Classical and quantum computation with small space bounds (PhD thesis)

In this thesis, we introduce a new quantum Turing machine (QTM) model that supports general quantum operators, together with its pushdown, counter, and finite automaton variants, and examine the computational power of classical and quantum machines using small space bounds in many different cases. The main contributions are summarized below. Firstly, we consider QTMs in the unbounded error setting: (i) in some cases of sublogarithmic space bounds, the class of languages recognized by QTMs is shown to be strictly larger than that of classical ones; (ii) in constant space bounds, the same result can still be obtained for restricted QTMs; (iii) the complete characterization of the class of languages recognized by realtime constant space nondeterministic QTMs is given. Secondly, we consider constant space-bounded QTMs in the bounded error setting: (i) we introduce a new type of quantum and probabilistic finite automata (QFAs and PFAs, respectively,) with a special two-way input head which is not allowed to be stationary or move to the left but has the capability to reset itself to its starting position; (ii) the computational power of this type of quantum machine is shown to be superior to that of the probabilistic machine; (iii) based on these models, two-way PFAs and two-way classical-head QFAs are shown to be more succinct than two-way nondeterministic finite automata and their one-way variants; (iv) we also introduce PFAs and QFAs with postselection with their bounded error language classes, and give many characterizations of them. Thirdly, the computational power of realtime QFAs augmented with a write-only memory is investigated by showing many simulation results for different kinds of counter automata. Finally, some lower bounds of realtime classical Turing machines in order to recognize a nonregular language are shown to be tight.

💡 Research Summary

The dissertation presents a comprehensive study of classical and quantum computation under stringent space constraints, introducing a versatile quantum Turing machine (QTM) model that accommodates arbitrary quantum operators and extending it to push‑down, counter, and finite‑automaton variants. In the unbounded‑error regime, the author demonstrates that QTMs strictly surpass probabilistic Turing machines when the available workspace is sublogarithmic, even achieving superiority in constant‑space settings for restricted QTMs. A full characterization of the languages recognized by realtime constant‑space nondeterministic QTMs is provided, establishing a clear separation from their classical counterparts.

In the bounded‑error setting, a novel class of automata is defined: both probabilistic and quantum finite automata equipped with a “reset” head that cannot stay stationary or move left but can instantly return to the start of the input. This reset capability enables efficient re‑initialization of quantum superpositions. The thesis proves that these reset‑enabled quantum automata are strictly more powerful than analogous probabilistic machines, and that two‑way versions of the models are exponentially more succinct than two‑way nondeterministic finite automata (NFA) and their one‑way variants.

The work further explores postselection—a hypothetical ability to condition on a particular measurement outcome—by introducing probabilistic and quantum automata with postselection. The resulting language classes (PPostS and QPostS) are characterized and shown to correspond respectively to classical complexity classes PP and BQP, providing new insights into the power of postselection in low‑space settings.

A major contribution is the analysis of realtime quantum finite automata augmented with a write‑only memory (WOM). Although the WOM cannot be read, it can be written to and used as a counter or stack. The author shows that such QFA‑WOMs can simulate various counter and push‑down automata, thereby recognizing non‑regular languages (e.g., aⁿbⁿcⁿ) while using only constant space on the work tape. This demonstrates that quantum devices can exploit write‑only storage to achieve capabilities beyond classical constant‑space machines.

Finally, the dissertation establishes tight lower bounds for realtime classical Turing machines required to recognize non‑regular languages, proving that at least logarithmic space is necessary for languages such as {aⁿbⁿ}. The same tight bounds are shown to hold for several variants, including multi‑head and restricted‑movement machines.

Overall, the thesis advances our understanding of how quantum resources interact with severe space limitations. By introducing reset heads, postselection, and write‑only memories, it reveals concrete scenarios where quantum computation outperforms classical probabilistic computation in terms of language recognition power and state‑complexity, while also delineating the fundamental space‑lower‑bound barriers that apply to both paradigms. This work lays a solid theoretical foundation for future research on space‑efficient quantum algorithms and the intrinsic limits of small‑space computation.