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 $p=Θ(1/n^3)$ 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 $p \in [Θ(1/n^3),1]$ and show there indeed exists a certain robustness of sparsification. We prove that with high probability, Gaussian states achieve only a $Θ(1/\sqrt{n})$-factor approximation to the true ground state energy of sparse SYK for all $p\geqΩ(\log n/n^2)$, and that Gaussian states cannot achieve constant-factor approximations unless $p \leq O(\log^2 n/n^3)$. Additionally, we prove that the quantum algorithm of Hastings--O'Donnell still achieves a constant-factor approximation to the ground state energy when $p\geqΩ(\log n/n)$. 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 whenever $p \geq Ω(\log n/n)$, extending the analogous $p=1$ result of Hastings--O'Donnell.

翻译：寻找强相互作用费米子系统的基态通常是深入理解量子化学与凝聚态系统的先决条件。Sachdev--Ye--Kitaev（SYK）模型是此类系统的典型代表；其特别引人关注不仅因为存在如Hastings--O'Donnell（STOC 2022）等能高效制备基态近似态的量子算法，还因已知许多经典拟设（ansatz）在制备低能态时存在不可行性。然而，当SYK模型被充分稀疏化时——即模型中的项以$1-p$的概率被舍弃，其中$p=Θ(1/n^3)$，$n$为系统尺寸——这种量子-经典分离被证明将不复存在。这引发了关于SYK模型的量子与经典计算复杂度对稀疏化的鲁棒性问题。本研究首次系统探讨$p \in [Θ(1/n^3),1]$区间的稀疏SYK模型，并证明稀疏化确实存在特定鲁棒性。我们证明：对于所有$p≥Ω(\log n/n^2)$，高斯态以高概率仅能获得稀疏SYK真实基态能量的$Θ(1/\sqrt{n})$倍近似；且除非$p \leq O(\log^2 n/n^3)$，高斯态无法实现常数倍近似。此外，我们证明当$p≥Ω(\log n/n)$时，Hastings--O'Donnell量子算法仍能实现基态能量的常数倍近似。综合这些结果可知：在$p ≥ Ω(\log n/n)$条件下，针对寻找稀疏SYK近似基态的目标，输出高斯态的经典算法与高效量子算法之间存在可证明的分离，这扩展了Hastings--O'Donnell在$p=1$情况下的对应结论。