Title: Multi-Stage Watermarking for Quantum Circuits

Abstract: Quantum computing represents a burgeoning computational paradigm that significantly advances the resolution of contemporary intricate problems across various domains, including cryptography, chemistry, and machine learning. Quantum circuits tailored to address specific problems have emerged as critical intellectual properties (IPs) for quantum computing companies, attributing to the escalating commercial value of quantum computing. Consequently, designing watermarking schemes for quantum circuits becomes imperative to thwart malicious entities from producing unauthorized circuit replicas and unlawfully disseminating them within the market.
Unfortunately, the prevailing watermarking technique reliant on unitary matrix decomposition markedly inflates the number of 2-qubit gates and circuit depth, thereby compromising the fidelity of watermarked circuits when embedding detectable signatures into the corresponding unitary matrices. In this paper, we propose an innovative multi-stage watermarking scheme for quantum circuits, introducing additional constraints across various synthesis stages to validate the ownership of IPs. Compared to the state-of-the-art watermarking technique, our multi-stage watermarking approach demonstrates, on average, a reduction in the number of 2-qubit gates by 16\% and circuit depth by 6\%, alongside an increase in the fidelity of watermarked circuits by 8\%, while achieving a 79.4\% lower probabilistic proof of authorship.