  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.
