Oracles and query lower bounds in generalised probabilistic theories

We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying three natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle to be well-defined. The three principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as the marginal of a pure state of a larger system), and strong symmetry existence of non-trivial reversible transformations). Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, then the corresponding lower bound in theories lying at the kth level of Sorkin's hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum oracle needed to solve certain problems are not optimal in the space of all generalised probabilistic theories, although it is not yet known whether the optimal bounds are achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue "Foundational Aspects of Quantum Information" in Foundations of Physic

Deriving Grover's lower bound from simple physical principles

Grover's algorithm constitutes the optimal quantum solution to the search problem and provides a quadratic speed-up over all possible classical search algorithms. Quantum interference between computational paths has been posited as a key resource behind this computational speed-up. However there is a limit to this interference, at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power? Sorkin has defined a hierarchy of possible interference behaviours---currently under experimental investigation---where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. In this work, we consider how Grover's speed-up depends on the order of interference in a theory. Surprisingly, we show that the quadratic lower bound holds regardless of the order of interference. Thus, at least from the point of view of the search problem, post-quantum interference does not imply a computational speed-up over quantum theory.Comment: Updated title and exposition in response to referee comments. 6+2 pages, 5 figure

How to understand the cell by breaking it: network analysis of gene perturbation screens

Modern high-throughput gene perturbation screens are key technologies at the forefront of genetic research. Combined with rich phenotypic descriptors they enable researchers to observe detailed cellular reactions to experimental perturbations on a genome-wide scale. This review surveys the current state-of-the-art in analyzing perturbation screens from a network point of view. We describe approaches to make the step from the parts list to the wiring diagram by using phenotypes for network inference and integrating them with complementary data sources. The first part of the review describes methods to analyze one- or low-dimensional phenotypes like viability or reporter activity; the second part concentrates on high-dimensional phenotypes showing global changes in cell morphology, transcriptome or proteome.Comment: Review based on ISMB 2009 tutorial; after two rounds of revisio

Higher-order interference in extensions of quantum theory

Quantum interference lies at the heart of several quantum computational speed-ups and provides a striking example of a phenomenon with no classical counterpart. An intriguing feature of quantum interference arises in a three slit experiment. In this set-up, the interference pattern can be written in terms of the two and one slit patterns obtained by blocking some of the slits. This is in stark contrast with the standard two slit experiment, where the interference pattern is irreducible. This was first noted by Rafael Sorkin, who asked why quantum theory only exhibits irreducible interference in the two slit experiment. One approach to this problem is to compare the predictions of quantum theory to those of operationally-defined `foil' theories, in the hope of determining whether theories exhibiting higher-order interference suffer from pathological--or at least undesirable--features. In this paper two proposed extensions of quantum theory are considered: the theory of Density Cubes proposed by Dakic et al., which has been shown to exhibit irreducible interference in the three slit set-up, and the Quartic Quantum Theory of Zyczkowski. The theory of Density Cubes will be shown to provide an advantage over quantum theory in a certain computational task and to posses a well-defined mechanism which leads to the emergence of quantum theory. Despite this, the axioms used to define Density Cubes will be shown to be insufficient to uniquely characterise the theory. In comparison, Quartic Quantum Theory is well-defined and we show that it exhibits irreducible interference to all orders. This feature of the theory is argued not to be a genuine phenomenon, but to arise from an ambiguity in the current definition of higher-order interference. To understand why quantum theory has limited interference therefore, a new operational definition of higher-order interference is needed.Comment: Updated in response to referee comments. 17 pages. Comments welcom

Towards a Quantum-Like Cognitive Architecture for Decision-Making

We propose an alternative and unifying framework for decision-making that, by using quantum mechanics, provides more generalised cognitive and decision models with the ability to represent more information than classical models. This framework can accommodate and predict several cognitive biases reported in Lieder & Griffiths without heavy reliance on heuristics nor on assumptions of the computational resources of the mind