Knapsack Problem variants of QAOA for battery revenue optimisation

We implement two Quantum Approximate Optimisation Algorithm (QAOA) variants for a battery revenue optimisation problem, equivalent to the weakly NP-hard Knapsack Problem. Both approaches investigate how to tackle constrained problems with QAOA. A first 'constrained' approach introduces a quadratic penalty to enforce the constraint to be respected strictly and reformulates the problem into an Ising Problem. However, simulations on IBM's simulator highlight non-convergent results for intermediate depth ($p\leq 50$). A second 'relaxed' approach applies the QAOA with a non-Ising target function to compute a linear penalty, running in time $O(p(\log_2 n)^3)$ and needing $O(n \log n)$ qubits. Simulations reveal an exponential improvement over the number of depth levels and obtain approximations about $0.95$ of the optimum with shallow depth ($p \leq 10$).