Hamiltonian and picking its eigenvectors at random. Here, we study the transition from stabilizer bases to random bases according to the Haar measure by T-doped random quantum circuits. We then compute the average value of the probes over ensembles of random spectra from Random Matrix Theory, the Gaussian Diagonal Ensemble and the Gaussian Unitary Ensemble, associated with non-chaotic and chaotic behavior respectively. We also study the behavior of such probes over the Toric Code Hamiltonian.

Figures (20)

• Figure 1: Comparison of $g_2(t)$, $\tilde{g}_3(t)$ and $g_4(t)$ averaged over the GDE (panel \ref{['fig:compare_GDE']}) and the GUE (panel \ref{['fig:compare_GUE']}) for $d=2^{16}$. One can observe in both cases how the Clifford spectral form factor $\tilde{g}_3(t)$ shares the same equilibrium value of $g_2(t)$. Moreover, one can observe the suppression of oscillations in the case of GUE.
• Figure 2: Illustration of Clifford \ref{['fig:clifford_circuit']} and T-doped \ref{['fig:t_doped_circuit']} circuits.
• Figure 3: Plot of the normalized versions of $\overline{g_2(t)}^{\rm GDE}$ ( \ref{['fig:g2_GDE']}) and $\overline{g_2(t)}^{\rm GUE}$ ( \ref{['fig:g2_GUE']}) for $d=2^{N}$.
• Figure 4: The Toric code. A qubit lives on each of the $2N^2$ edges of the lattice. For each vertex $v$ and facet $f$ one defines respectively a vertex $A_v$ and facet $B_f$ operator respectively. Notice that the lattice has the topology of a torus, i.e. one has periodic boundary conditions.
• Figure 5: Plot of the functions $g_2^{\rm Tor}(t)$ (panel \ref{['fig:g2Tor']}), $\tilde{g}_3(t)$ (panel \ref{['fig:g3tilde_Tor']}) and $g_4^{\rm Tor}(t)$ (panel \ref{['fig:g4Tor']}) for different lattice size $N$. One can once again observe how the behavior of $\tilde{g}_3(t)$ is in between the ones of $g_2(t)$ and $g_4(t)$.