Recent approaches to multilingual open-domain question answering (MLODQA) have achieved promising results given abundant language-specific training data. However, the considerable annotation cost limits the application of these methods for underrepresented languages. We introduce a \emph{few-shot learning} approach to synthesise large-scale multilingual data from large language models (LLMs). Our method begins with large-scale self-supervised pre-training using WikiData, followed by training on high-quality synthetic multilingual data generated by prompting LLMs with few-shot supervision. The final model, \textsc{FsModQA}, significantly outperforms existing few-shot and supervised baselines in MLODQA and cross-lingual and monolingual retrieval. We further show our method can be extended for effective zero-shot adaptation to new languages through a \emph{cross-lingual prompting} strategy with only English-supervised data, making it a general and applicable solution for MLODQA tasks without costly large-scale annotation.
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