量子元复杂性是你所需的全部：通过有界时间柯尔莫哥洛夫复杂性刻画单向谜题
Quantum Meta-Complexity Is All You Need: Characterizing One-Way Puzzles via Time-Bounded Kolmogorov Complexity
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中文总结 AI 辅助
该研究启动量子密码学的有界时间元复杂性计划，定义量子计划复杂性pK^q_t并证明两个无条件定理，还确定了多项式时间编码定理为关键开放猜想，刻画了单向谜题的存在条件。
中文摘要 AI 辅助
我们启动了量子密码学的有界时间元复杂性计划。近期研究通过量子可采样分布上的普通不可计算柯尔莫哥洛夫复杂性的平均情况硬度，刻画了单向谜题——量子密码学中无单向函数时的最小搜索本原；相比之下，Liu和Pass的经典计划则处于多项式时间界。我们为经典字符串定义了概率有界时间量子计划复杂性pK^q_t，并证明了两个无条件定理。第一，量子编码定理：量子多项式时间采样器以概率δ输出的任意字符串，存在信息论最优长度log(1/δ)加上对数项的描述，该描述可由量子机器在O(√(1/δ))乘以多项式的时间内解码，方法是对相干执行的采样器进行振幅放大。第二，亚指数时间下的精确刻画：当且仅当时间界为2^(n/2)poly(n)时，pK^q的间隙问题是弱量子平均情况难的，单向谜题才存在，这改进了普通复杂性的刻画。随后，我们将多项式时间编码定理确定为该计划唯一的关键开放猜想，证明其蕴含完整的多项式时间刻画，分析经典去随机化证明为何难以量子化，并提出一个相对化的障碍猜想，限定了单向谜题处的字符串值元复杂性，全文中猜想均按此标注。
英文摘要
We initiate the time-bounded meta-complexity program for quantum cryptography. Recent work characterizes one-way puzzles, the minimal search primitive of quantum cryptography without one-way functions, by the average-case hardness of approximating the plain, uncomputable Kolmogorov complexity over quantumly samplable distributions; the classical program of Liu and Pass, by contrast, lives at polynomial time bounds. We define a probabilistic time-bounded quantum program complexity pKq^t for classical strings and prove two unconditional theorems. First, a quantum coding theorem: any string output by a quantum polynomial-time sampler with probability delta admits a description of the information-theoretically optimal length log(1/delta) plus logarithmic terms, decodable by a quantum machine in time O(sqrt(1/delta)) times a polynomial, via amplitude amplification over the coherently executed sampler. Second, an exact characterization at subexponential time: one-way puzzles exist if and only if the gap problem for pKq at time bound 2^(n/2) poly(n) is weakly quantum-average-hard, refining the plain-complexity characterizations. We then isolate the polynomial-time coding theorem as the single load-bearing open conjecture of the program, prove that it implies the full polynomial-time characterization, analyze why the classical derandomization proof resists quantization, and formulate a relativized barrier conjecture delimiting string-valued meta-complexity at one-way puzzles. Conjectures are labeled as such throughout.
发表机构
- Sunway University（双威大学）
机构由 AI 辅助整理，请以论文原文为准。
