ConQuER: Modular Architectures for Control and Bias Mitigation in IQP Quantum Generative Models

TL;DR

ConQuER tackles the lack of controllability and the generation bias in IQP-based quantum generative models by introducing a lightweight distribution controller that couples to pre-trained IQP generators. By exploiting IQP gate commutativity, the controller achieves additive parameter control for conditional generation, while a data-driven, implicit embedding approach mitigates bias without retraining the base circuit, preserving polynomial-time classical training. Theoretical foundations include an additive effective parameter model $\theta_{\text{eff}}=\theta+\phi$, an efficient MMD loss decomposition $\text{MMD}=\sum_k \alpha_k \langle Z_{S_k} \rangle_{\text{combined}}$, and a CRZ-based gate representation; these ensure scalability to thousands of qubits. Empirically, ConQuER demonstrates precise control of Hamming-weight distributions and substantial bias reduction on Ising-like and structured binary data, with minimal controller overhead that grows sublinearly with system size, enabling practical train-on-classical, deploy-on-quantum workflows.

Abstract

Quantum generative models based on instantaneous quantum polynomial (IQP) circuits show great promise in learning complex distributions while maintaining classical trainability. However, current implementations suffer from two key limitations: lack of controllability over generated outputs and severe generation bias towards certain expected patterns. We present a Controllable Quantum Generative Framework, ConQuER, which addresses both challenges through a modular circuit architecture. ConQuER embeds a lightweight controller circuit that can be directly combined with pre-trained IQP circuits to precisely control the output distribution without full retraining. Leveraging the advantages of IQP, our scheme enables precise control over properties such as the Hamming Weight distribution with minimal parameter and gate overhead. In addition, inspired by the controller design, we extend this modular approach through data-driven optimization to embed implicit control paths in the underlying IQP architecture, significantly reducing generation bias on structured datasets. ConQuER retains efficient classical training properties and high scalability. We experimentally validate ConQuER on multiple quantum state datasets, demonstrating its superior control accuracy and balanced generation performance, only with very low overhead cost over original IQP circuits. Our framework bridges the gap between the advantages of quantum computing and the practical needs of controllable generation modeling.

Figures (5)

• Figure 1: ConQuER framework overview in the example of 4 qubits. (A) Original IQP circuit pre-trained on target data. (B) ConQuER controller trained independently with the pre-trained IQP parameters. (C) Direct connection approach: controller parameters $\theta$ are added to pre-trained parameters $X_n$, enabling conditional generation. While two consecutive H gates would cancel during transpilation in quantum deployment, we retain them here for visual clarity and modularity. (D) Implicit connection approach: controller structure is embedded within the IQP topology for bias mitigation. Both approaches preserve the IQP structure while achieving different distribution control objectives.
• Figure 2: Parameter influence analysis on 16-qubit systems. (a) Learned parameter magnitudes across qubit pairs in pre-trained IQP circuit. (b) Qubit-wise parameter importance for each of 8 binary patterns. Darker colors indicate larger values.
• Figure 3: Hamming weight (HW) control on 2D Ising model. (a) Baseline IQP shows bimodal distribution. (b-d) ConQuER controls output distributions: (b) Low HW mode (81.36% in 0-5 range), (c) High HW mode (80.5% in 11-16 range), (d) Balanced mode (centered at weight 8, $\sigma$=1.99).
• Figure 4: Bias mitigation on binary blob dataset. ConQuER (implicit connection) achieves more uniform distribution across 8 patterns compared to baseline IQP. ConQuER reduces pattern STD by 18.1%, max/min ratio by 17.6%, and total deviation by 10.7%.
• Figure 5: Controller overhead v.s. Ising model size for different numbers of control modes. Solid lines show measured data for 4×4 and 5×5 models, dashed lines show projections for larger systems. The main plot demonstrates logarithmic decrease in overhead percentage as system size increases. Inset shows the same data on a linear scale for the measured systems.