Let A and B be local operators in Hamiltonian quantum systems with N degrees of freedom and finite-dimensional Hilbert space. We prove that the commutator norm \(\Vert [A(t),B]\Vert \) is upper bounded by a topological combinatorial problem: counting irreducible weighted paths between two points on the Hamiltonian’s factor graph. Our bounds sharpen existing Lieb–Robinson bounds by removing extraneous growth. In quantum systems drawn from zero-mean random ensembles with few-body interactions, we prove stronger bounds on the ensemble-averaged out-of-time-ordered correlator \({\mathbb {E}}\left[ \Vert { [A(t),B]} \Vert _{\mathrm {F}}^2\right] \). In such quantum systems on Erdös–Rényi factor graphs, we prove that the scrambling time \(t_{\mathrm {s}}\), at which \(\Vert { [A(t),B]} \Vert _{\mathrm {F}}={\Theta }(1)\), is almost surely \(t_{\mathrm {s}}={\Omega }(\sqrt{\log N})\); we further prove \(t_{\mathrm {s}}={\Omega }(\log N) \) to high order in perturbation theory in 1/N. We constrain infinite temperature quantum chaos in the q-local Sachdev-Ye-Kitaev model at any order in 1/N; at leading order, our upper bound on the Lyapunov exponent is within a factor of 2 of the known result at any \(q>2\). We also speculate on the implications of our theorems for conjectured holographic descriptions of quantum gravity.

设A和B是具有N个自由度和有限维希尔伯特空间的哈密​​顿量子系统中的局部算符。我们证明了交换子范数\(\Vert [A(t),B]\Vert \)的上限是一个拓扑组合问题：计​​算哈密顿量因子图上两点之间的不可约加权路径。我们的边界通过去除无关的增长来锐化现有的 Lieb-Robinson 边界。在从具有少体相互作用的零均值随机系综绘制的量子系统中，我们证明了对系综平均的无序相关器\({\mathbb {E}}\left[ \Vert { [A (t),B]} \Vert _{\mathrm {F}}^2\right] \). 在 Erdös–Rényi 因子图上的此类量子系统中，我们证明了加扰时间\(t_{\mathrm {s}}\)，此时\(\Vert { [A(t),B]} \Vert _{ \mathrm {F}}={\Theta }(1)\)，几乎肯定是\(t_{\mathrm {s}}={\Omega }(\sqrt{\log N})\) ; 我们进一步证明\(t_{\mathrm {s}}={\Omega }(\log N) \)在 1/ N 的微扰理论中达到高阶。我们在q -local Sachdev-Ye-Kitaev 模型中以 1/ N 中的任意阶约束无限温度量子混沌；在领先的顺序，我们的李雅普诺夫指数的上限在任何\(q>2\)已知结果的 2 倍以内. 我们还推测了我们的定理对量子引力的推测全息描述的影响。