We study an architecture for fault-tolerant measurement-based quantum computation (FT-MBQC) over optically-networked trapped-ion modules. The architecture is implemented with a finite number of modules and ions per module, and leverages photonic interactions for generating remote entanglement between modules and local Coulomb interactions for intra-modular entangling gates. We focus on generating the topologically protected Raussendorf-Harrington-Goyal (RHG) lattice cluster state, which is known to be robust against lattice bond failures and qubit noise, with the modules acting as lattice sites. To ensure that the remote entanglement generation rates surpass the bond-failure tolerance threshold of the RHG lattice, we employ spatial and temporal multiplexing. For realistic system timing parameters, we estimate the code cycle time of the RHG lattice and the ion resources required in a bi-layered implementation, where the number of modules matches the number of sites in two lattice layers, and qubits are reinitialized after measurement. For large distances between modules, we incorporate quantum repeaters between sites and analyze the benefits in terms of cumulative resource requirements. Finally, we derive and analyze a qubit noise-tolerance threshold inequality for the RHG lattice generation in the proposed architecture that accounts for noise from various sources. This includes the depolarizing noise arising from the photonically-mediated remote entanglement generation between modules due to finite optical detection efficiency, limited visibility, and the presence of dark clicks, in addition to the noise from imperfect gates and measurements, and memory decoherence with time. Our work thus underscores the hardware and channel threshold requirements to realize distributed FT-MBQC in a leading qubit platform today -- trapped ions.

翻译：我们研究了一种在光网络囚禁离子模块上实现容错测量基量子计算（FT-MBQC）的架构。该架构采用有限数量的模块及每个模块中有限数量的离子，利用光子相互作用实现模块间的远程纠缠，并借助局部库仑相互作用实现模块内的纠缠门操作。我们重点研究生成具有拓扑保护的Raussendorf-Harrington-Goyal（RHG）晶格簇态，该态对晶格键失效和量子比特噪声具有鲁棒性，其中模块充当晶格位点。为确保远程纠缠生成速率超过RHG晶格的键失效容错阈值，我们采用了空间与时间复用技术。基于实际系统时序参数，我们估算了RHG晶格的代码循环时间以及在双层实现中所需的离子资源，其中模块数量与两个晶格层中的位点数量相匹配，且量子比特在测量后被重新初始化。对于模块间距离较大的情况，我们在位点间引入量子中继器，并分析了其在累积资源需求方面的优势。最后，我们推导并分析了该架构中RHG晶格生成的量子比特噪声容错阈值不等式，该不等式综合考虑了多种噪声源的影响。这包括由于有限光学探测效率、有限可见度及暗计数存在导致的模块间光子介导远程纠缠生成所产生的退极化噪声，以及非完美门操作与测量引入的噪声和随时间变化的存储器退相干。因此，我们的研究明确了在当前领先的量子比特平台——囚禁离子上实现分布式FT-MBQC所需的硬件与信道阈值条件。