对称初始化量子吉布斯采样：非阿贝尔非对称级联
Symmetry-initialized quantum Gibbs sampling: a non-Abelian asymmetry cascade
中文总结 AI 辅助
本研究针对混合瓶颈为弱破缺对称性的量子吉布斯采样器，基于表示论提出对称初始化方法，通过消除慢模式重叠实现混合时间渐近加速，在SU(2)戴维斯采样器中验证了加速级联的存在。
中文摘要 AI 辅助
对于混合瓶颈为弱破缺对称性的量子吉布斯采样器，我证明正确的初始化由表示论决定。我首先证明了一个通用的加速-前置因子二分性：通过精确消除慢模式重叠，我将名义上的前置因子缩减转化为系统混合时间的基本渐近加速。接着我表明，当瓶颈是弱破缺对称性时，原本指数级昂贵的瓶颈本征向量就是对称性本身。对于非阿贝尔群G，我证明正确的初始化目标是群平均非对称性，投影该非对称性需要G不变输入，部分子群不变输入会产生由子群格索引的混合时间加速级联。相反，仅匹配一阶矩已被证明无法实现加速，非对称性是可直接测量的初始化目标。我在SU(2)戴维斯采样器中通过解析和数值验证了这些结果，其中总自旋多重态构成慢模式，预测的加速级联出现，弛豫速率随对称性破缺参数线性缩放。最后，该模型划定了非对称性目标的边界，说明何时需要单独匹配守恒逻辑数据。
英文摘要
For a quantum Gibbs sampler whose mixing bottleneck is a weakly broken symmetry, I show that the correct initialization is determined by representation theory. I first prove a general speedup-versus-prefactor dichotomy: by exactly eliminating slow-mode overlap, I convert a nominal prefactor reduction into a fundamental, asymptotic acceleration of the system's mixing time. I then show that when the bottleneck is a weakly broken symmetry, the otherwise exponentially expensive bottleneck eigenvector is the symmetry charge.For a non-Abelian group $G$, I prove that the correct initialization target is the group-averaging asymmetry. Projecting this asymmetry out requires a $G$-invariant input. Partial, subgroup-invariant inputs produce a cascade of mixing-time speedups indexed by the subgroup lattice. Conversely, matching first moments alone is provably insufficient.The asymmetry is a directly measurable initialization target. I verify these results analytically and numerically in an $SU(2)$ Davies sampler where total-spin multiplets constitute the slow modes. The predicted speedup cascade emerges with relaxation rates scaling linearly with the symmetry-breaking parameter. Finally, the model maps the boundaries of the asymmetry target, illustrating where the separate matching of conserved logical data becomes necessary.
