精确局部最优性不可组合：时序实现的计算复杂性

中文摘要 AI 辅助

时序共享实现复杂性（CRC）是为使用一个共享的受控转移动力学族来复现一组未来响应菜单所需的最小归一化随机状态维度 $C_{\rm seq}$。我们关注秩紧区域，其中局部实现维度和独立查询静态载波宽度均等于 $K$，从而隔离了由时序一致性施加的额外维度和计算约束。该框架也作为顺序量子过程中状态维度界限的经典基线，其中随机动力学推广到完全正映射。我们在此区域建立了三个结果。首先，一个显式的载荷-延迟族表现出无界的乘法状态膨胀：$C_{\rm loc}=C_{\rm stat}=k$ 而 $C_{\rm seq}=k(L+1)$，隔离了共享时间拉回的固有状态成本。其次，对于固定五字母字母表上具有单个布尔终端效应的显式列出的有理有限菜单，精确共享可实现性是 $\exists\mathbb{R}$-完全的，并且零与逆多项式缺陷承诺问题是 $\mathsf{PromiseNP}$-完全的，局部和静态最优值固定在 $K$。第三，一个总五字母时序编译器将有界理性中间单纯形实例转换为同一固定字母表上的多项式指定的正则几何族，保持精确局部和静态宽度 $K$。编译后的族在强有界理性输入上产生共享实现的强 $\mathsf{PromiseNP}$-困难性，以及由精确多项式大小有限核心证明的 $\exists\mathbb{R}$ 上界。总之，这些结果表明，即使在秩紧区域，精确局部和静态最优性在时序共享下也未必组合。

英文摘要

We study a controlled, normalized multi-menu positive-realization problem. A normalized realization reproduces declared root--word responses using common row-stochastic transition matrices and a single terminal effect. We compare three realization complexities: independent local realizations, a static shared carrier with independent query effects, and chronological shared realizations. We construct response families for which the local and static optimum widths both equal $k$, isolating the additional cost imposed by chronological consistency. An explicit payload--delay family has exact width $k$ locally and statically but requires exact width $k(L+1)$ under shared chronological coupling, yielding an unbounded multiplicative separation. For explicitly listed rational menus, exact shared realizability is $\exists\mathbb{R}$-complete, while the promise problem of distinguishing zero defect from defect at least inverse-polynomial is $\mathsf{PromiseNP}$-complete. These finite-menu hardness results hold with five control letters and a single Boolean terminal effect. For regular response families generated by a geometric compiler, rank-tight realizability over the full infinite language is equivalent to realizability on a polynomial-size finite core. Consequently, exact rank-tight realizability for the compiled instances has an $\exists\mathbb{R}$ upper bound, complementing a strongly bounded-rational $\mathsf{PromiseNP}$-hardness result.

发表机构

• Beijing Key Laboratory of Fault-Tolerant Quantum Computing, Beijing Academy of Quantum Information Sciences（北京容错量子计算重点实验室，北京量子信息科学研究院）
• Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（凝聚态物理国家重点实验室，中国科学院物理研究所）
• University of Chinese Academy of Sciences（中国科学院大学）

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