Abstract： Circuit Quantum Electrodynamics (cQED) is a very promising approach for quantum information processing. Inevitable noise is still the main limitation for practical quantum computers. We can understand the quantum noise and decoherence through study of the quantum channels on the one hand, and on the other hand, we can protect quantum information by quantum encoding and quantum error correction (QEC). Those studies calls for high fidelity control and fast speed real-time feedback, which is very hard in experiment. Fortunately, Field-Programmable Gate Array (FPGA) is programmable and fast speed, more importantly, it can do real time feedback. Therefore, it has been widely used in cQED.

Logical qubit encoding and QEC have been experimentally demonstrated in various physical systems with multiple physical qubits. However, logical operations are challenging due to the necessary nonlocal operations. Alternatively, logical qubits with bosonic-mode-encoding are of particular interest because their QEC protection is hardware efficient. It requires the minimum hardware, just a microwave cavity coupled to an ancilla qubit, through which we can do all kind of operations to it. Moreover, a bosonic system has infinite dimensional Hilbert space, which not only is helpful in the study of the quantum channels and QEC, but also can be used to study the quantum machine learning and the application of quantum computers.

We realize an FPGA control system for a 3-dimensional (3D) two cavity one ancilla qubit cQED system, and provide the development methods. With this system, we experimentally demonstrate full control on a single logical qubit with a binomial bosonic code, including encoding, decoding, repetitive QEC, and high-fidelity (97.0 % process fidelity on average) universal quantum gate set on the logical qubit, and we demonstrate a Ramsey experiment on a protected logical qubit for the first time; simulate arbitrary quantum channels for an open quantum system, i.e. a single photonic qubit in a superconducting quantum circuit, realize an arbitrary Liouvillian for a continuous evolution of an open quantum system for the first time; report the first proof-of-principle experimental demonstration of quantum generative adversarial learning in a superconducting quantum circuit. Our results, represent an important step towards fault-tolerant quantum computation based on bosonic encoding; provides not only a testbed for understanding quantum noise and decoherence, but also a powerful tool for full control of practical open quantum systems; pave the way for experimentally exploring the intriguing long-sought-after quantum advantages in machine learning tasks with noisy intermediate-scale quantum devices \cite{Preskill2018}, respectively.