  We introduce the Backwards Quantum Propagation of Phase errors (Baqprop)
principle, a central theme upon which we construct multiple universal
optimization heuristics for training both parametrized quantum circuits and
classical deep neural networks on a quantum computer. Baqprop encodes error
information in relative phases of a quantum wavefunction defined over the space
of network parameters; it can be thought of as the unification of the phase
kickback principle of quantum computation and of the backpropagation algorithm
from classical deep learning. We propose two core heuristics which leverage
Baqprop for quantum-enhanced optimization of network parameters: Quantum
Dynamical Descent (QDD) and Momentum Measurement Gradient Descent (MoMGrad).
QDD uses simulated quantum coherent dynamics for parameter optimization,
allowing for quantum tunneling through the hypothesis space landscape. MoMGrad
leverages Baqprop to estimate gradients and thereby perform gradient descent on
the parameter landscape; it can be thought of as the quantum-classical analogue
of QDD. In addition to these core optimization strategies, we propose various
methods for parallelization, regularization, and meta-learning as augmentations
to MoMGrad and QDD. We introduce several quantum-coherent adaptations of
canonical classical feedforward neural networks, and study how Baqprop can be
used to optimize such networks. We develop multiple applications of parametric
circuit learning for quantum data, and show how to perform Baqprop in each
case. One such application allows for the training of hybrid quantum-classical
neural-circuit networks, via the seamless integration of Baqprop with classical
backpropagation. Finally, for a representative subset of these proposed
applications, we demonstrate the training of these networks via numerical
simulations of implementations of QDD and MoMGrad.
