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We implement two Quantum Approximate Optimisation Algorithm (QAOA) variantsfor a battery revenue optimisation problem, equivalent to the weakly NP-hardKnapsack Problem. Both approaches investigate how to tackle constrainedproblems with QAOA. A first 'constrained' approach introduces a quadraticpenalty to enforce the constraint to be respected strictly and reformulates theproblem into an Ising Problem. However, simulations on IBM's simulatorhighlight non-convergent results for intermediate depth ($ p\leq 50$). A second'relaxed' approach applies the QAOA with a non-Ising target function to computea 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 depthlevels and obtain approximations about $0.95$ of the optimum with shallow depth($p \leq 10$).

Knapsack Problem variants of QAOA for battery revenue optimisation