Ancilla-Shared Time Multiplexing of Three FCC Sheet Codes

Surface codes [5] pair a high circuit-level threshold with a planar layout, at a cost in rate: a

distance-d rotated surface code holds one logical qubit in d

structions such as the bivariate-bicycle family [6] reduce that overhead using weight-6 stabilizers

and non-planar connectivity.

weight-4 stabilizers and K = 4 connectivity. This construction is due to the author’s earlier work

their data qubits are distinct, but they share every vertex and every octahedral void. Those

are exactly the ancilla positions. A single ancilla set can therefore serve all three sheets if the

stabilizers are extracted in sequence rather than in parallel. This is an architectural claim, not

a coding-theoretic one, and it has a measurable cost: while one sheet is measured, the other two

idle and decohere.

its equivalence to a toric stack explicitly. Section 3 states the relation to stacked toric codes,

since the encoding-rate figures invite a comparison that does not hold up. Sections 4–7 report

the qubit saving, the threshold cost, and the sub-threshold distance scaling with uncertainties.

Section 8 compares against the rotated toric code, and Section 9 covers hardware feasibility.

sheet is an edge-disjoint K = 4 sublattice on the same vertices and octahedral voids. Figure 1

shows the construction.

The single-sheet code. Fix a sheet S and place a qubit on each of its L

are the weight-4 vertex parity checks over the sheet edges at each FCC vertex; X-stabilizers are

the weight-4 checks over the four sheet edges of each octahedral void. This gives a CSS code

with parameters [[L

Layer decomposition. The following is the structural fact this paper rests on. We state and

Every edge of S therefore lies inside a single plane of constant z, and there are L such planes,

each holding L

only the S-edges at that vertex. A void check uses only the four S-edges of the void, and those

share the void’s z coordinate. No check couples two planes, so the sheet is a direct sum of L

independent codes.

operators on a torus. That is the rotated toric code [[L

checks, and the per-sheet total is k = 2L.

three edge-disjoint groups, plus L

ancilla is physically shared by all three sheets, so the total is 4L

a three-round cycle, one sheet per round. From any one sheet’s perspective two rounds in three

are idle. If a round takes time t, a sheet’s data qubits idle for 2t between active measurements,

which we model as per-round depolarizing noise at rate 2p, additional to gate noise. In Stim this

is DEPOLARIZE1(2p) on all data qubits before each round.

form a cuboctahedron. (b) Those twelve bonds partition into three orthogonal triad sheets of

four bonds each. (c) The three-round multiplex cycle: one sheet is measured per round while the

other two idle and accumulate depolarizing noise at rate 2p. The shared vertex and void ancillas

serve all three sheets, which is what reduces the physical-qubit count from 6L

Combined parameters. The encoded subspace is the direct sum of the three sheets’ code

spaces, giving [[3L

rate, the three-sheet block error rate is 1 − (1 − p

The rate figures below invite a comparison that does not survive scrutiny, so we state the position

first.

Memory performance. Each sheet is L disjoint rotated toric codes by Proposition 1, and

the sheets fail independently, so the three-sheet code is 3L rotated toric codes sharing ancillas.

Every memory quantity follows: distance L, weight-4 stabilizers, K = 4 connectivity, and the

toric code’s threshold up to the idle penalty measured in Section 6. The suppression factors of

Section 7 are the toric code’s suppression factors.

Encoding rate. At d = 4 the three-sheet code has rate 24/192 = 12.5%. The rotated toric

code [[16, 2, 4]] has rate 2/16 = 12.5%. These are equal, and equal at every L: both sit at 2/L

Any claim that this construction has the highest rate at its distance would be false, and we make

none.

Figure 2: The trade quantified. (a) Ancilla sharing needs 4L

three separate blocks, a saving of one third at every L. (b) The measured circuit-level threshold

falls from 1.13% to 0.86% under the multiplex idle penalty, a factor of 1.3. Both are measured

with the full FCC sheet-code circuit; error bars are the spread across the three pairwise crossings.

Table 3 reports per-logical error rates for the full three-sheet circuit under the idle penalty, and

Table 4 the corresponding suppression factors. Figure 3 plots both.

this section is for.

Shot-floor limited points. At p = 10

errors across 30 000 shots and twelve observables, so its uncertainty is 70%. The L = 8 points

at p = 10

bounds. The largest suppression factor in Table 4 is Λ(4→ 6) = 40.8 ± 13.9 at p = 3 × 10

corresponds to currently demonstrated trapped-ion two-qubit gate fidelities and is the row with

adequate statistics at every size. Values at lower p are limited by the shot floor, and n/a marks

a ratio whose denominator observed no errors.

tion are not comparable to those measured on hardware near threshold. A surface-code simulation

at p = 10

measured near threshold, on a single logical qubit. The sheet code decomposes into toric codes,

so the values here are the toric code’s. We therefore make no claim of advantage over the surface

code from these numbers. What they establish is narrower and is what we intend: suppression

survives ancilla sharing and the idle penalty, on a memory carrying 24 to 48 logical qubits.

Table 5 compares at d = 4. The essential row is the rotated toric code, which is often omitted

in favor of the rotated surface code and which the three-sheet construction matches exactly.

Table 5: Comparison at distance d = 4, counting data qubits only. The three-sheet code and

the rotated toric code have identical rate, as the layer decomposition requires. The factor of

two over the rotated surface code is the periodic-boundary factor. The smallest bivariate-bicycle

instances are at d = 6 and d = 12, so no d = 4 comparison is possible.

code wins on per-logical fidelity at d = 4. The FCC three-sheet chip wins on physical footprint

per logical qubit. It does so by the periodic-boundary factor plus the ancilla sharing, not by

anything intrinsic to the lattice.

The three-sheet code at L = 4 needs 256 physical qubits at K = 4, and at L = 6 needs 864.

Neutral-atom platforms fit the protocol cleanly: optical-tweezer rearrangement lets one ancilla set

pair with different data groups across the three rounds without rewiring, and K = 4 connectivity

is native.

Phoenix processor. Running L = 4 first and then L = 6 on the same processor would give a

on any platform. The nearest prior multi-logical demonstration encoded 24 logical qubits with

the distance-2 [[4, 2, 2]] code on a 256-atom processor [7], which is error detection rather than

correction. We note that the ancilla saving is what brings L = 6 inside the machine at all: three

separate blocks would need 1296 physical qubits and would not fit.

Idle noise model. The 2p per-round idle rate is a model choice, not a measurement. Actual

idle decoherence during neutral-atom rearrangement and shelving is a separate experimental

question, and the threshold cost reported here scales with it.

Scope of the memory result. As Proposition 1 shows, the sheets are independent and each

decomposes into toric layers. Nothing in the distance-scaling data is specific to the FCC lattice

beyond the ancilla sharing and the idle penalty.

No logical operations. This paper concerns memory only. Operations between sheets would