Quantum-Circuit Framework for Two-Stage Stochastic Programming via QAOA Integrated with a Quantum Generative Neural Network

TL;DR

This work addresses two-stage stochastic programming under uncertainty by replacing explicit scenario enumeration with a quantum-circuit approach that encodes the uncertainty distribution in a quantum state via a pre-trained qGAN. The two-stage objective, including the expected recourse cost, is evaluated as the expectation value of a problem Hamiltonian and minimized using a two-stage QAOA framework, while non-anticipativity is preserved through measurement statistics. Key contributions include (i) a unified qGAN-QAOA workflow, (ii) a measurement-based formulation of non-anticipativity, (iii) a sparse Walsh–Hadamard based Pauli-$Z$ expansion of the random-variable operator yielding polylogarithmic scaling in the number of scenarios, and (iv) a detailed Hubbard-like HUBO mapping for a stochastic unit-commitment case under PV uncertainty with numerical demonstrations showing competitive performance against classical baselines. The results suggest a viable quantum-route to treat uncertainty as a distribution and to perform joint optimization of first- and second-stage decisions on a single variational circuit, with favorable scaling properties when scenario counts are large. Practical impact includes potential speedups for energy systems planning and other stochastic-design problems where scenario enumeration is a bottleneck, provided quantum hardware and error-mitigation advances align with the circuit requirements.

Abstract

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.

Figures (6)

• Figure 1: Quantum generator circuit $U_{\mathrm{qGAN}}(\boldsymbol{\theta})$ (TwoLocal circuit) used in qGAN. After applying an $H$ gate and a parameterized rotation $R_y(\theta)$ to each qubit, we alternate an entangling layer (a sequence of controlled gates) with an $R_y(\theta)$ layer. By stacking these layers, the circuit generates a quantum state whose amplitudes represent the scenario distribution parameterized by $\boldsymbol{\theta}$.
• Figure 2: Training procedure of qGAN. The quantum generator produces a probability distribution via the parameterized circuit $U_{\mathrm{qGAN}}(\boldsymbol{\theta})$, yielding the generated probability vector $\mathbf{p}_{\boldsymbol{\theta}}$. The classical discriminator $D_{\boldsymbol{\phi}}$ takes as input either the target distribution $\mathbf{p}_{\mathrm{data}}$ or the generated distribution $\mathbf{p}_{\boldsymbol{\theta}}$ and outputs the probability that the input originates from real data. Gradients of the loss are propagated to update the parameters $\bm{\theta},\bm{\theta}_{\mathrm{d}}$ iteratively.
• Figure 3: Schematic overview of the stochastic UCP for a power company owning uncertain PV generation. The power company supplies electricity to consumers by integrating generation from thermal power units (No. 1--3) and PV, while PV output is uncertain (pink), leading to an imbalance-adjustment power flow by the system operator (blue) and an imbalance cost (orange dashed line).
• Figure 4: Performance evaluation of the qGAN. (a) For each discretization (scenario) size $N=4,8,16,32$, we evaluate the agreement score based on the Jensen--Shannon (JS) divergence over five random seeds used to generate the real data; the median and the range from the minimum to the maximum are shown. (b) Comparison of the PV scenario probability distributions for $N=32$ between the real data (black line) and the generated data (blue line). We report the result with the highest agreement among the five qGAN training seeds.
• Figure 5: Evaluation of the optimization results obtained by qGAN-QAOA. (a) Marginal probability distribution of the first-stage measurement outcomes (interpreted as first-stage solutions) $\boldsymbol{x}_{\mathrm{1st}}$ obtained over 40 random seeds ($\lambda=30$). The horizontal axis denotes the bit string representing $\boldsymbol{x}_{\mathrm{1st}}$, where the bits from left to right indicate the on/off status of generators No. 1, 2, and 3, respectively (on$=1$/off$=0$), and the vertical axis denotes the measurement probability for each seed. The boxplot summarizes the distribution of the measurement probabilities (over 40 seeds) for each bit string: the box spans the first and third quartiles $Q_1$ and $Q_3$ (with $\mathrm{IQR}=Q_3-Q_1$), and the center line indicates the median. The whiskers extend to the minimum and maximum data points within the range $[Q_1-1.5\,\mathrm{IQR},\,Q_3+1.5\,\mathrm{IQR}]$, and points outside this range are shown as outliers (circles). The triangles indicate the mean measurement probability for each bit string. (b) Comparison of the expected cost as a function of the imbalance cost $\lambda$. The stochastic-programming baseline RP (black solid line) and EEV (black dashed line) are the evaluation values based on the $L_1$ formulation in Eq. \ref{['eq:uc_sp_obj_xi']} and Eq. \ref{['eq:eev']}, respectively. For qGAN-QAOA (green), for each $\lambda$ we fix the first-stage solutions obtained from 40 random seeds as representative solutions and compute the expected cost using the $L_1$ two-stage evaluation in Eq. \ref{['eq:Exp_cost_qGAN-QAOA']}; the mean (green solid line) and the range (minimum--maximum, shaded band) are shown. In the qGAN-QAOA optimization stage, for quantum-implementation reasons, we use the surrogate objective in Eq. \ref{['eq:HUBO_obj_core']}, where $\lvert \sigma \rvert$ is approximated by a quadratic penalty.

Theorems & Definitions (5)

• Proposition 3.1
• proof
• Proposition 3.2: Satisfaction of Req. \ref{['req:Quantum_non-anticipativity']}
• proof
• Proposition 6.1