Optimizing Sparse SYK

TL;DR

The work investigates ground-state energy approximation for sparsified SYK models, bridging the gap between fully dense and highly sparse interactions. It develops tight classical bounds for Gaussian-state ansätze and extends the Hastings–O'Donnell quantum algorithm to the sparse setting by reducing to a two-color SYK variant, proving a constant-factor quantum approximation for $p≥Ω(\log n/n)$. A universality result shows the maximal eigenvalue remains $Θ(√n)$ across sparsity, enabling robust comparisons between classical and quantum approaches. Overall, the paper demonstrates a provable quantum advantage in finding approximate ground states for a physically relevant sparse fermionic model, while clarifying the limits of Gaussian-state classical methods in this regime.

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 $p=Θ(1/n^3)$ 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.

Figures (5)

• Figure 1: We plot the maximum achievable energies on sparse SYK Hamiltonians for various classes of quantum states (up to logarithmic factors in $p$). Green represents arbitrary disorder-independent sets of $\exp(n^2\log n)$ states, including states of circuit complexity $\operatorname{\widetilde{\Theta}}\left(n^2\right)$; blue represents Gaussian states; and red represents general quantum states generated from known quantum algorithms. Solid lines represent upper bounds proved in this work, dots represent bounds shown in previous work HTS21hastings2023optimizingHerasymenko_2023, dashed lines denote conjectured upper bounds in this work, and shaded regions represent where the true energy achieved by the state class may lie, i.e., where there is no tight matching lower bound.
• Figure 2: Tensor networks calculating $\mathop{\mathrm{\mathrm{Tr}}}\limits(J_{1234}\gamma_{\{1,2,3,4\}} \rho) = J_{1234}(-\Gamma_{1,2}\Gamma_{3,4}+\Gamma_{1,3}\Gamma_{2,4}-\Gamma_{1,4}\Gamma_{2,3})$ (the signs are due to the antisymmetry of the SYK tensor $J$).
• Figure 3: Tensor network contraction representing $\left\langle\pi_L(\Gamma_{vec}^{\otimes 2})\right| J^{mat}_{1,3}\otimes J^{mat}_{3,1}\left|\pi_R(\Gamma_{vec}^{\otimes 2})\right\rangle$.
• Figure 4: The empirical scaling of the Lovász number of sparsified commutation graphs with respect to the sparsification parameter $p$ conforms well to a scaling of $\Theta(\sqrt{p})$. Here, we plot a comparison of the Lovász number of sparsified commutation graphs of the SYK model with their fit to a square root power law $c_1\sqrt{p}+c_2$ where $c_1$ and $c_2$ are the fit parameters. Error bars are equal to the standard deviation over values drawn from $10$ random sparsifications of the initial graph for each data point.
• Figure 5: Visualization of the Hastings--O'Donnell algorithm on sparse SYK (\ref{['alg:generalized_HO']}).

Theorems & Definitions (49)

• Theorem 1.1: Gaussian state energy upper bound
• Theorem 1.2: Sparse SYK quantum algorithm
• Definition 2.1: Weyl--Brauer matrices
• Definition 2.3: Fermionic Gaussian states
• Definition 2.4: Gaussian state covariance matrix
• Proposition 2.5: Wick's theorem for fermionic Gaussian states
• Definition 2.6: Gaussian approximation factor
• Proposition 2.7: Gaussian state net
• proof
• Definition 2.8: Commutation graph