发表机构
Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了SYK哈密顿量算子范数的严格期望界，证实了相关预测，推广到稀疏SYK模型，还证明了Basso等人的耗散量子算法可计算SYK模型基态能量的近似值。
AI 中文摘要
我们研究具有k体相互作用的n个Majorana模式上的Sachdev-Ye-Kitaev（SYK）哈密顿量$H_{\operatorname{SYK}}$，并证明对于超常数$k\leq o(\sqrt{n})$，有$\mathbb{E}\\|H_{\operatorname{SYK}}\\|_{\operatorname{op}} = (1 - o(1))\cdot\sqrt{2n}/k$，其中期望是对哈密顿量中的无序变量取的。这证实了Garcia-Garcia、Jia和Verbaarschot在2018年的预测，并回答了Feng、Tian和Wei在2019年提出的问题。我们的结果可推广到稀疏SYK哈密顿量。作为推论，我们得出Basso、Chen和Dalzell在2024年提出的耗散量子算法，对于所有$k < \sqrt{n}/4$，可证明能以$O(1)$的乘性因子计算SYK哈密顿量的基态能量。我们的关键技术思路是识别一个显式的确定性线性算子$\mathsf{x}$，使得对于任意n和k，$\mathsf{x}^{2\ell}$的固定二次型恰好等于SYK哈密顿量的期望迹矩。该线性算子可自然视为超图超边上的玻色子的“扭曲”模型，因此问题简化为确定$\mathsf{x}$的谱边，我们证明其由Johnson方案中自然的$\binom{n}{k}$维矩阵的谱主导，且可利用已知结果直接计算。为证明我们的界是严格的，我们构造了一个在$\mathsf{x}$上具有大二次型的见证态，并将其转化为$H_{\operatorname{SYK}}$上最大二次型下界的证明。
英文摘要
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 predictions due to Garcia-Garcia, Jia and Verbaarschot and answers a question posed by Feng, Tian, and Wei. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that a dissipative quantum algorithm due to Basso, Chen, and Dalzell 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 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 $H_{\operatorname{SYK}}$. Beyond the SYK Hamiltonian, our techniques also apply to the $k-$local quantum spin glass (QSG), yielding $\mathbb{E}\|H_{k-\operatorname{QSG}}\|_{\operatorname{op}}\leq O(\sqrt{n/k})$ for all $1\leq k\leq\sqrt{n}/2$, thus improving the standard $O(\sqrt{n})$ bound by a factor of $\sqrt{k}$.
CommentsAdded improved bounds for quantum spin glasses
