AI 中文总结
该研究提出基于非厄米哈密顿量EP编织的拓扑学习框架，以组合优化替代梯度下降，生成通用编织门，实现稳健可解释的拓扑保护神经形态计算。
AI 中文摘要
我们提出了一种基于非厄米Bogoliubov-de Gennes哈密顿量中例外点（EP）编织的拓扑学习框架。推导了EP超曲面的闭合代数方程；通过有限系统的动量量子化，该方程可精确预测实空间中所有EP的数量和参数位置，与系统尺寸无关。EP的拓扑由两个量子化不变量表征：态交换保真度和归一化贝里相位，对于拓扑EP，两者不能同时为零。完整的拓扑图显示所有EP都位于特定区域内。绝热环绕证实了稳健的态交换，并产生了通用的编织门集合，包括Pauli-X门、Pauli-Y门、Pauli-Z门、类Hadamard门、T门和SWAP操作，当参数a为0的特殊情况还提供额外的相位门。基于这些生成元，我们将学习重新表述为编织编程，即对编织群的离散搜索，用组合优化替代连续权重上的梯度下降。概念验证的遗传搜索成功发现了短编织字，以完美保真度重现了标准Hadamard门和H.Z门。该范式具有固有的抗噪声能力、通过组合连接防止灾难性遗忘，以及通过数学构造保证的泛化能力，确立了EP编织作为稳健、可解释且拓扑保护的神经形态计算的有潜力基底。
英文摘要
We present a framework for topological learning based on exceptional point (EP) braiding in a non-Hermitian Bogoliubov-de Gennes Hamiltonian. A closed algebraic equation for the EP super-surface is derived; through momentum quantisation in finite systems, it predicts the exact number and parameter positions of all EPs in real space, irrespective of system size. The EP topology is characterised by two quantised invariants the state-swap fidelity and the normalised Berry phase which cannot both be zero for a topological EP. A complete topological map shows that all EPs lie within a specefic region. Adiabatic encirclements confirm robust state swapping and yield a universal set of braid gates, including Pauli-X, Pauli-Y, Pauli-Z, a Hadamard-like gate, the T-gate, and a SWAP operation, with the special case where a is 0, providing additional phase gates. Building on these generators, we reformulate learning as braid programming a discrete search over the braid group that replaces gradient descent on continuous weights with combinatorial optimisation. A proof-of-concept genetic search successfully discovers short braid words that reproduce the standard Hadamard gate and the H.Z gate with perfect fidelity. This paradigm offers inherent noise immunity, catastrophic-forgetting prevention through compositional concatenation, and guaranteed generalisation by mathematical construction, establishing EP braiding as a promising substrate for robust, interpretable, and topologically protected neuromorphic computation.
Comments20 pages, 7 figures
