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))cdotsqrt{2n}/k for super-constant kleq 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}^2ell exactly equals the expected trace moments of the SYK Hamiltonian for every n and k. This linear operator can be naturally viewed as a 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 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}}.