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arXiv:2402.15029

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

Caleb Rotello, Peter A Graf, Reynolds, Matthew, Jones, Eric B., Cody James Winkleblack, Jones, Wesley

Published Feb 23, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2402.15029

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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.
Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.
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Copyright © 2020 by SIAM Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference.
Two-stage stochastic programming often discretizes uncertainty into scenarios, but scenario enumeration makes expected recourse evaluation scale at least linearly in the scenario count. We propose qGAN-QAOA, a unified quantum-circuit workflow in which a pre-trained quantum generative adversarial network encodes the scenario distribution and QAOA optimizes first-stage decisions by minimizing the full two-stage objective, including expected recourse cost. With the qGAN parameters fixed after training, we evaluate the objective as the expectation value of a problem Hamiltonian and optimize only the QAOA variational parameters. We interpret non-anticipativity as a condition on measurement outcome statistics and prove that the first-stage measurement marginal is independent of the scenario. For uniformly discretized uncertainty, the diagonal operator encoding the uncertainty admits a sparse Pauli-Z expansion via the Walsh--Hadamard transform, yielding polylogarithmic scaling of gate count and circuit depth with the number of scenarios. Numerical experiments on the stochastic unit commitment problem (UCP) with photovoltaic (PV) uncertainty compare the expected cost of the proposed method with classical expected-value and two-stage stochastic programming baselines, demonstrating the effectiveness of qGAN-QAOA as a two-stage decision model.