Space-bounded quantum computation

In this dissertation, we investigate the computational power of quantum Turing machines operating in bounded space. First, we consider space-bounds that are space-constructible and at least logarithmic in the input size. For such space-bounds, it is shown that quantum Turing machines and probabilistic Turing machines are equivalent in power in the unbounded-error setting, in the sense that each model may simulate the other with at most a constant factor increase in space. From this, it follows that any quantum Turing machine computation can be simulated deterministically with at most a quadratic increase in space, and can be simulated deterministically in time at most exponential in the space-bound. Several other facts regarding quantum complexity classes defined in terms of such space-bounds are also proved. Second, we consider the power of quantum Turing machines restricted to constant space. In this case, we first prove that quantum Turing machines having one-sided error and running in linear time are strictly more powerful than probabilistic Turing machines having either one-sided error or having two-sided bounded error and running in polynomial time. Second, we prove that exact (i.e., accepting with probability 0 or 1) constant-space quantum Turing machines upon which no restrictions on running time are placed can recognize languages that cannot be recognized by any bounded-error constant-space probabilistic Turing machine.