Distributed fault-tolerant quantum memories over a 2xL array of qubit modules

We propose an architecture for a quantum memory distributed over a $2 \times L$ array of modules equipped with a cyclic shift implemented via flying qubits. The logical information is distributed across the first row of $L$ modules and quantum error correction is executed using ancilla modules on the second row equipped with a cyclic shift. This work proves that quantum LDPC codes such as BB codes can maintain their performance in a distributed setting while using solely one simple connector: a cyclic shift. We propose two strategies to perform quantum error correction on a $2 \times L$ module array: (i) The cyclic layout which applies to any stabilizer codes, whereas previous results for qubit arrays are limited to CSS codes. (ii) The sparse cyclic layout, specific to bivariate bicycle (BB) codes. For the $[[144,12,12]]$ BB code, using the sparse cyclic layout we obtain a quantum memory with $12$ logical qubits distributed over $12$ modules, containing $12$ physical qubits each. We propose physical implementations of this architecture using flying qubits, that can be faithfully transported, and include qubits encoded in ions, neutral atoms, electrons or photons. We performed numerical simulations when modules are long ion chains and when modules are single-qubit arrays of ions showing that the distributed BB code achieves a logical error rate below $2 \cdot 10^{-6}$ when the physical error rate is $10^{-3}$.