AI 中文总结
研究结构化量子表示工作负载中Gram认证资源延续问题，通过构建Gram矩阵等方法进行迹和算子范数诊断，证明相关性质，给出稳定性测试，通过实验表明延续需满足特定条件，未确立通用40量子比特可模拟性等。
AI 中文摘要
n量子比特纯态的密集表示需要2^n个复振幅，这使得在n较大时无法进行密集的经典实现。我们为结构化量子表示工作负载开发了Gram认证资源延续，并研究在低成本资源模型下获得的解决方案何时仍然是更丰富模型的合理初始化。对于通过声明的等距连接的粗粒度和细粒度状态集合，细粒度、粗粒度和交叉复振幅重叠形成一个半正定块Gram矩阵。一个维度至多为样本数两倍的带符号算子具有细粒度密度减去提升后的粗粒度密度的非零带符号谱，从而无需构建任何密度算子即可进行迹和算子范数诊断。我们证明，具有目标次优性δ_c的粗加权谱标志在细粒度水平上的次优性至多为δ_c + 2ε，其中ε是经验迹距离；因子2是可达到的。我们区分编码器变化和精确可行族延拓，给出一个依赖间隙的子空间稳定性测试，并表明延续无法突破最终的施密特秩上限。在对8到40量子比特阶梯的确定性综合控制中，精确的辅助提升在数值精度上是一致的。转移初始化将最终梯级块更新从30减少到20，但完整的级联成本是直接求解最终梯级的4.80至5.43倍，且没有实质性的目标改进。重新排列八个贝尔对可将最大矩阵乘积态键从256减少到2。因此，只有当跨梯级不匹配、可行族包含、拓扑结构和总工作量共同满足预先指定的审计时，延续才是合理的。这些无噪声的经典结果既未确立通用的40量子比特可模拟性，也未声称硬件性能或量子优势。
英文摘要
Dense representation of an $n$-qubit pure state requires $2^n$ complex amplitudes, precluding dense classical materialization at large $n$. We develop Gram-certified resource continuation for structured quantum-representation workloads and ask when a solution obtained under a lower-cost resource model remains a justified initialization for a richer one. For coarse and fine state ensembles connected by a declared isometry, fine, coarse, and cross complex amplitude overlaps form a positive-semidefinite block Gram matrix. A signed operator of dimension at most twice the sample count has the nonzero signed spectrum of the fine density minus the lifted coarse density, yielding trace- and operator-norm diagnostics without constructing either density operator. We prove that a coarse weighted spectral flag with objective suboptimality $δ_c$ has fine-level suboptimality at most $δ_c+2\varepsilon$, where $\varepsilon$ is the empirical trace distance; the factor two is attainable. We distinguish encoder change from exact feasible-family prolongation, give a gap-dependent subspace-stability test, and show that continuation cannot overcome a final Schmidt-rank ceiling. In deterministic synthetic controls over an 8-to-40-qubit ladder, exact ancilla lifts agree to numerical precision. Transferred initialization reduces final-rung block updates from 30 to 20, but the complete cascade costs $4.80$--$5.43$ times a direct final-rung solve, without material objective improvement. Reordering eight Bell pairs reduces the maximum matrix-product-state bond from 256 to 2. Thus continuation is justified only when cross-rung mismatch, feasible-family inclusion, topology, and total work jointly satisfy prespecified audits. These noise-free classical results neither establish generic 40-qubit simulability nor claim hardware performance or quantum advantage.
Comments15 pages, 1 figure
