We actually had an engine which was based on neural networks, called Giraffe. It used an alpha-beta search with a neural network evaluation. The Computer Chess Rating List put it at 2500 Elo, which is a strong human level. It was very slow, searching thousands of positions per second compared to the millions that even weak programs can do, but it's widely agreed that the NNs were worth about 300-400 elo - if Giraffe could search at Stockfish speeds, and given the rule of thumb that a doubling of speed is worth 70 elo, that's 2500 + 70 * log2(3000000/3000) = 3270 elo. That puts it in the top 20.
Sadly the developer was pinched by Google to work in DeepMind. We suspect this was to help work on AlphaGo.
The author, Matthew Lai, was not part of the AlphaGo team.
Current chess algorithm to evaluate board position is already very fast. A deep learning version needs to be trained for a long time and might still not be as fast. Maybe one day when training can be done faster and cheaper, there will be more interest in deep learning chess engines.
PS: The estimated hardware cost to train AlphaGo Zero is around $25M (2000 TPU over 40 days, or 1700 GPU years).
Sadly the developer was pinched by Google to work in DeepMind. We suspect this was to help work on AlphaGo.