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Quantum technologies, such as quantum communication, sensing and imaging, need a platform which is flexible, miniaturizable and works at room temperature. Integrated photonics is a promising and fast-developing platform. This requires to develop the right tools to design and fabricate arbitrary photonic quantum devices. Here we present an algorithm which, starting from a $n$-mode transformation $U$, designs a photonic device implementing $U$. Using this method we design integrated photonic devices which implement quantum gates with high fidelity. Apart from quantum computation, future applications include the design of photonic subroutines or embedded quantum devices. These custom-designed photonic devices will implement in a single step a given algorithm and will be small, robust and fast compared to a fully-programmable processor.