A [[L3, 2L, L]] CSS Code from the FCC Lattice with Weight-4 Stabilizers and K=4 Connectivity

[[L

, 2L, L]]

CSS Code from the FCC Lattice

with Weight-4 Stabilizers and

K=4

Connectivity

Raghu Kulkarni

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

raghu@idrive.com

Abstract

We construct a CSS stabilizer code by restricting the

[[3L

, 2L

+2, 3]]

FCC lat-

tice code to a single triad sheet. On a torus, the resulting code has parameters

[[L

, 2L, L]]

at even

: growing distance

d = L

, uniform weight-

stabilizers, and

per-qubit connectivity

K = 4

. At

L = 4

, the code is

[[64, 8, 4]]

; at

L = 6

[[216, 12, 6]]

The construction yields three independent copies (one per triad sheet) on the same

lattice, encoding

logical qubits in

physical qubits at distance

. Com-

pared to the 3D toric code

[[3L

, 3, L]]

, one sheet encodes

logical qubits in

physical qubits at the same distance  a

-fold higher encoding rate. A planar

variant

[[L

, L, L]]

, using standard rotated surface code boundaries on each indepen-

dent layer, preserves the distance at the cost of halving the encoding rate. Under

circuit-level depolarizing noise with MWPM decoding, the single-layer threshold is

≈ 0.88%

; the

-layer sheet code threshold is

≈ 0.63%

(single-sheet opera-

tion) or

≈ 0.42%

(three-sheet time-multiplexing with idle decoherence), both above

or near the operating error rates of current quantum processors.

1 Introduction

The

[[3L

, 2L

+2, 3]]

CSS code on the Face-Centered Cubic (FCC) lattice [1] achieves a

67% encoding rate with weight-12 stabilizers, but its xed distance

d = 3

and

K = 12

connectivity requirement limit practical deployment. The 3D toric code on the cubic

lattice encodes only

k = 3

logical qubits but achieves growing distance

d = L

[2].

We show that restricting the FCC code to a single triad sheet produces a construction

that inherits the distance scaling of the 3D toric code while encoding substantially more

logical qubits. The CSS construction [3] on the sheet yields a valid stabilizer code. The

sheet code has weight-4 stabilizers and requires only

K = 4

per-qubit connectivitythe

same as the surface code [4].

2 Construction

2.1 The triad decomposition

The FCC lattice has

K = 12

nearest-neighbor vectors, which partition into three orthog-

onal sheets of 4:

: (±1, ±1, 0)

: (±1, 0, ±1)

(1)

: (0, ±1, ±1)

Each edge of the FCC lattice belongs to exactly one sheet. At lattice size

(even), each

sheet contains

edges. Figure 1 illustrates the decomposition.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

All K=12

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Sheet xy (K=4)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Sheet xz (K=4)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Sheet yz (K=4)

Figure 1: Triad sheet decomposition of the FCC lattice. Left: all

K = 12

bonds (colored

by sheet). Right three panels: each sheet individually, showing

K = 4

connectivity per

vertex.

2.2 The sheet code

Denition 1

(FCC sheet code)

Fix one triad sheet

(say

). Place one physical qubit

on each edge in

(

n = L

qubits). Dene:



-stabilizers: for each vertex

, apply

to the 4 edges of

incident to



-stabilizers: for each octahedral void

, apply

to the 4 edges of

connecting the

6 vertices surrounding

Both stabilizer types have uniform weight 4. Figure 2 illustrates both check types.

Proposition 1

(CSS validity)

The sheet code is a valid CSS code:

= 0

over

GF(2)

Proof.

Each edge in sheet

connects two vertices and participates in two octahedral voids

restricted to

. The overlap between any

-stabilizer (4 edges from an oct void in

) and

any

-stabilizer (4 edges from a vertex in

) is even, because the vertex and the oct void

share either 0 or 2 of the 4 sheet edges. Veried computationally at

L = 4

and

L = 6

1.5 1.0 0.5 0.0 0.5 1.0 1.5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Z-stabilizer: weight 4

(vertex checks 4 sheet edges)

Vertex (Z-ancilla)

Data qubit (edge)

1.5 1.0 0.5 0.0 0.5 1.0 1.5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

X-stabilizer: weight 4

(oct void checks 4 sheet edges)

Oct void (X-ancilla)

Data qubit (edge)

Vertex (neighbor)

Figure 2: Weight-4 stabilizers of the sheet code. Left:

-stabilizer (vertex check) acts on

4 data qubits (edges) in the sheet. Right:

-stabilizer (octahedral void check) acts on 4

data qubits. Both have uniform weight 4, matching the surface code.

3 Code Parameters

Theorem 1

(Sheet code parameters)

The FCC sheet code at even

has parameters:

[[n, k, d]] = [[L

, 2L, L]]

(2)

with

n = L

physical qubits,

k = 2L

logical qubits, and code distance

d = L

Computational verication:

L n rank(H

) rank(H

) k d

4 64 28 28 8 4 (exact)

6 216 102 102 12 6 (exact)

L = 4

k = 2×4 = 8

and

d = 4 = L

. The distance

d = 4

is proven by exhaustive elimi-

nation of all weight-

≤ 3

candidates (no weight-1, 2, or 3 vectors in

ker(H

)\rowspace(H

)

ker(H

) \ rowspace(H

)

) combined with constructive weight-4 logical operators (28

weight-4

-logicals and 24 weight-4

-logicals).

L = 6

k = 2 × 6 = 12

and

d = 6 = L

. Both

= 6

and

= 6

are conrmed

by exhaustive elimination of weight

≤ 5

and constructive weight-6 logicals (57 weight-6

-logicals found).

The stabilizer ranks satisfy

rank(H

) = rank(H

) = (L

− 2L)/2

at both tested sizes,

giving

k = L

− 2 × (L

− 2L)/2 = 2L

. We conjecture this holds for all even

4 Why the Distance Increases

The full FCC code has

d = 3

because weight-3 logical operators exist at tetrahedral

voids: one edge from each of the three triad sheets forms a triangle that commutes with

all weight-12 stabilizers.

Proposition 2

(Sheet restriction eliminates weight-3 logicals)

Within a single triad

sheet, no weight-3 logical operator exists.

Proof.

A weight-3 logical of the full FCC code has the triad structure: one edge from

one from

, one from

. Restricted to a single sheet

, only the

edge survives 

a weight-1 operator. Weight-1 operators are always detected by the vertex

-stabilizers

(each edge has two distinct endpoints). Therefore no weight-3 FCC logical survives the

sheet restriction.

The minimum-weight logical operators of the sheet code are non-contractible cycles within

the sheet, wrapping the torus. These have length

, matching the torus period.

Layer structure.

Each triad sheet decomposes further into

independent layers at each

value of the zero-displacement coordinate (

for

). Each layer is a

D toric code on a

rotated square lattice of linear dimension

, with

edges,

vertices, and parameters

[[L

, 2, L]]

. The sheet code

[[L

, 2L, L]]

is therefore

parallel copies of the

D toric code.

The three triad sheets yield

independent toric codes on the same FCC lattice, all at

K = 4

. The novelty lies not in the per-layer code (which is the standard toric code) but

in the decomposition: the FCC lattice accommodates

parallel toric codes encoding

logicals in

qubits, compared to

∼ 3L/2

surface code blocks encoding

∼ 3L/2

logicals

in the same qubit count on a square lattice  a

4×

improvement in logical qubit density.

5 Comparison with Known Codes

Table 1: The FCC sheet code compared with established codes at similar parameters.

Code

n k d

Rate

Surface code

−1

d ∼ 1/(2d

)

3D toric code

L ∼ 1/L

FCC sheet code

2L L 2/L

Full FCC code

3 2/3 12

L = 10

(

d = 10

): the surface code encodes 1 logical qubit in 199 physical qubits; the 3D

toric code encodes 3 in 3,000; three FCC sheets encode 60 in 3,000 (Table 1). Three sheets

encode

20×

more logicals than the 3D toric code at the same qubit count and distance.

For 60 logical qubits at

d = 10

, the surface code requires

60 × 199 = 11,940

qubits vs

3,000

for three FCC sheets: a

4×

savings. Figure 3 summarizes these comparisons.

6 Hardware Compatibility

The sheet code requires

K = 4

: each data qubit connects to 2 vertex

-ancillas and 2 oc-

tahedral

-ancillas. Each ancilla connects to 4 data qubits. This is the same connectivity

as the surface code. The

[[L

, 2L, L]]

parameters require periodic (torus) boundaries; on a

Surface

(n=3000)

3D Toric

(n=3000)

FCC Sheet×3

(n=3000)

Logical qubits (k)

Logical qubits at d=10, n=3000

4 6 8 10 12 14

Code distance d

Encoding rate k/n

Rate vs Distance

Surface code

3D Toric code

FCC Sheet ×3

Figure 3: Left: Logical qubits encoded in 3,000 physical qubits at distance

d = 10

for

three code families. The FCC sheet code (

×3

sheets) encodes

20×

more logicals than the

3D toric code. Right: Encoding rate

k/n

vs code distance

(

= L

for the toric and sheet

codes). The plotted scalings are

∼ 1/(2d

)

(surface),

∼ 1/d

(3D toric), and

∼ 2/d

(FCC

sheet

×3

sheets).

planar chip, the code becomes

[[L

, L, L]]

using standard rotated surface code boundaries

(Section 8), or alternatively, the toric parameters are preserved with

O(L)

long-range

wrap couplers per boundary.

Platform Connectivity

K = 4

compatible?

Google Willow

K = 4

(square) Yes

Google hex dynamic [5]

K = 3

(hex) Yes (with SWAP)

IBM Heron/Eagle

K = 3

(heavy-hex) Yes (with SWAP)

Any

K ≥ 4

chip

K ≥ 4

Yes

Figure 4 compares logical qubit counts achievable with torus topology on

K = 4

hardware.

7 Three Sheets on One Chip

The three triad sheets are edge-disjoint: no physical qubit belongs to more than one sheet.

They share vertex and octahedral void positions (the ancilla qubits), but the stabilizer

checks within each sheet are independent.

On a single chip, the three sheets can be interleaved:

data qubits partitioned into 3

groups of

, with ancilla qubits shared across sheets and measured in 3 sequential rounds

(one per sheet). This time-multiplexing mirrors the dynamic surface code approach [5],

where alternating circuit congurations achieve eective connectivity beyond the static

hardware wiring.

Idle-time penalty.

Time-multiplexing introduces a coherence cost. If one round takes

time

, a full three-sheet syndrome cycle takes

. Data qubits in sheets not currently

being measured sit idle for

, accumulating decoherence errors. This eectively triples

Google

Willow

Google

Hex Dynamic

IBM

Heron 133q

FCC Sheet

on K=4 chip

FCC Sheet×3

on K=4 chip

Logical qubits

k=1

(Surface d=5)

k=1

(Hex surface d=5)

k=1

(Heavy-hex d=3)

k=8

([[64,8,4]])

k=24

(3×[[64,8,4]])

Logical qubits on existing hardware (K 4, ~50-200 physical qubits)

Figure 4: Logical qubits on

K ≤ 4

hardware (torus topology). The surface code encodes 1

logical qubit per code block. A single FCC sheet encodes

k = 8

with no time-multiplexing

penalty. Three sheets (

k = 24

) require the time-multiplexed operation described in Sec-

tion 7, which triples the syndrome cycle time. On planar hardware, each sheet encodes

k = L

(Section 8).

the per-cycle physical error rate compared to a parallel (non-multiplexed) implementation,

lowering the code's threshold. For applications where cycle time is critical, operating a

single sheet (without multiplexing) avoids this penalty at the cost of encoding only

rather than

logical qubits.

8 Boundary Conditions

The parameters

[[L

, 2L, L]]

are proven for periodic boundary conditions (torus topology).

Real quantum processors are planar and do not naturally support periodic boundaries.

The planar sheet code.

Each layer of the sheet code is an independent 2D toric code

[[L

, 2, L]]

on a rotated square lattice. On a plane, the toric code becomes the rotated

surface code

[[L

, 1, L]]

using the standard construction [4]: weight-4 stabilizers in the

interior, weight-2 boundary stabilizers of alternating type (rough X-boundaries on two

sides, smooth Z-boundaries on the other two). Since the

layers are independent (no

shared edges or stabilizers), the boundary engineering applies to each layer separately.

The resulting planar sheet code has parameters:

[[L

, L, L]] (

planar boundaries

)

(3)

The distance

d = L

preserved

; the encoding rate drops by exactly a factor of 2 (from

k = 2L

k = L

), which is the standard toric-to-surface code cost.

We verify this construction numerically:

L n k

(planar)

3 27 3 3

4 64 4 4

5 125 5 5

6 216 6 6

CSS validity and

= d

= L

are conrmed at every

tested.

Comparison.

The toric-to-planar transition for the sheet code follows the same pattern

as for the ordinary toric code:

halves,

is unchanged. The planar sheet code

[[L

, L, L]]

still encodes

logical qubits at distance

in a

K = 4

planar layout 

L/2

times more

logical qubits than

L/2

independent surface codes at the same distance, using the same

number of physical qubits.

Periodic boundaries on planar hardware.

If the full

k = 2L

encoding is required,

periodic boundary conditions can be implemented on a planar chip using

O(L)

long-

range wrap couplers per sheet boundary, or via classical post-processing that identies

opposite-edge syndromes.

9 Error Threshold

Each layer of the sheet code is an independent surface code of distance

. We com-

pute the error threshold under two noise models using stim [6] for circuit generation and

PyMatching [7] for MWPM decoding.

9.1 Code-capacity model

Under code-capacity depolarizing noise (i.i.d. errors on data qubits, perfect syndrome

measurements), the single-layer threshold exceeds

12%

. The sheet code, with

layers

failing independently, has an eective threshold of

≈ 8.5%

(the point where the log-

ical error rate

sheet

= 1 − (1 − p

layer

)

begins increasing with

). The threshold ratio

sheet/single

≈ 71%

9.2 Circuit-level model

Under circuit-level noise (depolarizing errors after two-qubit gates, measurement ip er-

rors, reset errors,

syndrome extraction rounds), the single-layer threshold is:

layer

≈ 0.88%

(4)

For the

-layer sheet code, a logical failure occurs if any layer fails. The eective circuit-

level threshold is:

sheet

≈ 0.63% (

single-sheet operation

)

(5)

9.3 Three-sheet multiplexing penalty

When three sheets are time-multiplexed (Section 7), data qubits in idle sheets accumu-

late depolarizing noise during the two extra measurement rounds. We model this by

adding single-qubit depolarizing noise at rate

per round on all data qubits (via stim's

before_round_data_depolarization

). The

-layer threshold with multiplexing is:

multiplex

≈ 0.42%

(6)

Table 2 reports the per-layer logical error rates without idle noise, and Table 3 summarizes

all thresholds. Figure 5 shows the threshold curves.

Table 2: Logical error rate per layer (circuit-level, MWPM,

shots per point,

rounds,

no idle noise).

(%)

d = 3 d = 5 d = 7 d = 9 d = 11

Trend

0.1

4.3 × 10

−4

5 × 10

−5

< 10

−5

< 10

−5

< 10

−5

↓

0.3

4.2 × 10

−3

1.7 × 10

−3

6.8 × 10

−4

1.4 × 10

−4

6 × 10

−5

↓

0.5

1.1 × 10

−2

7.1 × 10

−3

4.0 × 10

−3

2.3 × 10

−3

1.1 × 10

−3

↓

0.8

2.6 × 10

−2

2.7 × 10

−2

2.4 × 10

−2

2.1 × 10

−2

1.9 × 10

−2

≈

1.0

3.7 × 10

−2

4.7 × 10

−2

5.1 × 10

−2

5.5 × 10

−2

6.1 × 10

−2

↑

Table 3: Threshold summary for all operating modes (circuit-level noise, MWPM de-

coder).

Operating mode

Willow compatible?

Single layer (no idle)

≈ 0.88%

Yes

-layer sheet (single sheet, no idle)

≈ 0.63%

Yes

-layer sheet (3-sheet multiplex,

idle)

≈ 0.42%

Marginal

9.4 Analysis

Below threshold, the per-layer logical error rate decreases exponentially:

layer

∼

exp(−αd)

. The

-layer sheet code logical error rate is

sheet

= 1 − (1 − p

layer

)

≈ L · p

layer

for small

. The linear prefactor

shifts the eective threshold downward but does not

change the asymptotic behavior: for any

p < p

layer

, the sheet code logical error rate still

vanishes as

L → ∞

At the Google Willow operating point (

p ≈ 0.3%

), the per-layer logical error rate at

d = 7

6.8 × 10

−4

. The 7-layer sheet code gives

sheet

≈ 4.8 × 10

−3

, which is below the

∼ 1%

regime where useful quantum computation becomes possible. Increasing to

d = 11

reduces the per-layer rate to

6×10

−5

and the sheet code rate to

6.6×10

−4

. These numbers

apply to single-sheet operation; three-sheet multiplexing at

p = 0.3%

gives per-layer rates

∼ 1.0%

d = 3

and

∼ 0.06%

d = 11

, indicating that multiplexing is viable at

p = 0.3%

but with reduced margin.

2 4 6 8 10 12

Physical error rate (%)

Logical error rate

(a) Single Layer Code Capacity

d = 3

d = 5

d = 7

d = 9

d = 11

2 4 6 8 10 12

Physical error rate (%)

Logical error rate

8.5%

(b) Sheet Code Code Capacity

L = 3

L = 5

L = 7

L = 9

L = 11

0.2 0.4 0.6 0.8 1.0 1.2

Physical error rate (%)

Logical error rate

0.88%

d = 3

d = 5

d = 7

d = 9

d = 11

0.2 0.4 0.6 0.8 1.0 1.2

Physical error rate (%)

Logical error rate

0.63%

Willow

range

(d) Sheet Code Circuit Level

L = 3

L = 5

L = 7

L = 9

L = 11

Figure 5: Error thresholds under code-capacity (top) and circuit-level (bottom) depolar-

izing noise. Left panels: single-layer logical error rate. Right panels: sheet code logical

error rate (

layers, any-layer failure). The circuit-level sheet code threshold

≈ 0.63%

(single-sheet operation) is above the operating range of Google Willow (

0.3



0.5%

, green

band in lower right). Three-sheet multiplexing reduces the threshold to

≈ 0.42%

due to

idle decoherence. MWPM decoding via PyMatching;

5 × 10



shots per point; code

distances

d = 3, 5, 7, 9, 11

10 Conclusion

The FCC sheet code

[[L

, 2L, L]]

occupies a new point in the rate-distance tradeo: grow-

ing distance with multiple logical qubits at the minimal

K = 4

connectivity of the surface

code. It is derived from the FCC lattice by a restriction (one triad sheet) that eliminates

the FCC code's weight-3 vulnerability while preserving useful encoding. Three sheets

on one chip encode

logical qubits at distance

physical qubits  a

-fold

eciency gain over the 3D toric code's

k = 3

at the same distance and qubit count (e.g.,

20×

L = 10

The planar variant

[[L

, L, L]]

uses standard rotated surface code boundaries on each inde-

pendent layer, preserving distance

d = L

at the cost of halving the encoding rate. Under

circuit-level depolarizing noise, the

-layer sheet code has a threshold of

≈ 0.63%

in single-sheet operation, which is above the operating error rates of current quantum

processors (

∼ 0.3



0.5%

on Google Willow). Three-sheet time-multiplexing reduces the

threshold to

≈ 0.42%

due to idle decoherence, making single-sheet operation the preferred

mode on near-term hardware.

One practical caveat: three-sheet time-multiplexing triples the syndrome cycle time, re-

ducing the threshold from

0.63%

0.42%

. Single-sheet operation avoids this penalty at

the cost of reduced encoding.

The core result  a CSS code with growing distance, weight-4 checks,

K = 4

connectivity,

k = 2L

(toric) or

k = L

(planar) logical qubits, and a competitive error threshold  is

established by construction and veried computationally at

L = 3, 4, 5, 6

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