Signatures of quantum chaos and complexity in the Ising model on random graphs

Abstract

We investigate signatures of quantum chaos in the mixed-field quantum Ising model on finite-size Erdős-Rényi graphs using probes scalable on near-term quantum devices. By tuning the graph connectivity, the system exhibits a crossover from a localized regime at low connectivity, through a chaotic regime at intermediate connectivity, to a permutation-symmetric integrable limit near all-to-all connectivity. This crossover has possible implications for the performance and trainability of variational algorithms such as QAOA. We characterize this crossover using complementary probes. First, deep thermalization of a projected ensemble starting from a product state reveals slow (fast) convergence to the Haar ensemble at extremal (intermediate) connectivities. Secondly, we analyze eigenstate and eigenvalue correlations using the partial spectral form factor, an experimentally scalable proxy for the spectral form factor with reduced resource overhead, and observe characteristic chaotic signatures at intermediate connectivities and distinct deviations at extremal connectivities. Finally, we explore the Krylov complexity of operators, a locality-independent diagnostic that, although not directly experimentally accessible, serves as a tool for quantifying scrambling. We show that it is maximized deep in the chaotic regime, corroborating the signatures observed through the experimentally scalable probes. Our results provide finite-size benchmarks demonstrating robust signatures of chaos in scalable probes and suggest that these diagnostics can be implemented in current quantum platforms to access regimes beyond classical simulation.

Figures (22)

• Figure 1: (a) A single realization of Erdos-Renyi graph with $L=8$ nodes and connectance, $\tilde{M}=0.6$ which implies number of connected edges, $M=17$. Interacting pairs are represented by connected edges. (b) Eigen-spectrum of Hamiltonian in Eq. \ref{['Hamiltonian']} for an instance at each $\tilde{M}$ with $L=8$. We have used darker shades to indicate degeneracy.
• Figure 2: An outline of QAOA protocol. The three-term variational circuit is parameterized by $\{\alpha_k,\beta_k,\gamma_k\}$. The entire circuit consists of $p$ such layers. The input state of the ansatz is taken as $|+\rangle^{\otimes L}$. The energy of the cost function is optimized with a classical optimizer and the parameters are iteratively updated using gradient descent.
• Figure 3: Trajectory of energy expectation value of instantaneous state in QAOA dynamics through the eigen-spectrum of instantaneous QAOA Hamiltonian. The thickness of dashes is proportional to the degeneracy of the eigenstates. The ansatz is a $4$-layer quantum circuit. The label at the bottom of each stack shows the instantaneous Hamiltonian and the corresponding layer (in superscript). The marker in the right-most stack marks the final energy expectation value of the optimized state, i.e., $\langle \psi_{\rm{opt}}|H_{\rm c}|\psi_{\rm{opt}} \rangle$. The color indicates the overlap between the optimized state and the instantaneous eigenspectrum. $L=8$. The problem graph has a connectivity $\tilde{M}_{\rm c}=0.4$.
• Figure 4: Approximation ratio of the optimized state for a QAOA circuit with $L=8,p=2$ (a,b) and $L=8,p=3$ (c,d). Keeping the problem $H_{\rm c}$ fixed, for each $\tilde{M}$, we show the QAOA optimized approximation ratio for $1000$ random choices of $H_{\rm d}$ at that $\tilde{M}$. The two columns describe data for two independently sampled problems $H_{\rm c}$ with $\tilde{M}_{\rm c}=0.4$.
• Figure 5: (a)-(b) Evolution of $\overline{\Delta^{(1)}}$ with time for $\tilde{M}=0.2$ and $0.4$, respectively. (c)-(d) Evolution of $\overline{\Delta^{(2)}}$ with time for $\tilde{M}=0.2$ and $0.4$, respectively. All data represented here are averaged over $500$ graphs. Power-law fits for $L=14$ are shown with black dashed lines, and the corresponding exponents are noted in the plot. Vertical black dotted lines indicate the region where the fitting was done to obtain the power-law exponents. (e)-(f) The slope of trace distance with time across $\tilde{M}$ (with estimated error in slope indicated). $L_A=2$. The parameters for Hamiltonian (Eq. \ref{['Hamiltonian']}) are taken as $g=1.0,h=0.1$.