Encoding dependent generalization bounds for parametrized quantum circuits

A large body of recent work has begun to explore the potential of parametrized quantum circuits PQCs as machine learning models, within the framework of hybrid quantum classical optimization. In particular, theoretical guarantees on the out of sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC based models that depend explicitly on the strategy used for data encoding. These imply bounds on the performance of trained PQC based models on unseen data. Moreover, our results facilitate the selection of optimal data encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC based models via trigonometric functions. Our generalization bounds emphasize the importance of well considered data encoding strategies for PQC based model

Encoding dependent generalization bounds for parametrized quantum circuits

A large body of recent work has begun to explore the potential of parametrized quantum circuits PQCs as machine learning models, within the framework of hybrid quantum classical optimization. In particular, theoretical guarantees on the out of sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC based models that depend explicitly on the strategy used for data encoding. These imply bounds on the performance of trained PQC based models on unseen data. Moreover, our results facilitate the selection of optimal data encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC based models via trigonometric functions. Our generalization bounds emphasize the importance of well considered data encoding strategies for PQC based model

Widespread colonisation of Tanzanian catchments by introduced Oreochromis tilapia fishes: the legacy from decades of deliberate introduction

From the 1950s onwards, programmes to promote aquaculture and improve capture fisheries in East Africa have relied heavily on the promise held by introduced species. In Tanzania these introductions have been poorly documented. Here we report the findings of surveys of inland water bodies across Tanzania between 2011 and 2017 that clarify distributions of tilapiine cichlids of the genus Oreochromis. We identified Oreochromis from 123 sampling locations, including 14 taxa restricted to their native range and three species that have established populations beyond their native range. Of these three species, the only exotic species found was blue-spotted tilapia (Oreochromis leucostictus), while Nile tilapia (Oreochromis niloticus) and Singida tilapia (Oreochromis esculentus), which are both naturally found within the country of Tanzania, have been translocated beyond their native range. Using our records, we developed models of suitable habitat for the introduced species based on recent (1960–1990) and projected (2050, 2070) East African climate. These models indicated that presence of suitable habitat for these introduced species will persist and potentially expand across the region. The clarification of distributions provided here can help inform the monitoring and management of biodiversity, and inform policy related to the future role of introduced species in fisheries and aquaculture