Quantum chaos in the sparse SYK model (2403.13884v2)

Abstract: The Sachdev-Ye-Kitaev (SYK) model is a system of $N$ Majorana fermions with random interactions and strongly chaotic dynamics, which at low energy admits a holographically dual description as two-dimensional Jackiw-Teitelboim gravity. Hence the SYK model provides a toy model of quantum gravity that might be feasible to simulate with near-term quantum hardware. Motivated by the goal of reducing the resources needed for such a simulation, we study a sparsified version of the SYK model, in which interaction terms are deleted with probability $1{-p}$. Specifically, we compute numerically the spectral form factor (SFF, the Fourier transform of the Hamiltonian's eigenvalue pair correlation function) and the nearest-neighbor eigenvalue gap ratio $r$ (characterizing the distribution of gaps between consecutive eigenvalues). We find that when $p$ is greater than a transition value $p_1$, which scales as $1/N^3$, both the SFF and $r$ match the values attained by the full unsparsified model and with expectations from random matrix theory (RMT). But for $p<p_1$, deviations from unsparsified SYK and RMT occur, indicating a breakdown of holography in the highly sparsified regime. Below an even smaller value $p_2$, which also scales as $1/N^3$, even the spacing of consecutive eigenvalues differs from RMT values, signaling a complete breakdown of spectral rigidity. Our results cast doubt on the holographic interpretation of very highly sparsified SYK models obtained via machine learning using teleportation infidelity as a loss function.

• The paper shows that modest sparsification of the SYK model retains quantum chaos as evidenced by spectral form factors and gap ratios.
• Critical sparsity thresholds are identified where reduced interactions compromise holographic duality and spectral rigidity.
• The work provides practical simulation guidelines to preserve gravitational properties while lowering computational complexity.

Quantum Chaos in the Sparse SYK Model

The paper "Quantum chaos in the sparse SYK model" presents a detailed exploration of a sparse variant of the Sachdev-Ye-Kitaev (SYK) model, investigating its quantum chaotic properties and its implications for a holographic dual description. The SYK model, notable for capturing features of quantum chaos and providing a simplified model of holography, is challenging to simulate due to its complexity, involving all possible quartic interactions among $N$ Majorana fermions. This research focuses on a sparsified version, where interaction terms are probabilistically deleted to reduce simulation resources, and examines the extent to which chaos and holographic properties persist.

Key Findings and Analysis

• Spectral Form Factor and Gap Ratio: The paper utilizes the spectral form factor (SFF), a Fourier transform of the eigenvalue pair correlation function, and the nearest-neighbor eigenvalue gap ratio $r$, to identify spectral properties indicative of quantum chaos. It is established that both quantities adhere to predictions from random matrix theory (RMT) when the SYK model is only moderately sparsified, demonstrating that the chaotic behavior is maintained.
• Threshold Sparsity Parameters: As the sparsity parameter $p$ falls below a critical threshold $p_1$, scaling as $1/N^3$, distinct deviations in SFF and $r$ emerge, marking a weakening of spectral rigidity and a departure from RMT correspondence. This suggests a compromise in the holographic duality. Below a further reduced parameter $p_2$, spectral rigidity completely vanishes, as indicated by significant deviations in nearest-neighbor eigenvalue distributions from RMT expectations.
• Practical Considerations for Simulation: The results provide guidelines for the practical sparsification of the SYK model to preserve its dual gravitational properties—a critical insight for experimental setups aiming to simulate aspects of quantum gravity on near-term quantum hardware. The report implies that simulations remain feasible at significantly reduced computational loads without compromising chaotic attributes, provided $p$ remains above the threshold $p_1$.

Implications and Theoretical Insights

The findings imply that quantum chaos, a prerequisite for holographic dual descriptions, is resilient to substantial reductions in interaction complexity. This strengthens the feasibility of utilizing sparsified quantum models in simulating theoretical constructs related to quantum gravity within the AdS/CFT framework.

Future explorations might involve:

• Investigating the effect of sparsification on other dynamic and static properties of the SYK model, such as correlation functions and thermal behavior.
• Developing algorithms that identify minimal sets of interaction terms necessary to retain desired chaotic and holographic features.
• Extending the sparsity paper to other values of $q$ (the number of fermionic operators per interaction) and analyzing convergence to classical gravity at large $N$.

The paper's computational approach also offers a robust methodology for evaluating spectral properties, potentially applicable to a wider class of quantum many-body systems exhibiting nontrivial chaotic dynamics yet maintaining theoretical interest due to their simplicity and analytical tractability.