We prove stability of the spectral gap for gapped, frustration-free Hamiltonians under general, quasi-local perturbations. We present a necessary and sufficient condition for stability, which we call "Local Topological Quantum Order" and show that this condition implies an area law for the entanglement entropy of the groundstate subspace. This result extends previous work by Bravyi et al., on the stability of topological quantum order for Hamiltonians composed of commuting projections with a common zero-energy subspace. We conclude with a list of open problems relevant to spectral gaps and topological quantum order.Comment: 21 pages, 1 figure, updated references, discussion includes note on stability of symmetry-protected and higher-energy sectors, and a path towards extending the result to frustrated, gapped Hamiltonians. Added new section on overview of the proof and updated the Introduction and Appendice

Increasing student retention (successfully finishing a particular course) and persistence (continuing through a sequence of courses or the major area of study) is currently a major challenge for universities. While students' academic and social integration into an institution seems to be vital for student retention, research into the effect of interpersonal interactions is rare. We use network analysis as an approach to investigate academic and social experiences of students in the classroom. In particular, centrality measures identify patterns of interaction that contribute to integration into the university. Using these measures, we analyze how position within a social network in a Modeling Instruction (MI) course-an introductory physics course that strongly emphasizes interactive learning-predicts their persistence in taking a subsequent physics course. Students with higher centrality at the end of the first semester of MI are more likely to enroll in a second semester of MI. Moreover, we found that chances of successfully predicting individual student's persistence based on centrality measures are fairly high-up to 75%, making the centrality a good predictor of persistence. These findings suggest that increasing student social integration may help in improving persistence in science, technology, engineering, and mathematics fields.

The Modeling Instruction (MI) approach to introductory physics manifests significant increases in student conceptual understanding and attitudes toward physics. In light of these findings, we investigated changes in student self-efficacy while considering the construct's contribution to the career-decision making process. Students in the Fall 2014 and 2015 MI courses at Florida International University exhibited a decrease on each of the sources of self-efficacy and overall self-efficacy (N ¼ 147) as measured by the Sources of Self-Efficacy in Science Courses-Physics (SOSESC-P) survey. This held true regardless of student gender or ethnic group. Given the highly interactive nature of the MI course and the drops observed on the SOSESC-P, we chose to further explore students' changes in self-efficacy as a function of three centrality measures (i.e., relational positions in the classroom social network): inDegree, outDegree, and PageRank. We collected social network data by periodically asking students to list the names of peers with whom they had meaningful interactions. While controlling for PRE scores on the SOSESC-P, bootstrapped linear regressions revealed post-self-efficacy scores to be predicted by PageRank centrality. When disaggregated by the sources of self-efficacy, PageRank centrality was shown to be directly related to students' sense of mastery experiences. InDegree was associated with verbal persuasion experiences, and outDegree with both verbal persuasion and vicarious learning experiences. We posit that analysis of social networks in active learning classrooms helps to reveal nuances in self-efficacy development.

Recent progress in building large-scale quantum devices for exploring quantum computing and simulation paradigms has relied upon effective tools for achieving and maintaining good experimental parameters, i.e., tuning up devices. In many cases, including in quantum-dot based architectures, the parameter space grows substantially with the number of qubits, and may become a limit to scalability. Fortunately, machine learning techniques for pattern recognition and image classification using so-called deep neural networks have shown surprising successes for computer-aided understanding of complex systems. In this work, we use deep and convolutional neural networks to characterize states and charge configurations of semiconductor quantum dot arrays when one can only measure a current-voltage characteristic of transport through such a device. For simplicity, we model a semiconductor nanowire connected to leads and capacitively coupled to depletion gates using the Thomas-Fermi approximation and Coulomb blockade physics. We then generate labeled training data for the neural networks, and find at least 90 % accuracy for charge and state identification for single and double dots purely from the dependence of the nanowire's conductance upon gate voltages. Using these characterization networks, we can then optimize the parameter space to achieve a desired configuration of the array, a technique we call 'auto-tuning'. Finally, we show how such techniques can be implemented in an experimental setting by applying our approach to an experimental data set, and outline further problems in this domain, from using charge sensing data to extensions to full one and two-dimensional arrays, that can be tackled with machine learning.