Finding Optimal Flows Efficiently

Among the models of quantum computation, the One-way Quantum Computer is one of the most promising proposals of physical realization, and opens new perspectives for parallelization by taking advantage of quantum entanglement. Since a one-way quantum computation is based on quantum measurement, which is a fundamentally nondeterministic evolution, a sufficient condition of global determinism has been introduced as the existence of a causal flow in a graph that underlies the computation. A O(n^3)-algorithm has been introduced for finding such a causal flow when the numbers of output and input vertices in the graph are equal, otherwise no polynomial time algorithm was known for deciding whether a graph has a causal flow or not. Our main contribution is to introduce a O(n^2)-algorithm for finding a causal flow, if any, whatever the numbers of input and output vertices are. This answers the open question stated by Danos and Kashefi and by de Beaudrap. Moreover, we prove that our algorithm produces an optimal flow (flow of minimal depth.) Whereas the existence of a causal flow is a sufficient condition for determinism, it is not a necessary condition. A weaker version of the causal flow, called gflow (generalized flow) has been introduced and has been proved to be a necessary and sufficient condition for a family of deterministic computations. Moreover the depth of the quantum computation is upper bounded by the depth of the gflow. However, the existence of a polynomial time algorithm that finds a gflow has been stated as an open question. In this paper we answer this positively with a polynomial time algorithm that outputs an optimal gflow of a given graph and thus finds an optimal correction strategy to the nondeterministic evolution due to measurements.Comment: 10 pages, 3 figure

RG Flow and Thermodynamics of Causal Horizons in AdS

Causal horizons in pure Poincare $AdS$ are Killing horizons generated by dilatation vector. Renormalization group (RG) flow breaks the dilatation symmetry and makes the horizons dynamical. We propose that the boundary RG flow is dual to the thermodynamics of the causal horizon. As a check of our proposal we show that the gravity dual of the boundary $c$-theorem is the second law of thermodynamics obeyed by causal horizons. The holographic $c$-function is the Bekenstein-Hawking entropy (density) of the dynamical causal horizon. We explicitly construct the $c$-function in a generic class of RG-flow geometries and show that it interpolates monotonically between the UV and IR central charges as a result of the second law.Comment: 15 pages, Latex, references added, figures added , relation to causal holographic information clarified, version accepted for publication in JHEP, references adde

Flow Decomposition for Multi-User Channels - Part I

A framework based on the idea of flow decomposition is proposed to characterize the decode-forward region for general multi-source, multi-relay, all-cast channels with independent input distributions. The region is difficult to characterize directly when deadlocks occur between two relay nodes, in which both relays benefit by decoding after each other. Rate-vectors in the decode-forward region depend ambiguously on the outcomes of all deadlocks in the channel. The region is characterized indirectly in two phases. The first phase assumes relays can operate non-causally. It is shown that every rate-vector in the decode-forward region corresponds to a set of flow decompositions, which describe the messages decoded at each node with respect to the messages forwarded by all the other nodes. The second phase imposes causal restrictions on the relays. Given an arbitrary set of (possibly non-causal) flow decompositions, necessary and sufficient conditions are derived for the existence of an equivalent set of causal flow decompositions that achieves the same rate-vector region

Causal connectivity of evolved neural networks during behavior

To show how causal interactions in neural dynamics are modulated by behavior, it is valuable to analyze these interactions without perturbing or lesioning the neural mechanism. This paper proposes a method, based on a graph-theoretic extension of vector autoregressive modeling and 'Granger causality,' for characterizing causal interactions generated within intact neural mechanisms. This method, called 'causal connectivity analysis' is illustrated via model neural networks optimized for controlling target fixation in a simulated head-eye system, in which the structure of the environment can be experimentally varied. Causal connectivity analysis of this model yields novel insights into neural mechanisms underlying sensorimotor coordination. In contrast to networks supporting comparatively simple behavior, networks supporting rich adaptive behavior show a higher density of causal interactions, as well as a stronger causal flow from sensory inputs to motor outputs. They also show different arrangements of 'causal sources' and 'causal sinks': nodes that differentially affect, or are affected by, the remainder of the network. Finally, analysis of causal connectivity can predict the functional consequences of network lesions. These results suggest that causal connectivity analysis may have useful applications in the analysis of neural dynamics