Hierarchical surface code for network quantum computing with modules of arbitrary size

The network paradigm for quantum computing involves interconnecting many modules to form a scalable machine. Typically it is assumed that the links between modules are prone to noise while operations within modules have a significantly higher fidelity. To optimize fault tolerance in such architectures we introduce a hierarchical generalization of the surface code: a small ``patch'' of the code exists within each module and constitutes a single effective qubit of the logic-level surface code. Errors primarily occur in a two-dimensional subspace, i.e., patch perimeters extruded over time, and the resulting noise threshold for intermodule links can exceed $\ensuremath{\sim}10$% even in the absence of purification. Increasing the number of qubits within each module decreases the number of qubits necessary for encoding a logical qubit. But this advantage is relatively modest, and broadly speaking, a ``fine-grained'' network of small modules containing only about eight qubits is competitive in total qubit count versus a ``course'' network with modules containing many hundreds of qubits.