Polynomial Equivalence of Complexity Geometries

This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group -- often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen -- and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds.Comment: v2: minor improvements and enhancement

A survey of business process complexity metrics.

Business processes have an inherent complexity which if not controlled can keep on increasing with time, thus making the processes error-prone, difficult to understand and maintain. In the last few years, several researchers have proposed a number of metrics which can be used to measure and therefore control the complexity of business processes. In this study, a survey of business process complexity metrics is conducted with the goal of investigating if there are any gaps in literature. Initially, a description of the process of metrics definition and validation is presented, followed by an analysis of business process complexity metrics that have appeared in literature in the last 5 years. The reviewers checked whether the identified metrics have any tool support, whether they have been validated and whether validation results are significant or not. Findings show that few business process complexity metrics have been proposed so far and that even fewer have been validated. In order to address these issues, some future research directions are proposed