nt - p h / 97 06 00 3 v 4 6 J ul 1 99 7 About the quantum mechanical speeding up of classical algorithms

1 Abstract This work introduces a relative diffusion transformation (RDT)-a simple unitary transformation which acts in a subspace, localized by an oracle. Such a transformation can not be fulfilled on quantum Turing machines with this oracle in polynomial time in general case. It is proved, that every function f : ω n −→ ω n , card(ω) = 4, computable in time T (n) and space S(n) on classical 1-dimensional cellular automaton, can be computed with certainty in time O(T 1/2 S) on quantum computer with RDTs over the parts of intermediate products of classical computation. This requires multiprocessor, which consists of √ T quantum devices P 1 , P 2 ,. .. , P √ T of O(S(n)) size, working in parallel-serial mode and interacting by classical lows. Such a function can be computed only in time O(S4 S/2 T /T 1) on a conventional quantum computer with oracle for intermediate results of the time T 1 of classical computations. Quantum mechanical computations (QC) are distinct in nature from the classical ones. The point is that a quantum system can be in different classical states simultaneously with the corresponding amplitudes. The vector, composed of these amplitudes completely determines the quantum state of system. The module squared of every amplitude is the probability of detecting the system in the corresponding classical state after observation. Such an observation is the only way to obtain a result of QC. An evolution of such a system is represented by the application of unitary transformation to its vector of amplitudes. Quantum computers became one of the most popular areas of investigations in theoretical computer science as well as in quantum physics because of that in the past 3-4 years considerable progress has been made in the theory of QC. Since that time when R.Feunmann in the work [Fe] proposed quantum mechanical (Q-M) computer, D.Deutsch in the work [De] gave the first formal model of computations on quantum Turing machine (QTM), and S.Lloyd in the work [Ll] presented the physical scheme of Q-M computational device, the advantages of quantum computations over the classical ones in a variety of particular problems became apparent from the sequence of results (look, for example, at [Be],[Sh], [BE]). Moreover, A.Berthiaume and G.Brassard in [BB] showed, that Q-M computations even can beat the nondeterministic ones in computations with oracles. But as for absolute (without oracles) computations, advantages of Q-M computers over …