Logical quantum circuits protected by the Steane code for specific noises in trapped ions

In the presence of physical noise of all platforms for quantum computation, quantum error correction (QEC) becomes a critical way to realize quantum algorithms with large quantum volumes. In order to understand the influence of quantum noise on QEC codes and further improve the performance of logical circuits, the noises should be accurately analyzed with proper models. Here we focus on the trapped-ion system. Fundamentally, we start from the laser pulses of the quantum gates in the circuits and extract the noise components from the complete evolution of the quantum states, beyond the standard depolarizing model and other simplified models. Our simulations indicate that the logical performance under real noises is significantly better than that predicted by previous models. Meanwhile, the advantage of QEC is shown in the levels of one, two, and more logical qubits. Moreover, we can increase the logical fidelity by the method of ion mapping, which is based on knowledge of the specific noise distribution of different ions. Some powerful evidence from the numerical results demonstrates the possibility to access fault-tolerant quantum computation with the trapped-ion system.