Inter-Sheet Joint Pauli Measurements on the FCC Lattice: A Cross-Block Fault-Tolerant Primitive at K = 4 Active Connectivity

Restricting the [[3L3,2L3+2,3]] FCC lattice code to a single triad sheet gives the [[L3,2L,L]] sheet code, with uniform weight-4 stabilizers and K=4 active connectivity. This code is isomorphic to L disjoint rotated 2D toric codes, so its memory is the toric code’s, and no density advantage is claimed. The contribution is a logical interconnect: native fault-tolerant joint Pauli measurement between selected logical qubits in independently encoded, co-located code blocks, with no dedicated routing region. Its natural use is entanglement generation and parity measurement between such blocks, since the ZZ and XX merges characterized here are exactly the two measurements a logical Bell measurement requires. Every FCC triangle has one edge in each triad sheet, so products of triangle measurements implement joint Pauli measurements across sheets while the merged code retains d=L. Finite-size crossing estimates under circuit-level depolarizing noise at L∈{4,6,8} are 1.07±0.05% for the ZZ-merge, 0.76±0.05% with planar boundaries, and 0.91±0.05% for the XX-merge. The value of the primitive is that, for the pairs it reaches, it needs no routing region. Surface-code surgery merges only adjacent patches, so joining arbitrary pairs costs a reservation of roughly 1.5 tiles per logical qubit; within the directly reachable set, the sheet architecture avoids that reservation, using L2 physical qubits per logical, against 1.5L2 for a bussed toric array. That set is structured and limited: at the verified sizes of L∈{4,6,8}, each basis logical reaches L partners in one other sheet, giving 3L2 directly measurable pairs, corresponding to about 17% of all logical pairs, and the same pattern is conjectured for general even L values. Three limits are stated rather than deferred: only the joint measurements are fault-tolerant primitives, the composed three-sheet CNOT being verified but not yet fault-tolerantly characterized; all thresholds are finite-size crossings at three lattice sizes, not asymptotic values; and pairs outside the reachable set would need routing as in any other architecture.