Autonomously Stabilizing Bell States of Remote Logical Qubits in Circuit QED

Reservoir engineering offers a powerful route to autonomous error correction and quantum state stabilization. Recent breakthrough experiments in circuit QED have achieved the dissipative stabilization of logical states in bosonic systems, as well as remote entanglement between transmon qubits. This naturally raises the question of whether one could autonomously stabilize remote pairs of logical qubits (e.g. cat quits, GKP qubits) into maximally-entangled Bell states. A priori, methods used for stabilizing entanglement of simple physical qubits do not seem feasible, given the difficult of realizing the needed interactions between structured dissipation and the logical qubits. Here, we present a solution to this problem using a modified approach. We propose a gate-based approach that uses weakly entangled dissipative ancilla qubits to mediate effective coupling between logical systems, enabling autonomous stabilization of logical Bell states. We demonstrate this method for both cat-Bell and GKP-Bell encodings, showing that it provides a universal and experimentally feasible framework for modular quantum computing with error-correcting codes.