Surface code error correction in a modular quantum computer

We consider error correction in a quantum computer formed by planar

modules connected along the edges, assuming that error probability for

two-qubit gates across the boundary be larger than that for

intra-modular gates. First, we prove a general structure theorem for

modular stabilizer codes: the total number of logical qubits supported

by individual modules after separation cannot exceed the dimension $k$

of the original code. Second, we use a statistical-mechanical map to

get semi-analytical estimates for the maximum-likelihood

error-correction threshold as a function of the module size $L$.

Namely, with ideal measurements, one gets a planar modulated

random-bond Ising model (MRBIM), and otherwise, assuming

phenomenological error model and repeated measurements, a

cubic-lattice MRBIM. We construct the corresponding phase diagrams

approximately using the mean-field effective couplings between block

spins, and exactly in 2D using the loop-counting technique. Finally,

we also locate the threshold using circuit simulations and

minimum-weight perfect matching decoder.