Encoding quantum information in non-classical states of harmonic oscillators (bosonic modes) was among the first conceptual schemes for the realization of quantum computers. In the past decade, quantum control and measurement of bosonic degrees of freedom have been reliably implemented in superconducting microwave circuits with Josephson junctions. Superconducting bosonic qubits are a leading platform for demonstrating error-correctable quantum logical memories in a hardware-efficient architecture, which opens a highly innovative subfield in superconducting quantum technology.

Compared to other architectures for fault-tolerant quantum processors - for instance, those based on the planar integration of two-level quantum systems (physical qubits) to achieve topological protection of quantum information (e.g., surface codes) - superconducting bosonic qubits show a list of main advantages:

• Bosonic modes provide more energy levels for the redundant encoding of quantum information, without introducing extra hardware elements or noise/decoherence channels
• Linear resonators typically have longer intrinsic physical lifetimes than nonlinear Josephson-junction elements
• The loss mechanism in bosonic quantum memories is dominated by photon dissipation, which can be detected and/or compensated through various quantum error correction protocols
• Superconducting bosonic qubits are compatible with autonomous (feedback-free) quantum-state stabilization through Hamiltonian engineering for driven-dissipative open quantum systems in addition to measurement-based quantum error correction schemes

Major challenges of superconducting bosonic qubits include designing and implementing more robust and efficient quantum error correction codes, improving single- and two-logical-qubit gate fidelities as well as the connectivity and scalability of logical quantum modules.