商空间上的量子自然梯度
Quantum Natural Gradient on Quotient Spaces
中文总结 AI 辅助
该研究在参数化量子电路的商空间上构建量子自然梯度（QNG），利用电路到轨道的转移原理、表示理论等推导相关性质，解决QFIM奇异问题，还验证了有限 shot 下的权衡规律。
中文摘要 AI 辅助
参数化量子电路常包含保持状态的冗余性，即使物理状态流形是正则的，也会使量子费舍尔信息矩阵（QFIM）奇异。我们在所得参数商上构建量子自然梯度（QNG），并证明当规定的冗余性耗尽费舍尔核时，Moore–Penrose更新是商黎曼梯度的最小范数水平提升。电路到轨道的转移原理将内在状态可区分性与电路坐标畸变分离，给出电路实现内在轨道-QNG方向的精确条件。表示理论进而在最高权flag轨道上得到根分辨的费舍尔尺度，以及Slater和费米子高斯流形的各向同性内在度量。在cominuscule嵌入下，内在保真度QNG保持主缺陷比并简化为一个标量方程；Lie-retracted步在η=2处局部立方收敛，稳定边界为η=4。对于有限 shot 实现，我们将精确规范移除与物理软模正则化分离，在结构投影下获得零累积规范漂移，并推导置信度控制的软模规则。在退极化下，逆费舍尔尺度仅通过放大波动恢复确定性尺度，因此无法恢复丢失的更新信噪比。一个冗余的Slater/Givens电路验证了转移定律和预测的有限 shot 权衡。
英文摘要
A parametrized quantum circuit reports its state geometry through a quantum Fisher information matrix (QFIM), often singular. A small Fisher value can reflect exact state-preserving redundancy, compression by the circuit chart, or weak intrinsic distinguishability, and these mechanisms call for different numerical treatments. We show that the circuit metric factors as $F=B^{*}MB$, where $B$ is the state-level circuit-to-orbit differential and $M$ is the intrinsic Fisher operator on the reachable orbit. The factorization identifies the exact kernel as $\ker B$, separates coordinate transfer from intrinsic geometry, and yields the condition for a circuit to realize an orbit-level quantum natural-gradient (QNG) direction. When the prescribed redundancy exhausts the Fisher kernel, the Moore--Penrose update is the minimum-norm horizontal lift of the quotient Riemannian gradient. At critical points with a locally diffeomorphic quotient-to-orbit map, chart singular values cancel from the linearized QNG operator while intrinsic anisotropy remains; in the trace-orthonormal full-control generator frame, excitation-gap anisotropy gives $κ_{\mathrm{QNG}}=κ_{\mathrm{Eucl}}$. Representation theory makes $M$ explicit on highest-weight, Slater, and fermionic-Gaussian orbits, and cominuscule fidelity flow becomes integrable, with conserved principal-defect ratios, cubic Lie-retracted convergence at $η=2$, and stability boundary $η=4$. Finite data impose a second boundary: an estimated QFIM and its confidence radius alone cannot distinguish an exact zero from a small physical mode, so the estimated spectrum alone cannot license hard projection. Under depolarization, inverse-Fisher scaling amplifies mean updates and fluctuations together and cannot restore update signal-to-noise. A redundant Slater/Givens circuit confirms exact transfer identities and illustrates finite-shot tradeoffs.
