Polynomial-time quantum Turing machines are provably superior to their classical counterparts within a common space bound in $o(\log \log n)$. For $Ω(\log \log n)$ space, the only known quantum advantage result has been the fact $\mathsf{BPTISP}(2^{O(n)},o(\log n))\subsetneq \mathsf{BQTISP}(2^{O(n)},o(\log n))$, proven by exhibiting an exponential-time quantum finite automaton (2QCFA) that recognizes $L_{pal}$, the language of palindromes, which is an impossible task for sublogarithmic-space probabilistic Turing machines. No subexponential-time quantum algorithm can recognize $L_{pal}$ in sublogarithmic space. We initiate the study of quantum advantage under simultaneous subexponential time and $Ω(\log \log n) \cap o(\log n)$ space bounds. We exhibit an infinite family $\mathcal{F}$ of functions in $(\log n)^{ω(1)}\cap n^{o(1)}$ such that for every $f_i\in\mathcal{F}$, there exists another function $f_{i+1}\in\mathcal{F}$ such that $f_{i+1}(n) \in o(f_{i}(n))$, and each such $f_i$ corresponds to a different quantum advantage statement, i.e. a proper inclusion of the form $\mathsf{BPTISP}(2^{O(f_i(n))},o(\log f_i(n)))\subsetneq \mathsf{BQTISP}(2^{O(f_i(n))},o(\log f_i(n)))$ for a different pair of subexponential time and sublogarithmic space bounds. Our results depend on a technique enabling polynomial-time quantum finite automata to control padding functions with very fine asymptotic granularity.

翻译：多项式时间量子图灵机在共同空间界为$o(\log \log n)$时已被证明优于其经典对应物。对于$Ω(\log \log n)$空间，目前已知的唯一量子优势结果是$\mathsf{BPTISP}(2^{O(n)},o(\log n))\subsetneq \mathsf{BQTISP}(2^{O(n)},o(\log n))$，该结果通过展示一个指数时间量子有限自动机（2QCFA）能够识别回文语言$L_{pal}$而得到证明，而亚对数空间概率图灵机无法完成此任务。任何亚指数时间量子算法均无法在亚对数空间内识别$L_{pal}$。我们首次在亚指数时间与$Ω(\log \log n) \cap o(\log n)$空间的双重约束下研究量子优势。我们构造了一个定义在$(\log n)^{ω(1)}\cap n^{o(1)}$上的无限函数族$\mathcal{F}$，使得对于每个$f_i\in\mathcal{F}$，都存在另一个函数$f_{i+1}\in\mathcal{F}$满足$f_{i+1}(n) \in o(f_{i}(n))$，且每个这样的$f_i$对应一个不同的量子优势命题，即形如$\mathsf{BPTISP}(2^{O(f_i(n))},o(\log f_i(n)))\subsetneq \mathsf{BQTISP}(2^{O(f_i(n))},o(\log f_i(n)))$的真包含关系，其中时间与空间界分别为不同的亚指数函数与亚对数函数。我们的结果依赖于一种使多项式时间量子有限自动机能够以极精细的渐近粒度控制填充函数的技术。