04421 Abstracts Collection -- Algebraic Methods in Computational Complexity

From 10.10.04 to 15.10.04, the Dagstuhl Seminar 04421 ``Algebraic Methods in Computational Complexity\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Quantifying the difference between many-body quantum states

The quantum state overlap is the textbook measure of the difference between two quantum states. Yet, it is inadequate to compare the complex configurations of many-body systems. The problem is inherited by the widely employed quantum state fidelity and related distances. We introduce the weighted distances, a new class of information-theoretic measures that overcome these limitations. They quantify how hard it is to discriminate between two quantum states of many particles, factoring in the structure of the required measurement apparatus. Therefore, they can be used to evaluate both the theoretical and the experimental performances of complex quantum devices. We also show that the newly defined "weighted Bures length" between the input and output states of a quantum process is a lower bound to the experimental cost of the transformation. The result uncovers an exact quantum limit to our ability to convert physical resources into computational ones.Comment: 4+2 pages, change from previous version: the contractivity of weighted distances holds only for single site operation

A random walk approach to quantum algorithms

The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial: pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e., when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk, due to interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point on average than a classical walker, and this forms the basis of a quantum speed up that can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, with even a small quantum computer available, development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems