使用量子算法计算两阶段随机优化程序的期望值函数
Calculating the expected value function of a two-stage stochastic optimization program with a quantum algorithm
AI总结：
针对两阶段随机规划中期望值函数的多维积分计算难题，提出利用Digitized Quantum Annealing (DQA) 并行求解场景最小值并结合Quantum Amplitude Estimation (QAE) 估计期望值的量子算法，实现了相对经典方法的多项式加速。
摘要：
两阶段随机规划是不确定性下决策的问题表述。在第一阶段，决策者在存在未来将解决的不确定量的情况下做出最佳的“此时此地”决策，这在目标函数中表示为期望值函数。该函数是第二阶段优化问题的多维积分，必须在所有可能的未来场景下求解。本研究使用量子算法来估计期望值函数，并实现了多项式加速。该算法的优势来源于以下两个观察。首先，通过将概率分布编码为辅助寄存器中的量子波函数，并使用该寄存器作为相位分离酉算子的控制逻辑，Digitized Quantum Annealing (DQA) 可以并行地收敛于随机变量中每个场景的最小值。其次，在 DQA 上进行 Quantum Amplitude Estimation (QAE) 可以计算这种按场景优化的波函数的期望值，从而产生期望值函数的估计。理论上，量子优化在组合优化问题上具有多项式加速，并且已知 QAE 的估计误差在样本数上呈逆线性收敛（相对于经典 Monte Carlo 最佳情况下的平方根倒数）。因此，假设概率分布波函数可以被高效地准备，我们得出结论：与估计期望值函数的经典方法相比，我们的方法具有多项式加速（加速程度因优化问题而异）。最后，我们通过在受天气不确定性下电网运行启发的随机规划问题上演示该算法作为总结。
英文摘要：
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.
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