Calculating the expected value function of a two-stage stochastic optimization program with a quantum algorithm

Two-stage stochastic programming is a problem formulation for decision-making under uncertainty. In the first stage, the actor makes a best "here and now" decision in the presence of uncertain quantities that will be resolved in the future, represented in the objective function as the expected value function. This function is a multi-dimensional integral of the second stage optimization problem, which must be solved over all possible future scenarios. This work uses a quantum algorithm to estimate the expected value function with a polynomial speedup. Our algorithm gains its advantage through the two following observations. First, by encoding the probability distribution as a quantum wavefunction in an auxilliary register, and using this register as control logic for a phase-separation unitary, Digitized Quantum Annealing (DQA) can converge to the minimium of each scenario in the random variable in parallel. Second, Quantum Amplitude Estimation (QAE) on DQA can calculate the expected value of this per-scenario optimized wavefunction, producing an estimate for the expected value function. Quantum optimization is theorized to have a polynomial speedup for combinatorial optimization problems, and estimation error from QAE is known to converge inverse-linear in the number of samples (as opposed to the best case inverse of a square root in classical Monte Carlo). Therefore, assuming the probability distribution wavefunction can be prepared efficiently, we conclude our method has a polynomial speedup (of varying degree, depending on the optimization problem) over classical methods for estimating the expected value function. We conclude by demonstrating this algorithm on a stochastic programming problem inspired by operating the power grid under weather uncertainty.