  Deep learning has rapidly become a widespread tool in both scientific and
commercial endeavors. Milestones of deep learning exceeding human performance
have been achieved for a growing number of tasks over the past several years,
across areas as diverse as game-playing, natural-language translation, and
medical-image analysis. However, continued progress is increasingly hampered by
the high energy costs associated with training and running deep neural networks
on electronic processors. Optical neural networks have attracted attention as
an alternative physical platform for deep learning, as it has been
theoretically predicted that they can fundamentally achieve higher energy
efficiency than neural networks deployed on conventional digital computers.
Here, we experimentally demonstrate an optical neural network achieving 99%
accuracy on handwritten-digit classification using ~3.2 detected photons per
weight multiplication and ~90% accuracy using ~0.64 photons (~$2.4 \times
10^{-19}$ J of optical energy) per weight multiplication. This performance was
achieved using a custom free-space optical processor that executes
matrix-vector multiplications in a massively parallel fashion, with up to ~0.5
million scalar (weight) multiplications performed at the same time. Using
commercially available optical components and standard neural-network training
methods, we demonstrated that optical neural networks can operate near the
standard quantum limit with extremely low optical powers and still achieve high
accuracy. Our results provide a proof-of-principle for low-optical-power
operation, and with careful system design including the surrounding electronics
used for data storage and control, open up a path to realizing optical
processors that require only $10^{-16}$ J total energy per scalar
multiplication -- which is orders of magnitude more efficient than current
digital processors.
