Logical qubit encoding and quantum error correction (QEC) protocols have been experimentally demonstrated in various physical systems with multiple physical qubits, generally without reaching the break-even point, at which the lifetime of the quantum information exceeds that of the single best physical qubit within the logical qubit. Logical operations are challenging, owing to the necessary non-local operations at the physical level, making bosonic logical qubits that rely on higher Fock states of a single oscillator attractive, given their hardware efficiency. QEC that reaches the break-even point and single logical-qubit operations have been demonstrated using the bosonic cat code. Here, we experimentally demonstrate repetitive QEC approaching the break-even point of a single logical qubit encoded in a hybrid system consisting of a superconducting circuit and a bosonic cavity using a binomial bosonic code. This is achieved while simultaneously maintaining full control of the single logical qubit, including encoding, decoding and a high-fidelity universal quantum gate set with 97% average process fidelity. The corrected logical qubit has a lifetime 2.8 times longer than that of its uncorrected counterpart. We also perform a Ramsey experiment on the corrected logical qubit, reporting coherence twice as long as for the uncorrected case.

逻辑量子位编码和量子纠错（QEC）协议已在具有多个物理量子位的各种物理系统中进行了实验证明，通常没有达到盈亏平衡点，在该点上，量子信息的寿命超过了其中单个最佳物理量子位的寿命。逻辑量子位。逻辑操作具有挑战性，因为在物理级别需要进行非本地操作，鉴于其硬件效率，使得依赖单个振荡器的较高Fock状态的Bosonic逻辑量子位具有吸引力。已经使用bosonic cat代码演示了达到收支平衡点和单个逻辑量子位操作的QEC。这里，我们通过实验证明了重复的QEC接近单逻辑量子位的收支平衡点，该混合系统使用二项式Bosonic代码在由超导电路和Bosonic腔组成的混合系统中进行了编码。这是在同时保持对单个逻辑量子位的完全控制的同时实现的，包括编码，解码和具有97％平均过程保真度的高保真通用量子门集。校正后的逻辑量子比特的寿命是未校正逻辑量子比特的2.8倍。我们还对校正后的逻辑量子位执行了Ramsey实验，报告了相干时间是未校正时的两倍。解码和具有97％平均过程保真度的高保真通用量子门装置。校正后的逻辑量子比特的寿命是未校正逻辑量子比特的2.8倍。我们还对校正后的逻辑量子位执行了Ramsey实验，报告了相干时间是未校正时的两倍。解码和具有97％平均过程保真度的高保真通用量子门装置。校正后的逻辑量子比特的寿命是未校正逻辑量子比特的2.8倍。我们还对校正后的逻辑量子位执行了Ramsey实验，报告了相干时间是未校正时的两倍。