Problem

Quantum circuits, serving as executable representations of quantum algorithms, hold significant intellectual property value, especially as quantum computing increasingly relies on cloud platforms. With the advent of generative artificial intelligence, the design paradigm for quantum circuits is shifting from manual construction to automated synthesis using Quantum Circuit Generative Models (QCGMs). This shift introduces new challenges for copyright protection, as individual circuit watermarking methods are insufficient and can compromise circuit quality or robustness. Existing post-hoc watermarking techniques, which modify circuit structures, struggle to integrate with generative processes, often failing to maintain stealthiness, functional correctness, and robustness against common post-generation manipulations like optimization or editing. Therefore, a novel watermarking approach is required that is inherently integrated with QCGMs to protect the generative process itself, ensuring all synthesized circuits carry consistent and verifiable ownership information.

Method

The Q-Tag framework introduces a novel approach to watermark Quantum Circuit Generative Models (QCGMs), consisting of three primary stages: watermark embedding, evaluation against watermark attacks, and watermark extraction. During the watermark embedding phase, a binary watermark message is first redundantly encoded using a combination of repetition coding and stream cipher encryption for error correction and pseudorandomness. This encoded sequence is then integrated into the latent input of the QCGM via a Symmetric Sampling Mechanism (SSM). The SSM is designed to ensure that the watermark information is injected while maintaining statistical alignment with the model's standard Gaussian prior, thereby preserving the fidelity and quality of the generated quantum circuits. The watermarked latent vector subsequently undergoes the QCGM's iterative denoising process, culminating in the generation of a quantum circuit with imperceptibly embedded ownership information.

For watermark extraction, the process begins by encoding a potentially attacked quantum circuit back into its latent representation. To counteract distortions introduced by structural modifications, a Synchronization Restoration Mechanism (SRM) is applied. This mechanism systematically inserts zero-initialized matrices at various column positions within the latent space, generating a set of candidate latents that are better aligned with the original watermark embedding structure. Following this, a Denoising Diffusion Implicit Model (DDIM) inversion procedure is used to approximate the original watermarked latent from these synchronized candidates. Finally, a reverse sampling procedure, mirroring the SSM, is performed, and error-correcting code decoding reconstructs the original binary watermark sequence. The extracted watermark is then compared against the original to verify ownership, with a predefined similarity threshold (e.g., 0.7916) determining successful identification.

Results

The Q-Tag framework demonstrates high fidelity and robust watermark detection across various conditions. Fidelity was theoretically proven and empirically confirmed through t-SNE visualizations, which showed no observable distributional bias between watermarked and non-watermarked latents, thus preserving the generation quality of quantum circuits. Robustness was primarily evaluated using the True Positive Rate (TPR).

Under clean conditions, the method achieved a consistently high TPR of at least 90%, with detection reliability increasing for circuits with more gates. Crucially, Q-Tag exhibited strong resilience against various adversarial attacks, including gate replacement, appending, insertion, and deletion. For gate replacement, TPR remained above 85%; for appending, it consistently exceeded 90%; for insertion, it was over 95%; and for deletion, it stayed above 90% across different configurations. Compared to existing structural watermarking methods like Roy et al., Q-Tag maintained watermark integrity and successful extraction under attack scenarios where other methods often failed. The framework also showed strong generalization, maintaining TPR above 91% even when varying the number of inference and inversion steps (10, 25, 50, 100) and remained robust with high TPR (>>95%) across different guidance scales (1.0, 2.5, 5.0, 7.5).

Regarding watermark capacity, increasing the embedding from 12 to 24 bits improved TPR; however, exceeding 24 bits led to a decline, attributed to reduced error correction redundancy. An ablation study specifically highlighted the critical contribution of the Synchronization Restoration Mechanism (SRM), showing that its inclusion significantly boosted TPR by over 40 percentage points under insertion and deletion attacks, underscoring its importance in maintaining detection reliability under adversarial conditions.

Implications

This work presents a foundational contribution to securing intellectual property in the evolving landscape of AI-powered quantum design by introducing the first watermarking framework specifically for Quantum Circuit Generative Models (QCGMs). Q-Tag successfully addresses the dual challenges of preserving generative fidelity through its symmetric sampling mechanism and ensuring robustness against structural perturbations via its synchronization restoration mechanism. Beyond its direct application to quantum circuits, the principle of resynchronizing misaligned generative trajectories offers a broadly applicable insight for enhancing watermark robustness in other structurally dynamic generative domains, such as image and video generation or neural program synthesis.

While establishing a strong baseline, the study also identifies avenues for future research, including the development of more adaptive and learning-based alignment strategies, more efficient and scalable embedding techniques, and cryptographically verifiable extraction protocols. Ultimately, this framework lays essential groundwork for fostering secure, verifiable, and accountable content creation as AI increasingly underpins quantum software development and design.