In many real-life situations in engineering (and in other disciplines), we need to solve an optimization problem: we want an optimal design, we want an optimal control, etc. One of the main problems in optimization is avoiding local maxima (or minima). One of the techniques that helps with solving this problem is annealing: whenever we find ourselves in a possibly local maximum, we jump out with some probability and continue search for the true optimum. A natural way to organize such a probabilistic perturbation of the deterministic optimization is to use quantum effects. It turns out that often, quantum annealing works much better than nonquantum one. Quantum annealing is the main technique behind the only commercially available computational devices that use quantum effects-D-Wave computers. The efficiency of quantum annealing depends on the proper selection of the annealing schedule, i.e., schedule that describes how the perturbations decrease with time. Empirically, it has been found that two schedules work best: power law and exponential ones. In this paper, we provide a theoretical explanation for these empirical successes, by proving that these two schedules are indeed optimal (in some reasonable sense).

No abstract

Many practical problems necessitate faster and faster computations. Simple physical estimates show that eventually, to move beyond a limit caused by the speed of light restrictions on communication speed, we will need to use quantum-or, more generally, reversible-computing. Thus, we need to be able to transform the existing algorithms into a reversible form. Such transformation schemes exist. However, such schemes are not very efficient. Indeed, in general, when we write an algorithm, we composed it of several pre-existing modules. It would be nice to be able to similarly compose a reversible version of our algorithm from reversible versions of these moduli-but the existing transformation schemes cannot do it, they require that we, in effect, program everything "from scratch". It is therefore desirable to come up with alternative transformations, transformation that transform compositions into compositions and thus, transform a modular program in an efficient way-by utilizing transformed moduli. Such transformations are proposed in this paper.

One of the main reasons for the current interest in quantum computing is that, in principle, quantum algorithms can break the RSA encoding, the encoding that is used for the majority secure communications-in particular, the majority of e-commerce transactions are based on this encoding. This does not mean, of course, that with the emergence of quantum computers, there will no more ways to secretly communicate: while the existing non-quantum schemes will be compromised, there exist a quantum cryptographic scheme that will enables us to secretly exchange information. In this scheme, however, there is a certain probability that an eavesdropper will not be detected. A natural question is: can we decrease this probability by an appropriate modification of the current quantum cryptography algorithm? In this paper, we show that such a decrease is not possible: the current quantum cryptography algorithm is, in some reasonable sense, optimal.

Need for faster and faster computing necessitates going down to quantum level -which means involving quantum computing. One of the important features of quantum computing is that it is reversible. Reversibility is also important as a way to decrease processor heating and thus, enable us to place more computing units in the same volume. In this paper, we argue that from this viewpoint, interval uncertainty is more appropriate than the more general set uncertaintyand, similarly, that fuzzy numbers (for which all alpha-cuts are intervals) are more appropriate than more general fuzzy sets. We also explain why intervals (and fuzzy numbers) are indeed ubiquitous in applications. Need for Quantum and Reversible ComputingNeed for quantum computing. Our current computers are very fast in comparison with what was available a few years ago. However, no matter how fast the computers, there are always computational tasks -from bioinformatics, from other disciplines -that necessitate even faster computers.