Relative diffusion transform and quantum speedup of computations

It is shown that every function computable in time T(n) and space S(n) on a classical one-dimensional cellular automaton can be computed with certainty in time O(T1/2S) and space n√T on a quantum computer with relative diffusion transforms (RDTs) on parts of intermediate products of classical computation. However, in the general case, RDTs cannot be implemented by the conventional quantum computer even with oracles for intermediate results. Such a function can be computed only in time O(S4S/2T/T1) on the conventional quantum computer with oracles for the intermediate results of classical computations with time T1. © 2000 MAIK "Nauka/Interperiodica".