Dinamica delle onde non lineari e computazione quantistica.

Computers have been one of the greatest discoveries of the last century. Start- ing from the first huge computing machines, today we have computers in very small dimensions and with great computing power. But still there exist physi- cal problems that need tremendous amounts of computer time to be solved, times that could be longer than many years. When we think about non linear equations involved in enviromental physics problems, we definitely know that we need to approximate equations and solutions in order to get a result coherent with the ex- perimental data in a reasonable time. We will see in this thesis that some equations involved in ocean dynamics are non linear and a complete solution would require a long time to be evaluated by a classical computer (e.g. the kind of computer we normally use today) running a simulation, for example, for a physical enviroment such as the study of the motion of deep-water waves in the ocean. It is necessary, in the future, to find a way to reduce the time needed to run sim- ulations on computers, in order to move toward a better knowledge of the phe- nomenon and to predict results with a better precision. The most interesting thing will be to understand how some solutions of these equa- tions appear to be correlated to the quantum mechanics world and its behaviour.How could we use this parallelism? Could it suggest a way to solve these equa- tions faster than on present day computers? For a number of years scientists have been investigating the possibility of devel- oping a new technology based on quantum mechanics, called quantum computers (from now on QC), a new type of computer that should be able to solve certain kinds of problems that, with a classical computer, would need a longer evaluation time. The point is that the computational power of a QC grows exponentially with the number of qubits, compared to a classical computer, which is polynomial in the number of bits. Can we use this technology (when it will be available) to sim- ulate non linear wave dynamics? The present thesis investigates this possibility. The aim of this work is therefore to show the possible link between the world of non linear waves and quantum computation. I will describe both the theory of waves and the functioning of a QC, understanding how and why it could be rea- sonable to look to QC as a good instrument for the study of non linear equations. Using FORTRAN 90, moreover, I have written a program which simulates a QC running a quantum Fourier transfom (QFT). The QFT is, today, an efficient al- gorithm which reproduces the Fourier transform using the quantum computation logic. I will explain the meaning of it and the problems related to the use of a QC, strictly related to quantum mechanics principles. There are still many disputes around quantum computation and the role it will have in the future of the sciences. Some scientists think it will never be possible to obtain a QC working with several qubits (we will see herein the meaning of this), some others are more positive and think it will not take too long before a working QC will be available. Others instead, believe that QC are a development which will occur in the near future.

La mia tesi descrive la fisica delle onde non lineari e cerca i collegamenti con la computazione quantistica, uno dei campi futuri della fisica. Tramite un programma da me scritto in FORTRAN 90 analizzo la trasformata di Fourier Quantistica risolvendo l'equazione di KdV lineare. Sono inoltre spiegati i motivi per cui ritengo che il computer quantistico possa essere in futuro lo strumento privilegiato per lo studio e l'analisi dei problemi di onde non lineari.