Quantum query complexity of state conversion

State conversion generalizes query complexity to the problem of converting between two input-dependent quantum states by making queries to the input. We characterize the complexity of this problem by introducing a natural information-theoretic norm that extends the Schur product operator norm. The complexity of converting between two systems of states is given by the distance between them, as measured by this norm. In the special case of function evaluation, the norm is closely related to the general adversary bound, a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. We thus obtain that the general adversary bound characterizes the quantum query complexity of any function whatsoever. This generalizes and simplifies the proof of the same result in the case of boolean input and output. Also in the case of function evaluation, we show that our norm satisfies a remarkable composition property, implying that the quantum query complexity of the composition of two functions is at most the product of the query complexities of the functions, up to a constant. Finally, our result implies that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem.Comment: 19 pages, 2 figures; heavily revised with new results and simpler proof

Logic Programming Approaches for Representing and Solving Constraint Satisfaction Problems: A Comparison

Many logic programming based approaches can be used to describe and solve combinatorial search problems. On the one hand there is constraint logic programming which computes a solution as an answer substitution to a query containing the variables of the constraint satisfaction problem. On the other hand there are systems based on stable model semantics, abductive systems, and first order logic model generators which compute solutions as models of some theory. This paper compares these different approaches from the point of view of knowledge representation (how declarative are the programs) and from the point of view of performance (how good are they at solving typical problems).Comment: 15 pages, 3 eps-figure

Lecture notes: Semidefinite programs and harmonic analysis

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.Comment: 31 page

The Depth and Breadth of John Bell's Physics

John Bell's investigations in foundations of quantum mechanics, particle physics, and quantum field theory are recalled.Comment: 46 pages, 1 figure, LaTeX; editorial corrections; email correspondence to R. Jackiw <[email protected]

Constructive Reasoning for Semantic Wikis

One of the main design goals of social software, such as wikis, is to support and facilitate interaction and collaboration. This dissertation explores challenges that arise from extending social software with advanced facilities such as reasoning and semantic annotations and presents tools in form of a conceptual model, structured tags, a rule language, and a set of novel forward chaining and reason maintenance methods for processing such rules that help to overcome the challenges. Wikis and semantic wikis were usually developed in an ad-hoc manner, without much thought about the underlying concepts. A conceptual model suitable for a semantic wiki that takes advanced features such as annotations and reasoning into account is proposed. Moreover, so called structured tags are proposed as a semi-formal knowledge representation step between informal and formal annotations. The focus of rule languages for the Semantic Web has been predominantly on expert users and on the interplay of rule languages and ontologies. KWRL, the KiWi Rule Language, is proposed as a rule language for a semantic wiki that is easily understandable for users as it is aware of the conceptual model of a wiki and as it is inconsistency-tolerant, and that can be efficiently evaluated as it builds upon Datalog concepts. The requirement for fast response times of interactive software translates in our work to bottom-up evaluation (materialization) of rules (views) ahead of time – that is when rules or data change, not when they are queried. Materialized views have to be updated when data or rules change. While incremental view maintenance was intensively studied in the past and literature on the subject is abundant, the existing methods have surprisingly many disadvantages – they do not provide all information desirable for explanation of derived information, they require evaluation of possibly substantially larger Datalog programs with negation, they recompute the whole extension of a predicate even if only a small part of it is affected by a change, they require adaptation for handling general rule changes. A particular contribution of this dissertation consists in a set of forward chaining and reason maintenance methods with a simple declarative description that are efficient and derive and maintain information necessary for reason maintenance and explanation. The reasoning methods and most of the reason maintenance methods are described in terms of a set of extended immediate consequence operators the properties of which are proven in the classical logical programming framework. In contrast to existing methods, the reason maintenance methods in this dissertation work by evaluating the original Datalog program – they do not introduce negation if it is not present in the input program – and only the affected part of a predicate’s extension is recomputed. Moreover, our methods directly handle changes in both data and rules; a rule change does not need to be handled as a special case. A framework of support graphs, a data structure inspired by justification graphs of classical reason maintenance, is proposed. Support graphs enable a unified description and a formal comparison of the various reasoning and reason maintenance methods and define a notion of a derivation such that the number of derivations of an atom is always finite even in the recursive Datalog case. A practical approach to implementing reasoning, reason maintenance, and explanation in the KiWi semantic platform is also investigated. It is shown how an implementation may benefit from using a graph database instead of or along with a relational database