Neural networks have proven effective for solving many difficult computational problems. Implementing complex neural networks in software is very computationally expensive. To explore the limits of information processing, it will be necessary to implement new hardware platforms with large numbers of neurons, each with a large number of connections to other neurons. Here we propose a hybrid semiconductor-superconductor hardware platform for the implementation of neural networks and large-scale neuromorphic computing. The platform combines semiconducting few-photon light-emitting diodes with superconducting-nanowire single-photon detectors to behave as spiking neurons. These processing units are connected via a network of optical waveguides, and variable weights of connection can be implemented using several approaches. The use of light as a signaling mechanism overcomes fanout and parasitic constraints on electrical signals while simultaneously introducing physical degrees of freedom which can be employed for computation. The use of supercurrents achieves the low power density necessary to scale to systems with enormous entropy. The proposed processing units can operate at speeds of at least $20$ MHz with fully asynchronous activity, light-speed-limited latency, and power densities on the order of 1 mW/cm$^2$ for neurons with 700 connections operating at full speed at 2 K. The processing units achieve an energy efficiency of $\approx 20$ aJ per synapse event. By leveraging multilayer photonics with deposited waveguides and superconductors with feature sizes $>$ 100 nm, this approach could scale to systems with massive interconnectivity and complexity for advanced computing as well as explorations of information processing capacity in systems with an enormous number of information-bearing microstates.