  Neural networks are one of the disruptive computing concepts of our time.
However, they fundamentally differ from classical, algorithmic computing in a
number of fundamental aspects. These differences result in equally fundamental,
severe and relevant challenges for neural network computing using current
computing substrates. Neural networks urge for parallelism across the entire
processor and for a co-location of memory and arithmetic, i.e. beyond von
Neumann architectures. Parallelism in particular made photonics a highly
promising platform, yet until now scalable and integratable concepts are
scarce. Here, we demonstrate for the first time how a fully parallel and fully
implemented photonic neural network can be realized using spatially distributed
modes of an efficient and fast semiconductor laser. Importantly, all neural
network connections are realized in hardware, and our processor produces
results without pre- or post-processing. 130+ nodes are implemented in a
large-area vertical cavity surface emitting laser, input and output weights are
realized via the complex transmission matrix of a multimode fiber and a digital
micro-mirror array, respectively. We train the readout weights to perform 2-bit
header recognition, a 2-bit XOR and 2-bit digital analog conversion, and obtain
< 0.9 10^-3 and 2.9 10^-2 error rates for digit recognition and XOR,
respectively. Finally, the digital analog conversion can be realized with a
standard deviation of only 5.4 10^-2. Our system is scalable to much larger
sizes and to bandwidths in excess of 20 GHz.
