We present new results on the landscape of problems that can be solved by quantum Turing machines (QTM's) employing severely limited amounts of memory. In this context, we demonstrate two infinite time hierarchies of complexity classes within the ``small space'' regime: For all $i\geq 0$, there is a language that can be recognized by a constant-space machine in $2^{O(n^{1/2^i})}$ time, but not by any sublogarithmic-space QTM in $2^{O(n^{1/2^{i+1}})}$ time. For quantum machines operating within $o(\log \log n)$ space, there exists another hierarchy, each level of which corresponds to an expected runtime of $2^{O((\log n)^i)}$ for a different positive integer $i$. We also improve a quantum advantage result, demonstrating a language that can be recognized by a polynomial-time constant-space QTM, but not by any classical machine using $o(\log \log n)$ space, regardless of the time budget. The implications of our findings for quantum space-time tradeoffs are discussed.

翻译：本文针对量子图灵机在内存严重受限条件下所能解决的问题领域提出了新的研究成果。在此背景下，我们证明了在"小空间"范围内存在两个无限的时间层次结构：对于所有$i\geq 0$，存在一种语言能被常数空间机器在$2^{O(n^{1/2^i})}$时间内识别，但无法被任何亚对数空间量子图灵机在$2^{O(n^{1/2^{i+1}})}$时间内识别。对于在$o(\log \log n)$空间内运行的量子机器，存在另一个层次结构，其每个层级对应不同正整数$i$的$2^{O((\log n)^i)}$期望运行时间。我们还改进了量子优势结果，证明存在一种语言能被多项式时间常数空间量子图灵机识别，但无法被任何使用$o(\log \log n)$空间的经典机器识别（无论时间预算如何）。最后讨论了我们的发现对量子时空权衡的启示。