Physical portrayal of computational complexity

Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized to complete in non-polynomial time (NP). The irreversible process with three or more degrees of freedom is found intractable because, in terms of physics, flows of energy are inseparable from their driving forces. In computational terms, when solving problems in the class NP, decisions will affect subsequently available sets of decisions. The state space of a non-deterministic finite automaton is evolving due to the computation itself hence it cannot be efficiently contracted using a deterministic finite automaton that will arrive at a solution in super-polynomial time. The solution of the NP problem itself is verifiable in polynomial time (P) because the corresponding state is stationary. Likewise the class P set of states does not depend on computational history hence it can be efficiently contracted to the accepting state by a deterministic sequence of dissipative transformations. Thus it is concluded that the class P set of states is inherently smaller than the set of class NP. Since the computational time to contract a given set is proportional to dissipation, the computational complexity class P is a subset of NP.Comment: 16, pages, 7 figure

Quantum Cryptography in Practice

BBN, Harvard, and Boston University are building the DARPA Quantum Network, the world's first network that delivers end-to-end network security via high-speed Quantum Key Distribution, and testing that Network against sophisticated eavesdropping attacks. The first network link has been up and steadily operational in our laboratory since December 2002. It provides a Virtual Private Network between private enclaves, with user traffic protected by a weak-coherent implementation of quantum cryptography. This prototype is suitable for deployment in metro-size areas via standard telecom (dark) fiber. In this paper, we introduce quantum cryptography, discuss its relation to modern secure networks, and describe its unusual physical layer, its specialized quantum cryptographic protocol suite (quite interesting in its own right), and our extensions to IPsec to integrate it with quantum cryptography.Comment: Preprint of SIGCOMM 2003 pape

Generalized Flow and Determinism in Measurement-based Quantum Computation

We extend the notion of quantum information flow defined by Danos and Kashefi for the one-way model and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the X-Y, X-Z and Y-Z planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the deterministic computation in the one-way model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly essential for the study of the algorithms and complexity in the one-way model.Comment: 16 pages, 10 figure

Generalized Flow and Determinism in Measurement-based Quantum Computation

We extend the notion of quantum information flow defined by Danos and Kashefi for the one-way model and present a necessary and sufficient condition for the deterministic computation in this model. The generalized flow also applied in the extended model with measurements in the X-Y, X-Z and Y-Z planes. We apply both measurement calculus and the stabiliser formalism to derive our main theorem which for the first time gives a full characterization of the deterministic computation in the one-way model. We present several examples to show how our result improves over the traditional notion of flow, such as geometries (entanglement graph with input and output) with no flow but having generalized flow and we discuss how they lead to an optimal implementation of the unitaries. More importantly one can also obtain a better quantum computation depth with the generalized flow rather than with flow. We believe our characterization result is particularly essential for the study of the algorithms and complexity in the one-way model.Comment: 16 pages, 10 figure

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008