Sharp Bounds on Ground State Energy of the SYK Model

We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian $H_{\operatorname{SYK}}$ on $n$ Majorana modes with $k$-body interactions, and prove that $\mathbb{E}\|H_{\operatorname{SYK}}\|_{\operatorname{op}} = (1 - o(1))\cdot\sqrt{2n}/k$ for super-constant $k\leq o(\sqrt{n})$, where the expectation is over the disorder variables in the Hamiltonian. This confirms the predictions due to Garcia-Garcia, Jia and Verbaarschot'18 and answers a question posed in Feng, Tian and Wei'19. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that the dissipative quantum algorithm of Basso, Chen and Dalzell'24 provably computes the ground state energy of the SYK Hamiltonian up to an $O(1)$-multiplicative factor for all $k < \sqrt{n}/4$. Our key technical idea is identifying an explicit, deterministic linear operator $\mathsf{x}$ such that a fixed quadratic form of $\mathsf{x}^{2\ell}$ exactly equals the expected trace moments of the SYK Hamiltonian for every $n$ and $k$. This linear operator can be naturally viewed as a \emph{twisted} model of bosons on the space of hyperedges of a hypergraph. The problem thus reduces to identifying the spectral edge of $\mathsf{x}$, which we show is dominated by the spectrum of a natural ${n \choose k}$-dimensional matrix from the \emph{Johnson} scheme and is straightforward to compute using known results. To show that our bound is sharp, we construct a witness state with a large quadratic form on $\mathsf{x}$ and transform it into a certificate of a lower bound on the largest quadratic form on $H_{\operatorname{SYK}}$.