Fine-grained quantum advantage beyond double-logarithmic space

Polynomial-time quantum Turing machines are provably superior to their classical counterparts within a common space bound in $o(\log \log n)$. For $\Omega(\log \log n)$ 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 $\Omega(\log \log n) \cap o(\log n)$ space bounds. We exhibit an infinite family $\mathcal{F}$ of functions in $(\log n)^{\omega(1)}\cap n^{o(1)}$ 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.