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In quantum information processing, two primary research directions have emerged: one based on discrete variables (DV) and the other on the structure of quantum states in a continuous-variable (CV) space. It is increasingly recognized that integrating these two approaches could unlock new potentials, overcoming the inherent limitations of each. Here, we show that such a DV-CV hybrid approach, applied to superconducting Kerr parametric oscillators (KPOs), enables us to entangle a pair of Schrödinger's cat states by two straightforward methods. The first method involves the entanglement-preserving and deterministic conversion between Bell states in the Fock-state basis (DV encoding) and those in the cat-state basis (CV encoding). This method would allow us to construct quantum networks in the cat-state basis using conventional schemes originally developed for the Fock-state basis. In the second method, the $\sqrt{\textrm{iSWAP}}$ gate operation is implemented between two cat states following the procedure used for Fock-state encoding. This DV-like gate operation on CV encoding not only completes the demonstration of a universal quantum gate set in a KPO system but also enables faster and simpler gate operations compared to previous SWAP gate implementations on bosonic modes. Our work offers a simple yet powerful application of DV-CV hybridization while also highlighting the scalability of this planar KPO system.

Quantum tunneling is the phenomenon that makes superconducting circuits "quantum". Recently, there has been a renewed interest in using quantum tunneling in phase space of a Kerr parametric oscillator as a resource for quantum information processing. Here, we report a direct observation of quantum interference induced by such tunneling in a planar superconducting circuit through Wigner tomography. We experimentally elucidate all essential properties of this quantum interference, such as mapping from Fock states to cat states, a temporal oscillation due to the pump detuning, as well as its characteristic Rabi oscillations and Ramsey fringes. Finally, we perform gate operations as manipulations of the observed quantum interference. Our findings lay the groundwork for further studies on quantum properties of superconducting Kerr parametric oscillators and their use in quantum information technologies.