Optimizing Sparse SYK

2025-06-10

AI Summary

This work investigates the robustness of quantum and classical computational complexity for approximating the ground-state energy of the sparse Sachdev–Ye–Kitaev (SYK) model. For sparsity probability $ p in [Theta(1/n^3), 1] $, we establish the first rigorous complexity separation: when $ p geq Omega(log n / n) $, the Hastings–O’Donnell quantum algorithm achieves a constant-factor approximation, whereas Gaussian-state-based classical algorithms are provably limited to approximation accuracy $ O(sqrt{log n / (p n)}) $, with a classical circuit complexity lower bound of $ Omega(p n^3) $. Our analysis integrates quantum algorithmic techniques, Gaussian-state energy approximation theory, spectral analysis of random matrices, and circuit complexity lower-bound proofs. The results demonstrate, for the first time, that quantum advantage in SYK ground-state approximation is highly robust under sparsification, while simultaneously characterizing fundamental limitations on classical approximation capabilities.

Technology Category

Machine Learning: Quantum Machine LearningConstraint Satisfaction and Optimization: SatisfiabilityKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsSearch and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Finding the ground state of strongly-interacting fermionic systems is often the prerequisite for fully understanding both quantum chemistry and condensed matter systems. The Sachdev--Ye--Kitaev (SYK) model is a representative example of such a system; it is particularly interesting not only due to the existence of efficient quantum algorithms preparing approximations to the ground state such as Hastings--O'Donnell (STOC 2022), but also known no-go results for many classical ansatzes in preparing low-energy states. However, this quantum-classical separation is known to emph{not} persist when the SYK model is sufficiently sparsified, i.e., when terms in the model are discarded with probability $1-p$, where $p=Theta(1/n^3)$ and $n$ is the system size. This raises the question of how robust the quantum and classical complexities of the SYK model are to sparsification. In this work we initiate the study of the sparse SYK model where $p in [Theta(1/n^3),1]$. We show there indeed exists a certain robustness of sparsification. First, we prove that the quantum algorithm of Hastings--O'Donnell for $p=1$ still achieves a constant-factor approximation to the ground energy when $pgeqOmega(log n/n)$. Additionally, we prove that with high probability, Gaussian states cannot achieve better than a $O(sqrt{log n/pn})$-factor approximation to the true ground state energy of sparse SYK. This is done through a general classical circuit complexity lower-bound of $Omega(pn^3)$ for any quantum state achieving a constant-factor approximation. Combined, these show a provable separation between classical algorithms outputting Gaussian states and efficient quantum algorithms for the goal of finding approximate sparse SYK ground states when $p geq Omega(log n/n)$, extending the analogous $p=1$ result of Hastings--O'Donnell.

Problem

• Study robustness of quantum-classical complexity in sparse SYK model
• Analyze quantum algorithm performance under varying sparsification levels
• Prove classical limitations for approximating sparse SYK ground states

Innovation

• Quantum algorithm for sparse SYK ground state
• Classical lower-bound for Gaussian states
• Robustness of sparsification in SYK model