Superconducting quantum circuits have emerged as one of the leading technology platforms for achieving a near‑term quantum advantage in various information processing tasks. This advancement has been based on the successful scaling of superconducting qubit numbers over a hundred transmons, improving superconducting circuit designs, and exploiting novel materials. As a result, several research teams have demonstrated reaching break-even points in quantum error corrections protocols or even beyond. At the same time, further progress in quantum computing using single-chip architectures is becoming more daunting because of qubit crosstalks as well as the spatial limitations of modern dilution fridges. A well-known recipe for solving these problems originates from the classical supercomputing architectures - to rely on a distributed, or modular, information processing approach. In application to quantum circuits, it means distributing individual quantum circuits with a moderate number of qubits, or other quantum systems, across individual cryogenic modules and, furthermore, in separate dilution fridges. The arising challenge is to find the most efficient ways of entangling and coherently manipulating these remote quantum modules across local area quantum networks. The task of establishing remote entanglement between individual qubits and implementing remote quantum gates represents a basic building block for future distributed quantum networks. Here, we intend to study and implement several key experiments towards realizing this task.

A particularly novel way to distribute entanglement between quantum nodes is to exploit Gaussian entangled states. In these protocols, one relies on a distributed two-mode entangled state, or bath, which may be off‑resonantly coupled to the remote qubits, thus avoiding some limitations of the QST approach. Moreover, theoretical works predict a possibility of entanglement distillation in this regime which allows to overcome moderate transmission losses. The necessary entangled two-mode states for these remote entanglement protocols can be conveniently generated with squeezed vacuum states generated by Josephson parametric amplifiers (JPAs) in the steady-state regime. Figure 1(a),(c) illustrates a particular flux-driven JPA design and the Wigner function of a related continuous-variable (CV) squeezed vacuum state. Discrete-variable (DV) Fock states correspond to the computational basis of modern gate-based quantum computing platforms and are routinely generated by superconducting qubits. Figure 1(b)(d) shows a particular flux qubit and the Wigner function of a single-photon DV state. A combination of the DV states emitted by qubits with the squeezed CV states forms the basis for the novel hybrid-variable regime of quantum communication. We imply that in hybrid-variable experiments, the corresponding CV and DV quantum  states may be freely superimposed or mixed during various operations and quantum gates. The HVMQC goal is to develop fundamental understanding how quantum correlations are converted between continuous and discrete degrees of freedom, how various quantum states thermalize in nonlinear noisy networks, with a possibility of finding quantum scar states in the microwave regime. From the practical side, the realization of hybrid-variable microwave quantum communication protocols is directly connected to implementation of distributed quantum computing architectures and promises to resolve the challenge of superconducting qubit scaling beyond the modern thresholds imposed by individual cryostats.