Ancilla-Shared Time Multiplexing of Three FCC Sheet Codes: A One-Third Qubit Saving at a 1.3× Threshold Cost

The three triad sheets of the face-centered cubic (FCC) lattice, at even lattice size L, are edge-disjoint but share their vertex and octahedral-void ancilla positions. Running the three sheet codes in a three-round time-multiplexed cycle on one chip needs 4L3 physical qubits against 6L3 for three separate blocks, a saving of exactly one third at every L, for instance, 864 qubits instead of 1296 at L=6. The schedule is implemented explicitly, with all three sheets on one shared ancilla set; comparing it against the effective single-sheet model used for the parameter sweeps shows agreement to within 5% at L=4 and 2.2% at L=6. The cost of the saving is that each sheet idles for two of every three sub-rounds, and the finite-size crossing estimate of the circuit-level threshold falls from 1.13±0.08% to 0.86±0.06%: a factor of 1.3, that is a 24% relative reduction, or 0.27 percentage points. Sweeping the idle-to-gate noise ratio r over 0,1,2,4 gives estimates decreasing monotonically from 1.13% to 0.71%. The dual X-basis memory gives 0.78±0.05%, overlapping the Z value. All values are finite-size crossings at L=4,6,8 from the full FCC circuit, not asymptotic thresholds, and the quoted spreads are dominated by finite-size drift rather than statistical error. Each sheet decomposes exactly into L independent rotated 2D toric codes (Proposition 1), so the memory is the toric code’s and no encoding-rate advantage is claimed; at d=4, the rate equals the rotated toric code’s. In the simulated noise model, where no channel couples two sheets, the sheets fail independently. Sub-threshold suppression survives the sharing at Λ(4→6)=12.3±1.1 and Λ(6→8)=5.6±1.2 at p=10−3, on a memory carrying 24 to 48 logical qubits. Block-level failure probabilities are reported alongside, since they grow with the logical count: at L=6 and p=10−3, the 36-logical chip loses at least one logical qubit in about one experiment in sixty. The L=6 instance, [[648,36,6]], needs 864 physical qubits, within the nominal atom count of current neutral-atom processors; no implementability claim is made beyond that count.