发表机构
California State University, Fresno（弗雷斯诺加州州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明归一化持续同调的加性误差估计对量子计算类BQP困难，通过有限证书核传递定理和八维标签寄存器构造，实现从量子验证器到团复形实例的归约。
AI 中文摘要
对于团复形 $X_1\subseteq X_2$，$d$ 次归一化持续同调定义为 $\operatorname{rank}[H_d(X_1)\to H_d(X_2)]/\dim H_d(X_1)$。估计该值需要精确的端点同调和包含诱导映射，即使端点拉普拉斯间隙为逆多项式。我们证明，加性误差 $1/24$ 的估计对 $\mathsf{BQP}_{1}^{G_2}$（精确门集 $G_2=\{X,\mathsf{CX},\mathsf{CCX},H\otimes H\}$ 上的完美完备性类）是困难的，因此对 $\mathsf{BQP}_{1}$ 在条目属于分圆域 $\mathbb{Q}(\zeta_{2^k})$ 的每个有限门集上也是困难的，即使对于无权团复形也是如此。主要工具是一个有限证书核传递定理。对于满足有限多个可精确检查的局部条件的加权团小工具固定调色板，完整几何复形的每个单位链 $x$ 满足 $\operatorname{dist}(x,K)^2\le C(t\lambda^2+\langle x,\Delta x\rangle/(g\lambda^{26}))$，其中 $K$ 是模拟投影哈密顿量的嵌入核，$g$ 是其间隙，$t$ 是小工具数量，$\lambda$ 是私有顶点权重。由于 $\lambda$ 独立于 $g$ 选择，几何拉普拉斯算子恰好有 $\dim K$ 个零模，并且在其上方有线性于 $g$ 的间隙。精确填充将端点同调与寄存器循环空间的商 $V/W_A$ 等同，嵌套项集诱导自然商满射，因此持续秩等于后面的核维数，无需选择兼容的调和代表。一个固定的八维标签寄存器将 $\mathsf{BQP}_{1}^{G_2}$ 验证器转化为实例，使得 $\beta_d(X_1)=8$ 且归一化持续同调为 $3/4$ 或 $1/8$，并且已建立的公共副本放大将所有内容转移到无权图。
英文摘要
For clique complexes $X_1\subseteq X_2$, normalized persistence in degree $d$ is $\operatorname{rank}[H_d(X_1)\to H_d(X_2)]/\dim H_d(X_1)$. Estimating it requires exact endpoint homology and the inclusion-induced map, even with inverse-polynomial endpoint Laplacian gaps. We prove that additive-error $1/24$ estimation is hard for $\mathsf{BQP}_{1}^{G_2}$, the perfect-completeness class over the exact gate set $G_2=\{X,\mathsf{CX},\mathsf{CCX},H\otimes H\}$, and hence for $\mathsf{BQP}_{1}$ over every finite gate set with entries in a cyclotomic field $\mathbb{Q}(ζ_{2^k})$, even for unweighted clique complexes. The main tool is a finite-certificate kernel-transfer theorem. For a fixed palette of weighted clique gadgets satisfying finitely many exactly checkable local conditions, every unit chain $x$ of the full geometric complex satisfies $\operatorname{dist}(x,K)^2\le C(tλ^2+\langle x,Δx\rangle/(gλ^{26}))$, where $K$ is the embedded kernel of the simulated projector Hamiltonian, $g$ its gap, $t$ the number of gadgets, and $λ$ the private vertex weight. Since $λ$ is chosen independently of $g$, the geometric Laplacian has exactly $\dim K$ zero modes and a gap linear in $g$ above them. Exact fillings identify the endpoint homology with a quotient $V/W_A$ of the register cycle space, and nested term sets induce the natural quotient epimorphisms, so the persistent rank equals the later kernel dimension without any choice of compatible harmonic representatives. A fixed eight-dimensional label register turns a $\mathsf{BQP}_{1}^{G_2}$ verifier into instances with $β_d(X_1)=8$ and normalized persistence $3/4$ or $1/8$, and an established common-copy blow-up transfers everything to unweighted graphs.
Comments23 pages; computational certificate data and verification scripts included as ancillary files
