AI 中文总结
研究超越轨道旋转的测量结构，利用代数与RANGE全局优化，证明关联块\(T\)等结论，给出宇称上限，定位高\(X\)秩对易族，应用框架降低认证成本，展示相关测量电路及成本情况。
AI 中文摘要
代数和RANGE全局优化在识别超越轨道旋转的测量结构中发挥互补且明确分开的作用。在每个自旋有两个空间轨道的固定(1,1)粒子扇区中，代数证明一个保持粒子数的轨道旋转上下文贡献一个秩为一的两体关联块\(T\)，可观测量至少需要\(\mathrm{rank}\,T\)个这样的上下文，其最佳\(K\)上下文关联块近似正是截断奇异值分解得到的埃卡特 - 扬奇异值尾部。连续的RANGE搜索独立证实了这种精确权衡。一个贝尔对角、海森堡型见证具有关联秩三，需要至少三个轨道旋转上下文，而一个明确的物理克利福德电路测量其对易的泡利代表。对于自旋守恒的约旦 - 维格纳分子哈密顿量，还证明了任何泡利子集的宇称上限\(r_X \le 2(N - 1)\)，即使在对易子集中也是紧的；\(X\)秩是一种路由诊断，严格分离由关联秩定理给出。RANGE的离散模式定位跨分子和生产f元素哈密顿量的高\(X\)秩对易族，找到CH4和NdO的饱和上限见证；除非达到已证明的上限，否则最好找到值。应用伴随证书框架，并通过完全对易、克利福德可及设置扩大乘积设置，在四个29 - 35量子比特f元素哈密顿量上，将认证的领先射击成本降低了31 - 70%，这是一个量子比特对易（QWC）与量子比特对易加完全对易（QWC + FC）的结果，而不是高斯与克利福德定价。哈达玛测试中的受控泡利插入是克利福德的；这些零\(T\)陈述仅涉及测量电路，而射击次数和状态制备保留其全部成本。
英文摘要
Algebra and RANGE global optimization play complementary, explicitly separated roles in identifying measurement structure beyond orbital rotations. In the fixed (1,1)-particle sector of two spatial orbitals per spin, algebra proves that one particle-number-preserving orbital-rotation context contributes a rank-one two-body correlation block $T$: an observable needs at least $\mathrm{rank}\,T$ such contexts, and its best $K$-context correlation-block approximation is exactly the Eckart-Young singular-value tail, attained by the truncated SVD. A continuous RANGE search over the physical rotation angles independently corroborates this exact trade-off. A Bell-diagonal, Heisenberg-type witness has correlation rank three: it needs at least three orbital-rotation contexts, while one explicit physical Clifford circuit measures its commuting Pauli representatives. For spin-conserving Jordan-Wigner molecular Hamiltonians we also prove the parity ceiling $r_X \le 2(N-1)$ for any Pauli subset, tight even within commuting subsets; $X$-rank is a routing diagnostic, and the strict separation is carried by the correlation-rank theorem. The discrete mode of RANGE locates high-$X$-rank commuting families across molecular and production f-element Hamiltonians, finding ceiling-saturating witnesses for CH4 and NdO; values are best found unless a proved ceiling is attained. Applying the companion certificate framework, enlarging product settings by fully commuting, Clifford-accessible settings reduces the certified leading shot cost by 31-70% on four 29-35-qubit f-element Hamiltonians, a QWC-versus-QWC+FC result rather than a Gaussian-versus-Clifford pricing. Controlled-Pauli insertions in Hadamard tests are Clifford; these zero-$T$ statements concern measurement circuitry only, while shot counts and state preparation retain their full costs.
Comments13 pages; companion paper to Certified Optimal Measurement Reduction over Quantum Context Landscapes
