qGAN-QAOA: Quantum Two-Stage Framework

• qGAN-QAOA is a unified quantum circuit framework that integrates a pre-trained quantum GAN for scenario encoding with QAOA for optimizing first-stage decisions.
• It leverages a two-stage variational quantum circuit to exactly realize expected recourse objectives while enforcing non-anticipativity between decisions and scenarios.
• The framework achieves polylogarithmic scaling in scenario encoding using efficient Walsh–Hadamard transforms and outperforms classical baselines on stochastic unit commitment problems.

qGAN-QAOA is a unified quantum circuit framework for two-stage stochastic programming, integrating a pre-trained @@@@1@@@@ (qGAN) for scenario encoding and the Quantum Approximate Optimization Algorithm (QAOA) for variational optimization of first-stage decisions. This architecture directly addresses the classical bottleneck where explicit scenario enumeration yields linear scaling in the number of samples, by compressing scenario dependence to polylogarithmic scaling under uniform variable discretization. The method provides exact circuit realizations for expected recourse objectives while enforcing non-anticipativity at the quantum level. The framework is demonstrated on the stochastic unit commitment problem (UCP) with non-Gaussian (Beta-distributed) renewable uncertainty and benchmarked against classical stochastic programming baselines (Kuroiwa et al., 27 Dec 2025).

1. Circuit Architecture and Workflow

The qGAN-QAOA workflow comprises a data-driven scenario encoder and a two-stage quantum variational circuit:

• Scenario Encoding: A discrete empirical scenario distribution $p_s = \mathbb{P}_{\mathrm{data}}(\xi_s)$ over $N=2^{n_\xi}$ outcomes is encoded via a qGAN generator $U_{\mathrm{qGAN}}(\theta)$ acting on $n_\xi$ qubits. Training follows the adversarial minimax loss

$$\min_\theta \max_\phi \mathcal{L}(\theta,\phi) = \mathbb{E}_{p \sim p_{\mathrm{data}}}[\log D_\phi(p)] + \mathbb{E}_{p \sim p_\theta}[\log(1 - D_\phi(p))],$$

with $$p_\theta(\xi_s) = |\langle b_s^\xi| U_{\mathrm{qGAN}}(\theta)|0\rangle^{\otimes n_\xi}|^2$$. After convergence, the generator is fixed at $\theta^*$, so $$U_{\mathrm{qGAN}}(\theta^*)|0\rangle^{\otimes n_\xi} = \sum_{s=0}^{N-1} \sqrt{p_s}\,|b_s^\xi\rangle$$.

• QAOA Variational Circuit: Three quantum registers are used: scenario ($\mathcal{H}_\xi$: $n_\xi$ qubits), first-stage decision ($\mathcal{H}_{\mathrm{1st}}$: $n_{\mathrm{1st}}$ qubits), and second-stage recourse ($\mathcal{H}_{\mathrm{2nd}}$: $n_{\mathrm{2nd}}$ qubits). First-stage and second-stage variables (continuous or discrete) are encoded with fixed-point and on/off binary expansions.
• Hamiltonians: The overall cost objective is encoded as a problem Hamiltonian $H_P$ composed of first-stage and recourse terms. Scenario dependence is localized via a diagonal operator on the scenario register,

$$\hat{\xi} = \sum_{s=0}^{N-1} \xi_s |b_s^\xi\rangle\langle b_s^\xi|.$$

Mixing Hamiltonians act locally on each register.

• QAOA Layers: The first-stage register is optimized independently, after which second-stage layers entangle decision and scenario registers. Measurement yields a superposition whose outcome statistics reconstruct the two-stage cost.

The combined variational state after all circuit layers is

$$|\Psi(\theta)\rangle = \sum_{k,s} \alpha_k \sqrt{p_s}\,|\psi_{s,k}^{\mathrm{2nd}}(\theta)\rangle \otimes |b_k^{\mathrm{1st}}\rangle \otimes |b_s^\xi\rangle,$$

with $\alpha_k$ determined by first-stage QAOA parameters and $|\psi_{s,k}^{\mathrm{2nd}}\rangle$ by second-stage/recourse subroutines.

2. Two-Stage Objective as Quantum Expectation

The classical two-stage stochastic programming (SP) objective,

$$C_{\mathrm{2stage}}(x) = c_1(x) + \mathbb{E}_\omega \big[ c_2(x, y(\omega)) \big],$$

is mapped to a quantum circuit expectation: $$\mathcal{L}(\theta) = \langle\Psi(\theta)| H_P | \Psi(\theta)\rangle.$$ Explicitly,

$$\langle\Psi|H_P|\Psi\rangle = \sum_k |\alpha_k|^2 \left[ H_P^{1\mathrm{st}}(x_k) + \sum_s p_s\, Q(x_k, \xi_s) \right],$$

where $$Q(x_k, \xi_s) = \langle\psi_{s,k}^{\mathrm{2nd}} | H_P^{2\mathrm{nd}} (y, x_k, \xi_s) | \psi_{s,k}^{\mathrm{2nd}}\rangle$$ is the quantum recourse cost under scenario $s$. Measurement over the registers reconstructs the joint distribution $(x, \xi, y)$, allowing classical sampling or quantum amplitude estimation (QAE) to converge to $\mathbb{E}[c_1(x) + c_2(x, y(\omega))]$.

3. Quantum Non-Anticipativity

Non-anticipativity in two-stage SP requires first-stage decisions to be scenario-independent. In the qGAN-QAOA circuit, this is realized as statistical independence between the first-stage measurement outcomes $B^{\mathrm{1st}}$ and the scenario register $B^\xi$: $$\forall\,k,s:~\mathbb{P}\big(B^{\mathrm{1st}}=b_k^{\mathrm{1st}} \mid B^\xi=b_s^\xi\big) = \mathbb{P}\big(B^{\mathrm{1st}}=b_k^{\mathrm{1st}}\big).$$ This follows because the first-stage QAOA block does not act on the scenario register, and second-stage/recourse blocks entangle the registers only after first-stage amplitudes are fixed. Thus,

$$\mathbb{P}\big(B^{\mathrm{1st}}=b_k \wedge B^\xi=b_s\big) = |\alpha_k|^2 p_s,$$

implying $$\mathbb{P}(B^{\mathrm{1st}}, B^\xi) = \mathbb{P}(B^{\mathrm{1st}})\,\mathbb{P}(B^\xi)$$ and enforcing non-anticipativity by construction.

4. Scenario Representation via Walsh–Hadamard Transform

Efficient scenario encoding is achieved using the Walsh–Hadamard transform (WHT) to sparsify the diagonal uncertainty operator. Any $N \times N$ diagonal matrix (such as $\hat\xi$) on $n_\xi = \log_2 N$ qubits has a Pauli-Z expansion: $\hat\xi = \sum_j c_j P_j,$ where $P_j$ are tensor products of $Z$ (Pauli Z) operators, and $c = \frac{1}{N} H_{n_\xi} \xi$ with $\xi = (\xi_0, ..., \xi_{N-1})^T$. If $\xi_s$ is uniformly discretized ($\xi_s = \xi_{\min} + s \Delta\xi$), all coefficients $c_j$ with Hamming weight $w(j) > 1$ vanish; only $n_\xi + 1 = O(\log N)$ Pauli-Z strings are nonzero: $$c_0 = \xi_{\min} + \Delta\xi\frac{2^{n_\xi} - 1}{2},\quad c_{2^i} = -\Delta\xi\,2^{i-1},\quad i = 0, ..., n_\xi - 1.$$ Each such operator is efficiently synthesized with $O(m)$ gates (for string weight $m$). This allows all scenario-dependent Hamiltonian operations to admit $O(\mathrm{poly}(\log N))$ gate scaling, instead of $O(N)$, subject to uniform discretization.

5. Numerical Experiments: Stochastic Unit Commitment

Numerical demonstration uses the three-unit stochastic UCP with Beta-distributed photovoltaic (PV) uncertainty:

• Setup: $M = 3$ thermal units (on/off first-stage variables $x_i$), deterministic demand $D = 2500$ kWh, and PV capacity factor $CF \sim$ Beta$(3,7)$. The scenario variable $\xi$ is discretized into $N=2^{n_\xi}$ bins ($n_\xi=2$ to 5; $N = 4,8,16,32$). First-stage feasible sets are binary; recourse includes discrete $y_{i,s}$ and slack $\sigma_s$ penalized quadratically ($\lambda \sigma_s^2$).
• qGAN Training: For each $N$, the qGAN is trained on 2000 continuous Beta samples binned into $N$ scenarios, using 10,000 shots per epoch. The final amplitude encoding achieves Jensen–Shannon divergence agreement $>0.994$ for all $N$.
• QAOA Optimization: The circuit consists of 11 qubits (5 scenario, 3 first-stage, 3 second-stage), QAOA depths $p_1 = p_2 = 4$. Optimization is performed via COBYLA (maximum 400 iterations, 50,000 shots per evaluation) across $40$ seeds and $\lambda$ values from 30 to 200.
• Results: First-stage marginals $P(x)$ are sharply concentrated on the two most plausible solutions, with moderate initialization sensitivity. Fixing $x_\mathrm{opt}$ at each seed and evaluating the classical expected cost over 200 out-of-sample scenarios, qGAN-QAOA costs closely agree with the recourse problem optimum and consistently outperform the classical expected-value evaluation (EEV). Gate count and transpiled depth for all scenario-related blocks exhibit $O(\mathrm{poly}(\log N))$ scaling.

6. Summary of Capabilities and Computational Scaling

qGAN-QAOA enables two-stage stochastic optimization on quantum circuits with:

• Data-driven, non-Gaussian scenario distributions encoded via amplitude amplification in $O(\mathrm{poly}(n_\xi))$ gates.
• Exact realization of expected-recoursed costs as a quantum circuit expectation.
• Structural enforcement of non-anticipativity, with provable scenario independence of first-stage marginals.
• Polylogarithmic ($O(\log N)$) scaling in the number of scenarios for uncertainty operators, enabled by the Walsh–Hadamard transform.
• Empirical convergence to near-optimal first-stage decisions and expected costs on stochastic UCP benchmarks, with moderate dependence on initialization and favorable gate complexity scaling (Kuroiwa et al., 27 Dec 2025).