集合覆盖的紧凑哈密顿量编码中的联合对称性与动力学可达性
Orbit-resolved spectra and dynamical accessibility in multi-register covering Hamiltonians
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中文总结 AI 辅助
该研究针对最小集合覆盖的紧凑多寄存器编码，分析了联合对称性与动力学可达谱结构，给出了多项式绝热运行时间的严格界，未提出量子加速主张。
中文摘要 AI 辅助
当初始态和插值过程均保持若干对称性时，哈密顿量谱的哪一部分具有物理相关性？我们针对最小集合覆盖的紧凑多寄存器编码解决该问题。所表示的对称性结合了寄存器置换与入射自同构群的忠实基作用。我们将其固定的对称允许空间与由协议生成的通常更小的循环空间区分开来，并定义该空间中相对于孤立能带的能隙。 sector 分解的舒尔补界证明，当扇区包含可行覆盖且具有正的无效态分离时，覆盖测量概率成立；扇区动能底是关键。精确轨道商揭示了对称协议不可见的扇区间重合，而稳定性定理表明，横向暗交叉在小的保对称扰动下仍持续存在。在偶环族上，原始线性插值具有精确的全局多重性闭合，而联合固定激发能隙保持恒定。对于同一族，我们构造了从 Dicke 态到 Gibbs 振幅态的 Johnson/Metropolis 父路径，其覆盖概率为 $1-O(n^{-5})$。零范围比较和对数凹三箱耦合给出了均匀循环能隙证书 $\tilde{\theta}(n^{-13})$。定量导数界随后在抽象哈密顿量访问模型中产生多项式绝热运行时间，条件是 Dicke 态制备和对父哈密顿量的访问。我们不提出量子加速主张：该结果是全局、对称允许和动力学可达谱结构的严格分离。改变初始态或破坏保留的对称性会改变该可达谱。
英文摘要
Which part of a Hamiltonian spectrum is relevant when the initial state and interpolation preserve multiple symmetries? We study this question for a multi-register encoding of fixed-cardinality Set Cover. Register permutations and incidence automorphisms define a joint action whose orbits are unlabelled candidate covers, with an exact hopping Hamiltonian on the orbit quotient. We distinguish the symmetry-fixed space from the generally smaller common invariant cyclic envelope generated by the initial state and endpoint Hamiltonians; a prescribed trajectory may span less still. We prove a sector-resolved Schur-complement bound on cover-measurement probability that includes the kinetic floor induced by sector projection. For a retained-walk linear path, a general hopping-cancellation point makes the spectrum exactly orbit-resolved. Multiplicity outside the fixed sector is then dynamically dark, while transverse dark crossings satisfy a separate stability theorem. On even cycles the fixed-sector excitation gap at cancellation equals $λ(n-1)/(2n-1)$ despite global multiplicity closure. Separately, for the same instances, we construct a distinct Johnson--Metropolis parent path on the injective hard-core space, from a Dicke state to a Gibbs-amplitude state. Its cover-failure probability is $O(n^{-5})$ and its cyclic-envelope gap is at least $1024\,n^{-13}$ uniformly along the path, yielding a conditional polynomial adiabatic runtime in an abstract Hamiltonian-access model. We claim no quantum speedup; the result is a protocol-dependent separation of global, symmetry-fixed, cyclic-envelope, and tracked spectra.
