  Quantum Annealing (QA) is one of the most promising frameworks for quantum
optimization. Here, we focus on the problem of minimizing complex classical
cost functions associated with prototypical discrete neural networks,
specifically the paradigmatic Hopfield model and binary perceptron. We show
that the adiabatic time evolution of QA can be efficiently represented as a
suitable Tensor Network. This representation allows for simple classical
simulations, well-beyond small sizes amenable to exact diagonalization
techniques. We show that the optimized state, expressed as a Matrix Product
State (MPS), can be recast into a Quantum Circuit, whose depth scales only
linearly with the system size and quadratically with the MPS bond dimension.
This may represent a valuable starting point allowing for further circuit
optimization on near-term quantum devices.
