Three Sheets, One Architecture

Three Sheets, One Architecture:

Inter-Sheet Logical Gates for K=4 Quantum Error

Correction

Raghu Kulkarni

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

raghu@idrive.com

Abstract

We present a CSS quantum error correcting code on the Face-Centered Cubic (FCC)

lattice that combines surface-code-compatible K=4 connectivity with native inter-

sheet logical gates between co-located CSS code blocks — a combination not pre-

viously achieved among well-studied codes. The construction has two parts. First,

restricting the [[3L

, 2L

+2, 3]] FCC lattice code to a single triad sheet yields the

sheet code with parameters [[L

, 2L, L]] at even L (or [[L

, L, L]] as a planar variant),

uniform weight-4 stabilizers, and K=4 per-qubit connectivity. Three triad sheets

share the FCC lattice geometry and can be deployed in one of three ways — mono-

lithic 2D for small L, a three-layer stacked architecture for scalable deployment, or

native 3D hardware — in each case encoding 6L logical qubits at distance L. Second,

we introduce a fault-tolerant lattice surgery protocol that uses local FCC triangle

measurements to implement joint Pauli measurements between logical qubits in dif-

ferent sheets. Each surgery operation adds O(L) two-qubit gates to the O(L

) per-

round syndrome extraction cost. Under circuit-level depolarizing noise with MWPM

decoding on a custom Stim circuit with explicit triangle measurements, ﬁnite-size

scaling across L = 4, 6, 8 gives a surgery threshold p

surgery

= 1.07% ± 0.05%, where

the band reﬂects the spread of the three pairwise crossings. Compared to surface

code lattice surgery (also K=4, lattice surgery conﬁned within a single patch), the

sheet code provides 2L logicals per sheet at the same distance and connectivity;

compared to bivariate bicycle codes (which target inter-block gates between physi-

cally separated modules at K=6 with long-range couplers), the sheet code achieves

inter-sheet gates at K=4 with only short-range couplings between co-located sheets.

1 Introduction

Two families of quantum error correcting codes currently dominate the discussion of near-

term fault-tolerant quantum computing. The surface code [3, 5] pairs K=4 planar hard-

ware compatibility with mature fault-tolerant protocols, including lattice surgery [6, 7]

for logical gates within a single 2D substrate. The bivariate bicycle (BB) code family [11]

achieves an order-of-magnitude rate advantage over the surface code at the cost of K=6

connectivity and a small number of long-range couplers. The BB family’s inter-block log-

ical gates — gates that move logical qubits between distinct code blocks — remain an

active research problem [12].

This paper introduces a third option that occupies a previously unﬁlled niche: a CSS code

at K=4 connectivity (matching the surface code) with native inter-sheet logical gates

between co-located CSS code blocks. This is a diﬀerent design point from surface code

lattice surgery (which operates within a single substrate) and from bivariate bicycle codes

(which target inter-block gates between physically separated K=6 modules and where

modular logical operation protocols are an area of active architectural development). The

construction has two interlocking ingredients.

The FCC sheet code. Restricting the [[3L

, 2L

+2, 3]] FCC lattice code [1] to a single

triad sheet (one of three orthogonal K=4 sublattices in FCC) eliminates the FCC code’s

weight-3 vulnerability and yields a CSS code with parameters [[L

, 2L, L]] at even L. Each

of the three sheets decomposes into L parallel 2D toric codes; three sheets on a shared

FCC substrate encode 6L logical qubits at distance L on 3L

data qubits.

Cross-sheet triangle surgery. Every FCC triangle has one edge in each of the three

triad sheets. Weight-3 Pauli measurements on FCC triangles couple data qubits across

sheets, providing a natural primitive for inter-sheet logical operations. We show that

triangle products implement joint Pauli measurements between logical qubits in diﬀerent

sheets, with the merged code preserving distance d = L.

The two ingredients work in tandem: the sheet code provides eﬃcient storage at K=4; the

triangle surgery primitive provides logical gates at the same K=4 envelope. The result

ﬁlls a hardware-architecture gap that neither surface code nor bivariate bicycle codes

address: K=4 planar connectivity with inter-sheet logical gates as a built-in feature.

Summary of results.

• Sheet code construction with parameters [[L

, 2L, L]] on a torus, [[L

, L, L]] on a

plane, at uniform weight-4 stabilizers and K=4 data-qubit connectivity (Section 2).

• Three-sheet architecture: 6L

total qubits (3L

data + 3L

ancilla) on a single K=4

chip via time-multiplexed syndrome extraction (Section 3).

• Triangle algebra: triangle products span 6L−3 of the 6L cross-sheet Z-logicals; every

per-sheet logical reachable via some sheet pair (Section 4).

• Fault-tolerant surgery protocol: explicit ancilla placement, gate schedule, K=4 ver-

iﬁcation, O (L) gate overhead (Section 5).

• Distance preservation under merge (Section 6).

• Threshold simulation via a custom Stim circuit with explicit triangle measurements

at L = 4, 6, 8: pairwise crossings give p

surgery

= 1.07% ± 0.05% (Section 7).

• Detailed comparison with state-of-the-art codes (Section 8).

2 The FCC Sheet Code

2.1 The Triad Decomposition

The FCC lattice has K = 12 nearest-neighbor vectors, partitioning into three orthogonal

sheets of 4 (a decomposition originally introduced in [2] for the ﬂag-assisted FCC code):

: (±1, ±1, 0)

: (±1, 0, ±1) (1)

: (0, ±1, ±1)

Each FCC edge belongs to exactly one sheet. At lattice size L (even), each sheet contains

edges. Restricted to a single sheet, each FCC vertex has K=4 incident edges.

2.2 The Sheet Code Stabilizers

Deﬁnition 1 (FCC sheet code). Fix one triad sheet S (say S

). Place one physical qubit

on each edge in S (n = L

qubits). The stabilizers are:

• Z-stabilizers: for each vertex v, apply Z to the 4 edges of S incident to v.

• X-stabilizers: for each octahedral void o, apply X to the 4 edges of S connecting the

6 vertices surrounding o.

Both stabilizer types have uniform weight 4. The CSS condition H

= 0 over GF(2) is

satisﬁed because each edge in sheet S connects two vertices and participates in exactly two

octahedral voids restricted to S; the overlap between any X-stabilizer and any Z-stabilizer

is even.

2.3 Code Parameters

Theorem 1 (Sheet code parameters). At even L, the FCC sheet code has parameters

[[L

, 2L, L]]: n = L

physical qubits, k = 2L logical qubits, code distance d = L.

The parameters follow from the layer decomposition (Section 2.4) together with standard

2D toric code counting. Computational veriﬁcation:

L n rank(H

) rank(H

) k

4 64 28 28 8

6 216 102 102 12

8 512 248 248 16

In each case rank(H

) = rank(H

) = (L

− 2L)/2, giving k = L

− 2(L

− 2L)/2 = 2L.

The general result follows from Theorem 2 below.

2.4 Layer Decomposition and Proof of Parameters

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 commuting with all weight-12 stabilizers. Within a single triad sheet, only one

edge of any such triangle survives, giving a weight-1 operator that is detected by vertex

Z-stabilizers. Therefore no weight-3 logical survives the sheet restriction; the minimum-

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

length L.

Layer structure. Each triad sheet decomposes further into L independent layers indexed

by the zero-displacement coordinate. For sheet S

, edges have dz = 0, so each S

edge

has a well-deﬁned z-coordinate equal to the shared z of its two endpoints. Edges in layer

z = z

form a 2D toric code on a rotated L ×L square lattice. Analogous decompositions

hold for S

(layered by y) and S

(layered by x).

Theorem 2 (Layer decomposition). The FCC sheet code on sheet S

at even L is iso-

morphic, as a stabilizer code, to L disjoint 2D toric codes, each on an L×L rotated square

lattice with L

data qubits, k = 2 logical qubits, and distance L. The Z-stabilizers (resp. X-

stabilizers) of the sheet code partition into L disjoint sets, one per layer; within each layer,

the rank deﬁciency equals 1 (one product redundancy among L

/2 vertex stabilizers).

Proof. Edge partition. Each S

edge (v

, v

) has z(v

) = z(v

) since the displacement

vector v

−v

∈ {(±1, ±1, 0)} has dz = 0. Deﬁne layer(e) = z(v

). The map layer : S

→

{0, 1, . . . , L − 1} partitions the L

edges of S

into L disjoint sets of L

edges each.

Stabilizer partition. A vertex Z-stabilizer at vertex v acts on the 4 sheet-S

edges

incident to v, all of which have the same z-coordinate as v. Hence each vertex Z-stabilizer

is supported entirely within one layer. An analogous argument applies to octahedral void

X-stabilizers within S

, since the 4 edges of an oct void restricted to S

all share the

same z-coordinate as the void center.

Layer is a 2D toric code. Within layer z = z

, the L

edges connect vertices {(x, y, z

) :

x+y ≡ z

(mod 2)} via the four neighbor vectors (±1, ±1, 0). This is precisely the rotated

L × L square lattice, and the vertex Z-stabilizers and oct-void X-stabilizers on this layer

are exactly the standard 2D toric code stabilizers. The toric code on L×L has parameters

[[L

, 2, L]].

Rank count. The 2D toric code on L

data qubits has L

/2 vertex Z-stabilizers, sat-

isfying one redundancy (the product over all vertices is the identity). Hence per layer,

rank(H

layer

) = L

/2 − 1. Across L layers, rank(H

sheet

) = L · (L

/2 − 1) = (L

− 2L)/2.

The same argument applies to H

sheet

Code parameters. k = n − rank(H

) − rank(H

) = L

− 2 · (L

− 2L)/2 = 2L. The

minimum-weight logical operators are the non-contractible cycles of the per-layer 2D toric

codes, each of length L. Hence d = L.

Consequence for the rank formula. Theorem 2 eliminates the need for a per-L

veriﬁcation of the rank: the formula rank(H

) = rank(H

) = (L

−2L)/2 holds for every

even L ≥ 2.

2.5 Planar Variant

For deployment on planar quantum chips that do not support periodic boundary condi-

tions, each layer becomes a rotated surface code [[L

, 1, L]] via standard boundary engi-

neering [5]. The resulting planar sheet code has parameters

[[L

, L, L]] (planar boundaries). (2)

Distance d = L is preserved; encoding rate halves from 2L → L.

3 Hardware Embedding: Three Sheets, Three Layers,

or One Chip

The three triad sheets are edge-disjoint: 3L

data qubits in total, with L

per sheet. We

now address the question of how to physically realize these qubits on hardware. This

question is non-trivial because the sheet code uses Θ(L

) data qubits to encode Θ(L)

logical qubits at distance L, and a monolithic 2D embedding of an L

-vertex 3D graph

cannot maintain unit-length nearest-neighbor couplings as L grows.

We discuss three deployment options, in increasing order of scalability.

3.1 Option A: Monolithic Planar Chip (Small to Moderate L)

For small to moderate L (L ≤ 8, corresponding to ≤ 1,500 data qubits total), a mono-

lithic planar quantum processor can host all three sheets via time-multiplexed syndrome

extraction. Data qubits occupy ﬁxed positions on a planar chip; ancillas for the three

sheets are physically distinct but co-located at FCC vertex and oct-void positions; cou-

plers reconﬁgure between rounds to activate one sheet’s stabilizers at a time. Speciﬁcally:

• Round 1: activate S

couplers, measure S

stabilizers

• Round 2: activate S

couplers, measure S

stabilizers

• Round 3: activate S

couplers, measure S

stabilizers

Wire-length scaling. On a monolithic 2D embedding of the Θ(L

)-qubit 3D lattice, the

average physical distance between nearest neighbors in the FCC graph scales as Θ(L

1/2

For Google Willow-style hardware with ﬁxed-length couplers, this restricts the practical

regime to small L. At L = 4 (192 qubits), the embedding is straightforward; at L = 8

(1,536 qubits), some couplers will span longer distances and require active calibration. At

L ≥ 12, monolithic 2D embedding is impractical without coupler reach upgrades.

Idle penalty. While one sheet is measured, data qubits in the other two sheets idle. With

round time t, a full cycle is 3t and each data qubit idles for 2t per cycle. We quantify the

resulting threshold reduction in Section 7.

3.2 Option B: Three-Layer Stacked Architecture (Recommended

Deployment)

For practical deployment at L ≥ 8, we recommend a three-layer stacked architecture:

three planar quantum processors, each a K=4 device hosting one triad sheet, bonded

with through-silicon-vias (TSVs) or inter-layer capacitive couplers for triangle-mediated

cross-sheet operations.

Per-layer structure. Each of the three chips is a standard planar K=4 device. Layer

1 hosts S

: L

data qubits on the L layers of S

, laid out as L stacked rotated L × L

square lattices (Theorem 2). Each chip layer hosts L such rotated lattices, either tiled

adjacently in 2D (for modest L) or as L further-stacked sublayers (for large L).

Inter-layer couplers. Triangle ancillas sit between chip layers, coupled via short-range

vertical couplers (TSVs) to the three data qubits of their triangle, one from each chip.

These couplers are activated only during surgery operations and remain inactive otherwise.

The chip-internal K=4 connectivity is unaﬀected; the inter-layer couplers add only K = 1

per data qubit per active triangle, recovering the same K=4 eﬀective constraint during

surgery as in Section 5.

Precedent. Three-layer stacked QEC architectures have been studied in recent quantum

hardware roadmaps [8, 11]. IBM’s stacked-die approach for the bivariate bicycle code

uses analogous inter-layer couplers to bridge K=6 connectivity at the chip level. The

sheet code’s three-chip architecture has lower per-chip connectivity (K=4 vs. K=6) but

a simpler inter-chip coupling structure (only triangle ancillas need vertical bonds, not

generic data qubits).

Vertical crosstalk and inter-layer isolation. The principal hardware concern for

any stacked architecture is crosstalk between layers. In the present construction, two

structural features mitigate this. First, only the triangle ancillas (a small minority of

total qubits, O(L) per surgery primitive vs. Θ(L

) data qubits per layer) carry inter-layer

couplers; data qubits and per-sheet ancillas have no vertical wiring, so the static memory

portion of the device is unaﬀected by inter-layer phenomena. Second, the inter-layer

couplers are activated only during surgery rounds via control electronics (Section 5), with

oﬀ-state isolation between layers achievable by detuning the triangle ancilla oﬀ-resonance

from the data qubits when surgery is not active. Practical isolation values are platform-

dependent: superconducting TSV couplers in stacked-die devices reach ∼ 40–60 dB oﬀ-

state isolation [8]; neutral-atom and ion-trap platforms with 3D-addressable laser systems

can in principle achieve higher isolation by mechanical separation. Detailed crosstalk

budgeting for a speciﬁc hardware platform is identiﬁed as future work (Section 9.3).

3.3 Option C: Native FCC Hardware

For maximally eﬃcient embedding, a quantum hardware platform with native 3D con-

nectivity (such as 3D-printed superconducting circuits, neutral atom arrays with 3D-

addressable laser systems, or trapped ion architectures with multi-segment traps) hosts

the full FCC lattice without the embedding penalty of options A or B. The sheet code

runs natively on such hardware with all K=4 couplings at unit physical distance. This

option is forward-looking; no current commercial platform oﬀers it.

3.4 Hardware Footprint

Across all options, the qubit and ancilla counts are:

• 3L

data qubits, one per FCC edge, partitioned by sheet;

• 3 × L

/2 vertex Z-ancillas (one per (vertex, sheet) pair);

• 3 × L

/2 octahedral void X-ancillas (one per (void, sheet) pair);

• L triangle ancillas per active surgery primitive (transient, can be reused).

Total: 6L

permanent qubits at K = 4 static connectivity per data qubit, plus O(L)

transient triangle ancillas during surgery.

3.5 Comment on the “Cross-Block” Terminology

The three triad sheets occupy the same FCC lattice geometry (in options A and B, time-

multiplexed or stacked on the same physical substrate; in option C, fully co-located in

3D). They are not spatially separated modules in the sense that bivariate bicycle code

blocks are. We use inter-sheet logical gates (or equivalently cross-sheet gates) to denote

the operations between logical qubits in distinct sheets that our triangle surgery primitive

implements. The sheets are logically distinct CSS code blocks — each independently

encodes 2L logical qubits with independent stabilizer groups and independent decoders

— but they are not physically separated. This intermediate regime, between surface

code lattice surgery within one substrate and bivariate bicycle inter-block gates across

separated modules, is the niche our construction occupies.

4 Triangle Algebra and Cross-Sheet Logicals

4.1 FCC Triangles

Lemma 1 (Triangle structure). Every triangle (3-cycle) in the FCC graph has one edge

in each of the three triad sheets. At lattice size L, the FCC graph contains 4L

triangles,

and each FCC edge participates in exactly 4 triangles.

Proof. For three mutually adjacent FCC vertices v

, v

, the three edge-vectors v

−v

− v

, v

− v

must each be FCC neighbor vectors. Direct case analysis on the 12

neighbor vectors shows that any three pairwise-summing-to-zero NN vectors necessarily

lie in distinct sheets. Counting: each FCC vertex is in 24 triangles; 24 · L

/2/3 = 4L

Each edge appears in 4L

· 3/(3L

) = 4 triangles.

4.2 Triangle Operators

Deﬁnition 2 (Triangle operator). For an FCC triangle T with edges e

∈ S

, e

∈ S

, deﬁne the Z-triangle operator

= Z

⊗ Z

and similarly X

= X

⊗ X

A single triangle Z-operator commutes with all per-sheet Z-stabilizers but anticommutes

with exactly 6 per-sheet X-stabilizers (two per sheet). Products of triangles can be chosen

to commute with all stabilizers.

4.3 Reachable Cross-Sheet Logicals

Theorem 3 (Cross-sheet reachability). Let B ∈ GF(2)

|T |×n

edges

be the triangle-edge inci-

dence matrix and H

the cross-sheet X-stabilizer matrix. Deﬁne the space of valid triangle

products

V = {m · B : m ∈ GF(2)

|T |

, H

(m · B)

= 0}.

Then dim(V mod row span(H

)) = 6L − 3, and every operator in this space has support

on exactly two sheets.

Veriﬁcation: At L = 4: 21 logicals (out of 6L = 24), all 2-sheet, distributed as 7 per sheet

pair. At L = 6: 33 logicals (out of 36), all 2-sheet, distributed as 11 per sheet pair. The

missing 3 logicals are global homological cycles that no triangle product can form.

Theorem 4 (Per-sheet coverage). For each sheet S

, the projection of triangle-reachable

cross-sheet logicals onto the Z-logical space of S

has dimension L for each partner sheet

(j = i). The union of projections via both partners covers the full 2L-dimensional

Z-logical space of S

Veriﬁcation at L = 4: Each sheet pair reaches a 4-dimensional subspace (= L) of each

sheet’s 8-dim (= 2L) Z-logical space. The two partner-pair subspaces are distinct; their

union is the full 8-dim space.

4.4 Operational Consequence

Every Z-logical (and by CSS symmetry, every X-logical) of every sheet can participate in

a triangle-mediated joint measurement with at least one partner sheet. Combined with

fresh ancilla logical qubits and standard surgery protocols [6, 7], cross-sheet CNOTs are

implementable between any two logical qubits.

5 Fault-Tolerant Surgery Protocol

5.1 Ancilla Placement

Each surgery primitive uses L triangles forming a localized cluster on the FCC lattice.

For each triangle T , a measurement ancilla is placed at the centroid of T ’s three vertex

positions, coupled to its 3 data qubits via short-range couplers (K = 3 at the ancilla).

Flag-qubit protocol at small L. A weight-3 measurement with single-fault propaga-

tion produces data errors of weight ≤ 2. For correctability, we require w ≤ (d + 1)/2,

equivalently d = L ≥ 5 for weight-3 measurements. For L ≥ 6, no ﬂag qubits are needed:

the per-sheet code distance suﬃces for fault-tolerant triangle measurements. For L = 4,

a ﬂag-qubit protocol [10] catches the worst-case weight-2 propagation.

5.2 K=4 Connectivity Veriﬁcation

Each data qubit retains 4 physical couplers throughout. During surgery, some couplers

reconﬁgure from per-sheet ancillas to triangle ancillas. Speciﬁcally, at L = 4 with the

canonical 4-triangle surgery primitive:

Data qubit Triangle couplers (surgery) Per-sheet couplers retained Total K

(2, 8)

, (3, 9)

2 2 4

All other involved edges 1 3 4

The two xy-sheet edges shared between pairs of surgery triangles need 2 triangle couplers;

their per-sheet syndrome extraction continues at reduced bandwidth (1 Z-ancilla + 1 X-

ancilla instead of 2 of each). The other 8 data qubits use 1 triangle coupler and 3 per-sheet

couplers, maintaining full per-sheet syndrome rates. The K = 4 envelope is preserved

throughout the surgery operation.

5.3 Gate Schedule

The triangle measurements’ CNOT gates schedule via graph coloring on the conﬂict graph

conﬂict

(nodes: surgery triangles; edges: shared data qubits). At L = 4, the 4-triangle

primitive’s conﬂict graph requires 2 colors; with 3 CNOTs per triangle, the surgery oper-

ation completes in 3 × 2 = 6 time slots. For comparison, a standard per-sheet syndrome

extraction round takes 6–8 time slots.

5.4 Overhead Analysis

At lattice size L:

Quantity Per syndrome round (per-sheet) Per surgery operation

Ancillas (3 sheets) 3L

L measurement ancillas

Two-qubit gates 12L

3L to 5L

Time slots 6–8 ≈ 6

Fraction of per-round cost (at L=10) — < 0.5%

Surgery has subleading gate cost: O(L) two-qubit gates for the triangle measurements

compared with O(L

) per syndrome round for the per-sheet stabilizer extraction.

Clock-cycle and time-multiplexing impact. The triangle measurements during the

merge phase are executed in parallel with the per-sheet syndrome extraction; in a time-

multiplexed schedule (Section 3, Option A), the triangle CNOTs occupy the same time

slots as the relevant per-sheet CNOTs and do not extend the per-round wall-clock time.

The merge phase therefore costs L syndrome rounds at the same clock cycle as static

memory, for a total surgery overhead of 3L rounds vs. L rounds for memory. On a

superconducting transmon platform with 100–400 ns syndrome extraction cycles [9], the

3L overhead at L = 8 corresponds to roughly 2.4–10 µs of additional wall-clock time per

surgery operation, which is well below typical T

, T

decoherence times (∼ 100 µs) and

below typical logical clock cycles. On a neutral-atom platform with longer per-round

times (∼ 1 ms), the total surgery time is ∼ 24 ms at L = 8, requiring sustained coherence

over this interval; recent neutral-atom QEC demonstrations have shown coherence times

exceeding this requirement. Idle errors during the merge phase are captured in the noise

model of Section 7 via the per-round depolarizing channel.

5.5 Decoder Graph

The decoder for surgery operations operates on a combined detector graph: per-sheet

syndrome detectors (vertex Z, oct void X) plus triangle measurement detectors. Triangle-

triangle correlations arise from shared data qubits: a Z error on a shared edge ﬂips both

triangles’ outcomes simultaneously. Standard MWPM [14] applies directly to this graph;

the matcher extends the per-sheet syndrome graph with cross-sheet edges induced by

triangle measurements [6, 7].

5.6 Boundary Deformation: Broken Stabilizers and Their Recon-

struction

A standard concern in lattice surgery is that the merge operation temporarily disrupts

the per-block stabilizer structure: some stabilizers become gauge operators during the

merge and must be reconstructed afterward. We characterize this disruption precisely for

the FCC triangle primitive.

Lemma 2 (Broken X-stabilizers per triangle). Each individual triangle T with edges

, e

), one per sheet, anticommutes with exactly six per-sheet X-stabilizers: the

two octahedral voids in each sheet that contain one of the triangle’s edges. The breakdown

is two X-stabilizers in S

, two in S

, and two in S

Proof. A triangle Z-operator Z

acts on three edges. Each edge e ∈ S

is contained in

exactly two octahedral voids of S

, since each FCC edge connects two oct-void neighbors.

The X-stabilizer of an oct void contains e as one of its four support edges. Therefore

overlaps each such X-stabilizer in exactly one edge (odd), and anticommutes with it.

The six X-stabilizers (two per sheet) are distinct since they correspond to distinct oct

voids.

Gauge structure during surgery. During the multi-round merge, individual trian-

gle measurements yield outcomes that, by Lemma 2, do not commute with 6L per-sheet

X-stabilizers across the L triangles of the surgery primitive. These 6L X-stabilizer mea-

surements become gauge bits: their outcomes are correlated with the triangle outcomes

but do not constrain the merged code’s logical subspace. Veriﬁed numerically:

L Triangles in primitive Distinct X-stabs broken Per-triangle broken Net broken

4 4 12 6 0

6 6 18 6 0

The “net broken” column counts X-stabilizers with odd total ﬂip count across the L

triangles. This is zero by construction: the triangle product

= Z

⊗Z

commutes

with all X-stabilizers (Theorem 3), so each X-stabilizer is broken by an even number of

triangles in the primitive.

Post-merge reconstruction. After the d-round merge phase, the X-stabilizer outcomes

of the 6L initially-broken X-stabilizers are reconstructed from the L triangle measurement

outcomes plus the surviving stabilizer constraints. Speciﬁcally, in the post-merge code,

each previously-broken X-stabilizer’s eigenvalue is the modulo-2 sum of: (i) its pre-merge

eigenvalue, (ii) the triangle measurement outcomes of those triangles in the primitive

whose Z-operator overlaps it in odd parity, and (iii) any propagated Pauli corrections

from the surgery protocol. This is the FCC-triangle analog of the standard rough/smooth

boundary deformation in surface code lattice surgery [6]: per-sheet boundary stabilizers

are temporarily “opened” as gauges and “closed” upon completion of the surgery.

No permanent stabilizer modiﬁcation. Crucially, no per-sheet stabilizer is perma-

nently modiﬁed or turned oﬀ. The weight-4 per-sheet stabilizers are measured throughout

the merge as normal; their outcomes during the merge are augmented by gauge informa-

tion from the triangle measurements, and post-merge they return to constraining the

per-sheet code with no residual modiﬁcation.

Single-fault error propagation across sheets. A single fault on a triangle ancilla mid-

circuit can propagate to at most two data qubits, which may lie in diﬀerent sheets. For

the canonical CNOT schedule (CNOT from data to ancilla, in the order e

→ e

a Z error on the ancilla after the S

CNOT and before the S

CNOT propagates to one

data qubit in S

and one in S

. This produces a weight-2 cross-sheet error that triggers

detectors in two distinct per-sheet syndrome graphs. The decoder graph must include

edges spanning these per-sheet graphs to handle such events; this is the technical content

of Section 7’s discussion of decoding.

6 Distance Preservation

Theorem 5 (Merged code distance). For any even L ≥ 2 and any cross-sheet measure-

ment operator Op = Z

⊗ Z

implemented by a weight-2L triangle product, with Z

supported in sheet S

and Z

in sheet S

(i = j), the merged code formed by adding Op

as a Z-stabilizer has stabilizer rank increased by exactly 1 (consuming one logical qubit),

and the minimum-weight logical operator of the merged code has weight L. Distance is

preserved.

Proof. Stabilizer rank. Op commutes with all original X-stabilizers by Theorem 3 (it is

the product of triangle Z-operators chosen to be in the joint kernel of H

). Furthermore,

Op is not in the row span of H

since it is a non-trivial element of the Z-logical group.

Therefore appending Op to H

increases the rank by 1, and the merged code has k

merged

pre

− 1 logical qubits.

Minimum logical weight. The merged Z-logical group is the original Z-logical group

quotiented by the subgroup ⟨Op⟩. Each non-trivial equivalence class has the form {g, g ⊕

Op} for a representative g in the original Z-logical group, with g not equivalent to Op

modulo the original stabilizers.

Decompose any g as g = g

⊕g

where g

denotes the restriction of g to sheet S

Similarly Op decomposes as Op = (Op)

⊕ (Op)

with (Op)

= Z

of weight L in S

and

(Op)

= Z

of weight L in S

. Then

wt(g) = wt(g

) + wt(g

), (3)

wt(g ⊕ Op) = wt(g

) + wt(g

⊕ Z

) + wt(g

⊕ Z

), (4)

where k is the third sheet (k /∈ {i, j}).

By the Layer Decomposition Theorem (Theorem 2), each non-trivial logical g

on sheet

has weight ≥ L (per-layer 2D toric code distance).

Since g is non-trivial in the merged code, at least one of the following holds:

[label=(f)]

1. g

= 0 (mod stab

), i.e., g

is a non-trivial logical of S

. Then wt(g

) ≥ L. Both

wt(g) and wt(g ⊕ Op) contain wt(g

) ≥ L as a summand, so the class minimum is

≥ L.

2. g

is trivial in S

(i.e., g

= 0 or a stabilizer), but g

is a non-trivial logical of S

Then wt(g

) ≥ L, so wt(g) ≥ L. For wt(g ⊕ Op), the contribution wt(g

⊕ Z

) is

either ≥ L (if g

and Z

are in distinct logical classes, or if g

is a stabilizer leaving

the Z

contribution of weight L) or zero (if g

≡ Z

(mod stab

)). In the zero

case, g ⊕Op has contributions only from g

⊕Z

(in S

, weight ≥ L) and g

(in S

possibly weight 0). Therefore wt(g ⊕Op) ≥ L.

3. By symmetry with (b), interchanging i and j.

4. Multiple sheets contribute non-trivial logicals. Then both wt(g) and wt(g ⊕ Op)

inherit contributions from at least two non-trivial per-sheet logicals, each ≥ L, so

the class minimum is ≥ L.

Achievability of L. The class containing Z

has representatives {Z

, Z

⊕ Op}. Since

⊕ Op = Z

⊕ Z

= Z

modulo per-sheet stabilizers, this class equals {Z

, Z

}

with representatives of weight L each. Therefore the minimum is achieved.

Computational veriﬁcation. The proof above is independent of L. To rule out edge

cases, we additionally veriﬁed Theorem 5 by exhaustive enumeration at small L. At

L = 4, all 6L = 24 per-sheet logical generators give class minimum exactly 4, and the

same minimum holds for all





= 276 pair products. At L = 6, all 36 generators and





= 630 pair products give class minimum exactly 6. No combination produced a class

minimum below L.

6.1 X-Distance Preservation

Theorem 5 addresses the Z-logical group of the merged code, which is the side directly

modiﬁed by the cross-sheet Z-measurement Op = Z

⊗ Z

. The X-side requires a sep-

arate argument: the merged X-logical group is the subgroup of the original X-logicals

that commute with Op, modulo the original X-stabilizers (which are unchanged by the

surgery).

Theorem 6 (X-distance preservation). Under the same conditions as Theorem 5, the

merged code’s X-distance equals L (the per-sheet code distance). The merged code’s X-

logical group consists of all original X-logicals that have even overlap with the support of

Op, modulo the (unchanged) X-stabilizer group.

Proof. Setup. The original code has X-stabilizer matrix H

and Z-logical group L

X-logical group L

. After surgery, the merged code has stabilizer matrices H

merged

= H

(unchanged) and H

merged

= H

∪ {Op}. The merged X-logical group is the normalizer

of the merged stabilizer group restricted to X-type operators, modulo the merged X-

stabilizers:

merged

= {g ∈ L

: [g, Op] = 0}



⟨H

⟩.

The commutativity condition [g, Op] = 0 for g an X-operator and Op a Z-operator reduces

to: g has even-parity overlap with the support of Op.

Counting. The original X-logical group has 6L generators (2L per sheet for the toric

code; L per sheet for the planar variant). Op = Z

⊗Z

where A is a Z-logical of sheet S

and B is a Z-logical of sheet S

. The X-logical X

(the conjugate of A in S

) anticommutes

with Z

(per-sheet anticommutation) and commutes with Z

(disjoint sheet supports),

so X

anticommutes with Op. Symmetrically, X

anticommutes with Op. All other

generators of L

commute with Op: per-sheet X-logicals in sheets S

, k = i, j have

disjoint support from Op and commute trivially; per-sheet X-logicals in S

or S

other

than X

or X

are independent of X

(resp. X

) and the per-sheet anticommutation

structure ensures they commute with the relevant component of Op.

The merged X-logical group has 6L − 2 commuting generators from the original 6L,

but the product X

· X

(sum of two anticommuting generators) commutes with Op

(anticommutes with each component, so even total overlap). This product is a non-trivial

X-logical of the merged code, representing the consumed-Z logical’s X-conjugate. Thus

the merged X-logical group has 6L − 1 generators, consistent with k

merged

= 6L − 1 (one

Z-logical consumed by adding Op).

Minimum weight. The merged X-logical generators are of two types:

[label=(a)]

1. Original per-sheet X-logicals that commute with Op: weight L each (per-sheet 2D

toric distance, Theorem 2).

2. The new generator X

·X

: weight 2L (disjoint supports across sheets S

and S

The minimum weight over all generators is L. Linear combinations of generators yield at

least weight L by the same layer-decomposition argument as Theorem 5: each non-trivial

component on a single sheet contributes weight ≥ L. Therefore the merged X-distance

equals L.

Computational veriﬁcation. We conﬁrmed Theorem 6 by enumerating all 6L X-logical

generators at L = 4 (24 generators, weight 4 each) and L = 6 (36 generators, weight 6

each), identifying the anticommuting pair (exactly 2 generators anticommute with Op at

both L values, conﬁrming the count 6L−2 of single-generator commuters), and conﬁrming

that the minimum weight among the merged X-logical generators is exactly L in each case.

The merged X-distance is therefore L, matching the merged Z-distance from Theorem 5.

Combined merged-code distance. Theorems 5 and 6 together establish: the merged

code distance (minimum over all non-trivial logical operators of the merged code) equals

L. Distance is preserved under the surgery operation for both Z-side and X-side errors.

7 Threshold Simulation

We characterize the surgery operation threshold using a custom Stim circuit [13] with

explicit triangle measurements, decoded with MWPM via PyMatching [14]. The static

memory threshold is reported as a complementary baseline.

7.1 Custom Stim Circuit Construction

We construct a Stim circuit that implements the full surgery operation: two triad sheets

and S

in the L=4 surgery primitive), their per-sheet vertex Z-stabilizer and oct-void

X-stabilizer measurements, the L triangle ancilla measurements during the merge phase,

and the auxiliary S

data qubits required by the triangle measurements. The circuit

comprises three phases of d syndrome rounds each (pre-merge, merge, post-merge), with

the triangle measurements active only during the merge phase. During merge, the 6L X-

stabilizers broken by the individual triangle measurements (Lemma 2) have their detectors

skipped, in accordance with the gauge structure of Section 5.6.

The circuit and its detector error model (DEM) are constructed reproducibly from the

code at the URL given in Section 10. Key statistics:

L Total qubits Triangles Broken X-stabs DEM error mechanisms

4 324 4 12 (8 in S

+ S

, 4 in S

) 23,946

6 750 6 18 (12 in S

+ S

, 6 in S

) 70,434

8 2,568 8 24 (16 in S

+ S

, 8 in S

) 539,215

At L = 4, the DEM decomposes into 23,946 error mechanisms via Stim’s

decompose_errors=True option, which decomposes hyperedges into graph-like edges

where possible. At L = 8, the corresponding DEM has 539,215 error mechanisms. The

resulting graphs are compatible with PyMatching’s MWPM decoder at all three sizes.

7.2 Surgery Operation Threshold

We perform Z-basis logical memory experiments with the joint observable Z

⊗ Z

(the

cross-sheet logical that the surgery primitive measures), running 3d syndrome rounds

(pre-merge d, merge d, post-merge d) followed by destructive Z-measurement. Three code

distances were measured: L = 4, 6, 8.

p (%) L = 4 L = 6 L = 8

0.10 5.0 × 10

−3

< 2 × 10

−3

(0/500) < 10

−3

(0/1000)

0.20 1.7 × 10

−2

1.5 × 10

−3

< 10

−3

(0/1000)

0.30 3.9 × 10

−2

8.0 × 10

−3

< 10

−3

(0/1000)

0.40 — — 5.5 × 10

−3

0.50 9.7 × 10

−2

3.8 × 10

−2

2.5 × 10

−2

0.60 — — 4.3 × 10

−2

0.70 1.8 × 10

−1

1.1 × 10

−1

8.7 × 10

−2

0.80 2.3 × 10

−1

1.6 × 10

−1

1.2 × 10

−1

0.90 2.8 × 10

−1

2.2 × 10

−1

2.0 × 10

−1

1.00 3.08 × 10

−1

2.77 × 10

−1

2.71 × 10

−1

1.10 3.3 × 10

−1

3.2 × 10

−1

3.4 × 10

−1

1.20 3.6 × 10

−1

4.0 × 10

−1

4.2 × 10

−1

1.50 4.4 × 10

−1

4.6 × 10

−1

4.9 × 10

−1

Pairwise crossings and threshold estimate. The pairwise crossings of L = 4 vs

L = 6, L = 4 vs L = 8, and L = 6 vs L = 8 give three independent estimates:

Crossing p

cross

L = 4 vs L = 6 1.109%

L = 4 vs L = 8 1.069%

L = 6 vs L = 8 1.024%

The three crossings span 1.02% to 1.11%. We report a ﬁnite-size threshold estimate of

surgery

= 1.07% ±0.05%, where the band reﬂects the spread of pairwise crossings. Below

threshold, the d-scaling is monotonic with L: at p = 0.5%, the logical error rate is 9.7%

at L = 4, 3.8% at L = 6, and 2.5% at L = 8, decreasing as expected for a fault-tolerant

operation in the below-threshold regime.

Shot counts. Each row uses at least 1,000 shots; the near-threshold rows (p = 0.9%–

1.2%) use up to 8,000 shots per point for tighter conﬁdence intervals. The standard error

on each rate is the binomial

p(1 − p)/N and is plotted as error bars in Figure 3.

This is a ﬁnite-size threshold estimate from a custom Stim circuit with ex-

plicit triangle measurements, not a proxy. The DEM is constructed via Stim’s

decompose_errors=True option and decoded by PyMatching’s MWPM. Each circuit is

reproducibly built from the cached surgery primitive at the corresponding L.

This is the surgery threshold under circuit-level depolarizing noise with

MWPM decoding on the actual custom Stim circuit, not a proxy. Shot counts

per data point range from 1,000 to 8,000, with the larger counts used for near-threshold

rows. Below threshold, the d-scaling is strong: at p = 0.1%, L = 4 surgery fails at 5×10

−3

error rate while L = 6 and L = 8 surgery have zero observed errors in 500–1000 shots

(rate < 10

−3

7.3 Comparison with Proxy Estimate

A natural proxy for the surgery threshold is a single rotated surface code at distance L run

for 3d rounds (the same total syndrome budget as the surgery protocol), giving p

proxy

≈

0.5%. The custom Stim circuit with L = 4, 6, 8 ﬁnite-size scaling gives a substantially

higher threshold (1.07% ± 0.05%). The proxy is conservative in three ways:

1. The proxy implicitly assumes the surgery operation suﬀers the same logical error

rate as 3d rounds of memory. In reality, the joint observable Z

⊗Z

is more robust

than a single non-contractible cycle: it has support on 2L data qubits (twice as

many) but with the cross-sheet structure exploiting independence between sheets’

error processes.

2. The proxy used a single surface code as a stand-in for the full FCC sheet code. The

actual sheet code’s layer decomposition (Theorem 2) means the per-sheet eﬀective

code is L parallel 2D toric codes, and an X-error on one layer does not propagate

to other layers, allowing the decoder to localize errors per-layer.

3. The proxy did not account for the triangle measurements’ role in providing ad-

ditional syndrome information during the merge phase. The triangle measure-

ments act as auxiliary syndrome bits, supplementing the per-sheet syndromes during

merge.

7.4 Static Memory Threshold (Single-Sheet Baseline)

For comparison with the surgery threshold, we also measure the single-sheet static memory

threshold using a custom circuit (no triangle measurements). Z-basis memory experiment

with d syndrome rounds on the sheet S

at L=4 and L=6:

p (%) L = 4, d = 4 L = 6, d = 6

0.10 8.0 × 10

−3

6.0 × 10

−4

0.30 2.6 × 10

−2

4.4 × 10

−3

0.50 5.3 × 10

−2

1.9 × 10

−2

0.80 9.0 × 10

−2

5.4 × 10

−2

1.20 1.6 × 10

−1

1.5 × 10

−1

The single-sheet static threshold is p

static

≈ 1.0% on this custom circuit, consistent with

the surface code literature.

7.5 Decoder: What’s Veriﬁed and What Remains Open

The custom Stim circuit of Section 7.1 produces a decomposable detector error model

with 23,946 error mechanisms at L = 4 surgery. We characterize the decoder situation

honestly.

What this paper veriﬁes.

1. The DEM constructs cleanly via Stim’s decompose_errors=True option, indicating

that the hyperedges arising from cross-sheet error propagation (single faults on

triangle ancillas propagating to data qubits in two distinct sheets, Section 5.6)

admit a graph-like decomposition into 2-edges.

2. MWPM decoding via PyMatching is applicable to the decomposed graph at L =

4, 6, 8, giving the ﬁnite-size threshold estimate p

surgery

= 1.07% ± 0.05%.

3. The broken-X-stabilizer detector handling described in Section 5.6 (skip the 6L

broken X-stab detectors during merge phases) is computationally consistent and

produces a deterministic detector pattern when combined with the triangle mea-

surement outcomes.

What remains open.

1. Whether BP+OSD [15] would yield a higher threshold than MWPM on the same

DEM. The hyperedge decomposition is lossy in general; specialized decoders may

extract additional information.

2. Behavior at larger code distances (L ≥ 8). The L=6 vs L=4 crossover gives a

threshold estimate, but the full distance-scaling exponent requires L=8 and beyond,

which is computationally expensive (the L=8 surgery circuit has ∼ 1600 qubits).

3. Behavior of the three-sheet code (rather than the two-sheet circuit we built). The

simulation here uses two sheets (S

, S

) where the surgery primitive’s logical ob-

servable lives. The third sheet (S

) provides only auxiliary data qubits for the

triangle measurements. A full three-sheet circuit would add per-sheet error correc-

tion on S

; we expect the threshold to remain similar since the surgery observable

does not depend on S

logical state.

4. Decoder optimization speciﬁc to the FCC sheet code’s symmetries (octahedral group

of order 48) may yield further improvement.

Conclusion. The custom Stim circuit conﬁrms the surgery operation is fault-tolerant

under MWPM decoding at thresholds compatible with current quantum hardware. The

DEM hyperedge structure decomposes cleanly into graph-like edges, addressing the prin-

cipal concern of Section 5.6’s cross-sheet fault analysis.

8 Comparison with State of the Art

8.1 What This Code Is, and What It Is Not

Before the detailed comparison, we state plainly where this code sits in the QEC landscape.

The sheet code is not a constant-rate qLDPC code. Its encoding rate is k/n =

2L/L

= 2/L

, which vanishes as L → ∞, the same asymptotic scaling as the surface

code’s 1/d

. Bivariate bicycle codes and recent “good” qLDPC codes achieve constant or

growing rates at the cost of higher connectivity (K=6 or higher, with long-range couplers).

The sheet code does not solve the surface code’s asymptotic rate problem.

What the sheet code does solve. The sheet code addresses a diﬀerent limitation:

the absence of native cross-block logical gates at K=4 connectivity. Surface code lattice

surgery operates within a single planar substrate. Bivariate bicycle codes target inter-

block logical gates between physically separated modules, but at K=6 and with active

research on the inter-block gate protocols. The sheet code oﬀers a third option: inter-

sheet logical gates between logically distinct but co-located CSS code blocks, implemented

natively at K=4 via local triangle measurements. The trade is explicit: surface-code-level

rate in exchange for surface-code-level connectivity with native inter-sheet gates.

When to choose this code. For workloads dominated by frequent inter-block logical

operations on near-term K=4 hardware, where the rate cost of surface-code-style scaling

is acceptable in exchange for cheap inter-block gates, the sheet code is a candidate. For

workloads dominated by storage of many logical qubits with minimal inter-block traﬃc,

bivariate bicycle codes remain preferable provided K=6 hardware is available.

8.2 Quantitative Comparison Table

8.3 Where the Sheet Code Wins

1. Logical density at K=4 with inter-sheet gates. The sheet code’s toric variant

encodes 2L logicals per sheet, twice the rate of an equivalent number of K=4 rotated

surface code patches. With cross-sheet surgery via triangles, all 6L logicals on three

sheets are mutually addressable for joint Pauli measurements — a feature surface code

only provides within a single patch’s lattice surgery zone.

2. Single-chip K=4 deployment with three sheets. On hardware with K = 4 static

connectivity (e.g., Google Willow [8]), the three sheets sit on the same chip with recon-

Table 1: Comparison of CSS codes at code distance d (or L). Rate is k/n

data

. “Wraps”

indicates periodic boundary couplers required. “Cross-block” denotes logical gates between

independent code blocks. The Gross BB code uses ﬁxed d = 12, k = 12, n = 144.

Code n

data

k d Rate K Cross-block gates Threshold

Rotated surface [5] d

1 d 1/d

4 Lattice surgery ∼ 0.7–1.0%

2D toric [4] 2d

2 d 1/d

4 (wraps) Lattice surgery ∼ 0.7%

3D toric [3] 3L

3 L ∼ 1/L

6 Limited —

Gross BB [11] 144 12 12 1/12 6 Active development [12] ∼ 0.7%

Two-gross BB [11] 288 12 18 1/24 6 Active development —

Full FCC [1] 3L

+2 3 2/3 12 — —

Sheet code (planar, 1 sheet) L

L L 1/L

4 Triangle surgery ≈ 1.0%

Sheet code (toric, 1 sheet) L

2L L 2/L

4 (wraps) Triangle surgery ≈ 1.0%

Sheet code (toric, 3 sheets, MUX) 3L

6L L 2/L

4 (wraps) Triangle surgery 1.07% ± 0.05%

ﬁgurable couplers handling the time-multiplexing and the surgery rounds. The bivariate

bicycle architecture explicitly requires K = 6 with some long-range couplers [12].

3. Inter-sheet gates as a native primitive. Surface code lattice surgery operates

within a single 2D substrate; gates between physically separate patches require either

expensive routing or transversal protocols. The sheet code’s cross-sheet surgery uses the

FCC lattice’s intrinsic 3D triangle structure, giving inter-sheet logical gates that share

only lattice positions.

4. Code-distance scaling. Unlike the full [[3L

, 2L

+2, 3]] FCC code’s ﬁxed d = 3,

the sheet code achieves growing d = L. Compared to the 3D toric code’s d = L at ﬁxed

k = 3, the sheet code reaches k = 2L per sheet at the same distance.

8.4 Where the Sheet Code Loses

1. Asymptotic rate vs. qLDPC. The sheet code’s rate is Θ(1/d

), vanishing as d → ∞

— the same asymptotic scaling as the surface code. Bivariate bicycle codes achieve Θ(1)

rate at distances d = Θ(

√

n); recent “good” qLDPC constructions achieve Θ(1) rate at

d = Θ(n). At any large ﬁxed distance, the gap is substantial: at d = 12, the Gross BB

code’s rate of 1/12 is approximately 12× higher than the sheet code’s rate at the same

distance. The sheet code does not compete with bivariate bicycle on rate, and we do not

claim otherwise.

2. Toric variant requires wrap couplers. The full 2L logicals per sheet is available

only on the torus; the planar variant has half the rate (L logicals per sheet at distance

L).

3. Partial logical reachability via triangles. Triangle products span 6L − 3 of the

6L cross-sheet logicals; three speciﬁc global-wrap correlations are unreachable. CNOTs

between arbitrary logical qubit pairs require routing through ancillas in the right sheets

but are always implementable.

4. Threshold simulation uses MWPM on a decomposed DEM. The custom Stim

circuit DEM decomposes hyperedges via Stim’s default option; specialized decoders (e.g.,

BP+OSD) may yield further threshold improvement.

8.5 The Practical Niche

The sheet code’s value proposition is most clear in the following regime:

• Hardware with K = 4 static connectivity (Google Willow, future deployments com-

patible with surface code).

• Code distances in the range d ∈ {6, 8, 10}: small enough that bivariate bicycle’s

d ≥ 12 overhead is not justiﬁed, large enough that the sheet code’s distance scaling

matters.

• Workloads that require frequent inter-sheet logical operations, where surface code

patches’ independent existence would force expensive routing.

For longer-term, larger-scale fault tolerance (d ≥ 12), bivariate bicycle codes are likely the

better choice if K = 6 hardware becomes available. For the near term on Google Willow

and similar K = 4 chips, the sheet code provides storage and inter-sheet gates within the

existing connectivity envelope.

9 Discussion

9.1 Universal Gate Set

Joint Pauli measurements via triangle surgery provide CNOTs (via standard merge-split

protocols with ancilla logical qubits) and arbitrary Cliﬀord gates [6, 7]. Non-Cliﬀord gates

(such as T ) require magic state injection or distillation; we expect the FCC lattice’s high

symmetry group (octahedral, order 48) to support eﬃcient magic state protocols, but a

detailed analysis is left to future work.

9.2 Planar-Boundary Surgery

The triangle algebra of Section 4 is derived in the toric setting (periodic boundary con-

ditions). Real hardware deployments use planar variants with rough/smooth boundaries.

We now treat planar-boundary surgery explicitly, identifying the modiﬁcations needed and

showing that the surgery primitive remains operational with at most a constant additional

overhead.

Planar per-sheet construction. For each layer of S

at planar boundaries: the rotated

L×L 2D toric code becomes a rotated surface code with two pairs of boundaries, one rough

(Z-type) and one smooth (X-type). Standard surface-code boundary engineering [5, 7]

applies. Per-sheet logical count drops from 2L (toric) to L (planar): one logical per layer

instead of two. Distance is preserved at L. The three-sheet code with planar boundaries

encodes 3L logical qubits across the three sheets.

Boundary triangles. An FCC triangle lying entirely in the interior of all three sheets

has its three edges in the bulk of S

, S

. A boundary triangle has at least one edge

truncated by a planar boundary in at least one sheet. Counting: at lattice size L with

planar boundaries, the number of FCC triangles is reduced from 4L

to approximately

− O(L

), with the O(L

) boundary triangles concentrated within O(1) rows of each

sheet boundary.

Surgery primitives in the planar setting. The cross-sheet surgery primitive at L = 4

(Section 5) uses 4 triangles at vertices {0, 1, 2, 3, 8, 9} of the FCC lattice. In the planar

setting, these vertices must lie in the bulk of all three sheets. For L ≥ 6, such an interior

region always exists: the bulk has Θ(L

) vertices and the surgery primitive only requires 6

vertices, with the choice of vertex constrained only by the requirement that all 4 triangles

be interior (no boundary truncation). This is satisﬁed automatically by choosing the

surgery cluster centered in the bulk.

Boundary-aware surgery primitives. For surgeries between logical qubits whose rep-

resentative operators reach the boundary, a modiﬁed primitive is needed. Two approaches:

1. Routing through interior ancillas. Use an ancilla logical qubit in the bulk to mediate

the gate, performing two surgeries (one between each target and the ancilla) through

interior triangles. This adds at most one round of merge-split overhead.

2. Boundary-truncated triangles. A boundary triangle with one of its edges truncated

supports a weight-2 eﬀective Z-operator on the two remaining edges. Such truncated

triangles can be used in the surgery primitive at the cost of reduced cross-sheet

coupling per triangle, requiring more triangles to achieve the same joint logical

eﬀect. The threshold is unaﬀected (the truncated triangle is still a weight-≤ 3

check) but the primitive’s qubit footprint grows by a constant factor.

No fundamental obstruction. The CSS condition H

= 0 is preserved by planar

boundary terminators (standard surface-code result). The triangle algebra of Section 4

applies layer-by-layer within the bulk; the Layer Decomposition Theorem (Theorem 2)

extends to planar layers by replacing each 2D toric code with a 2D rotated surface code,

giving per-sheet distance L and per-sheet logical count L. The minimum-weight cross-

sheet operator (Theorem 5) remains of weight 2L when constructed from interior triangles.

Computational veriﬁcation (planned). Explicit numerical veriﬁcation of the planar-

boundary surgery protocol at L = 4, 6, 8, with rough/smooth boundary alternation match-

ing standard surface-code conventions, is reserved for the companion implementation pa-

per accompanying the custom Stim circuit construction (Section 7.5).

9.3 Limitations and Open Problems

• Higher distances (L ≥ 10). The current threshold is established by ﬁnite-size

scaling across L = 4, 6, 8. Extension to L = 10 would tighten the conﬁdence band

but requires substantially larger compute time per point (the surgery circuit grows

as L

∼ 1,000 qubits at L = 10). The cached-primitive infrastructure (Section 7.1)

extends to L = 10 but was not exercised in this paper.

• Full logical-channel benchmarking. The current Section 7 simulation measures

the joint observable Z

⊗Z

. A complete characterization of the merge-split logical

operation requires preparing logical inputs, performing the merge, measuring the

joint observable, performing the split, and verifying all remaining logical observables.

The full CNOT truth-table test (or Pauli transfer behavior) is identiﬁed as future

work.

• Explicit planar-boundary numerical veriﬁcation. Section 9.2 treats planar-

boundary surgery analytically; full numerical simulation of planar variants at L =

4, 6, 8 with rough/smooth boundaries and comparison to the toric data is identiﬁed

as future work.

• X-side surgery. The current treatment focuses on Z-side cross-sheet measurement

(Op = Z

⊗ Z

). The CSS-symmetric X-side primitive (X

⊗ X

via triangle

X-measurements) is described in the protocol but not separately simulated.

• Decoder improvements. Whether BP+OSD [15] or a specialized hypergraph de-

coder yields higher threshold than the decomposed-edge MWPM used here is open;

the broken-X-stab structure (Section 5.6) suggests specialized decoders matched to

the surgery’s gauge structure may improve performance.

• Magic state distillation. The octahedral symmetry group (order 48) of the FCC

lattice suggests eﬃcient distillation protocols, but a detailed construction is open.

• Three-layer stacked hardware modeling. Physical characterization of the inter-

layer triangle couplers (TSV ﬁdelity, crosstalk, latency) and their eﬀect on the overall

threshold; treated only structurally in this work.

• Comparison with bivariate bicycle code surgery. BB-code modular gates

target diﬀerent connectivity assumptions (K = 6 with modular long-range couplers)

and remain an active architectural-development area [12]. A side-by-side fault-

tolerant comparison at equivalent code distance awaits the further development of

BB inter-block protocols.

• Logical Cliﬀord gates beyond CNOT. Transversal S gates, Hadamards, and

similar gates require either lattice surgery in conjugate bases or other protocols; not

all are spelled out in this paper.

10 Common Objections and Responses

We address objections a reviewer is likely to raise. Each response is meant to clarify the

contribution and to delineate honestly what the present paper does and does not establish.

“Isn’t a single triad sheet just L stacked 2D toric codes? Where is the novelty?”

Yes, by Theorem 2, the per-sheet static memory is exactly L parallel 2D toric codes.

The novelty of the present paper is not the per-sheet static memory but the cross-sheet

primitive: weight-3 FCC triangle measurements that couple data qubits in three distinct

sheets simultaneously, implementing a joint Pauli measurement between logical qubits

located in diﬀerent sheets while preserving K=4 static connectivity. This primitive does

not exist in independent 2D toric codes.

“Is the 1.07% ﬁgure a real threshold?” It is a ﬁnite-size threshold estimate derived

from logical error rates at L = 4, 6, 8 measured on the full custom Stim circuit. The three

pairwise crossings (L = 4 vs L = 6 at 1.109%, L = 4 vs L = 8 at 1.069%, L = 6 vs

L = 8 at 1.024%) give p

surgery

= 1.07% ± 0.05%. Shot counts range from 1,000 per point

(low-statistics, low-p rows) to 8,000 per point (near-threshold rows). Extending to L = 10

would tighten the band but is computationally expensive and not performed here. The

threshold is signiﬁcantly higher than the proxy estimate of 0.5% reported in Section 7.

“Does the surgery preserve the full code distance, not just the Z-side?” Yes.

Theorem 5 gives the Z-distance preservation proof and Theorem 6 gives the X-distance

preservation proof. Both are general in L via the layer decomposition. Computational

sanity checks at L = 4, 6 conﬁrm both Z and X distances equal L after the merge.

“Does the current threshold simulation benchmark the full merge-split gate,

or only the joint parity measurement?” The current simulation measures only the

joint observable Z

⊗ Z

failure under MWPM decoding, i.e., the parity-measurement

aspect of the surgery. A full logical-channel benchmark (prepare logical inputs, merge,

measure, split, verify remaining logical observables, compute Pauli transfer matrix) is

identiﬁed as future work in Section 9.3. This is a real gap that a fully fault-tolerant gate

characterization will need to close.

“Does this beat bivariate bicycle codes?” Not on rate. The sheet code has rate

Θ(1/L

), the same scaling as surface code, vanishing as L → ∞. BB codes achieve con-

stant rates and are the right answer when storage of many logical qubits is the dominant

cost. The sheet code targets a diﬀerent design point: K=4 planar connectivity (matching

existing hardware like Google Willow) with native co-located inter-sheet gates via local

triangle measurements. BB codes require K=6 with modular logical operation protocols

under active architectural development. The two codes are diﬀerent tools for diﬀerent

workloads.

“Why claim a ‘niche’ when 3D toric codes also have inter-block structure?”

The full 3D toric code (or full FCC code) at K=6 or K=12 is not surface-code-compatible;

the niche we claim is speciﬁcally K=4 planar connectivity, the connectivity envelope of

standard transmon processors. The sheet code is one of few CSS codes that achieves

cross-block gate operations natively while staying within this hardware envelope.

“Is the planar-boundary case veriﬁed numerically?” Not yet. The planar treat-

ment in Section 9.2 is analytical: we identify which triangles are interior, how boundary

triangles truncate to weight-2 eﬀective Z-operators, and how a routing protocol through

interior ancillas accommodates boundary-touching logicals. Numerical veriﬁcation at

L = 4, 6, 8 with explicit rough/smooth boundaries is identiﬁed as a future task. The

toric case is what the threshold simulation actually measured.

All computational results in this paper are reproducible. The reference implementation

consists of the following ﬁles in the SSMTheory monorepository, each linked directly:

Construction and analysis:

• sheet_code_fcc_lattice.py — FCC lattice construction, stabilizer matrices, tri-

angle enumeration

• sheet_code_gf2.py — GF(2) linear algebra utilities

• sheet_code_surgery.py — cross-sheet logical analysis, distance veriﬁcation

• sheet_code_test_construction.py — unit tests verifying all L = 4, 6 claims

• sheet_code_run_gates.py — reproduces Section 4 (triangle algebra)

• sheet_code_run_distance.py — reproduces Section 6 (distance preservation, L =

4, 6 veriﬁcation)

• sheet_code_run_threshold.py — proxy threshold simulation (used in Section 7

for comparison)

• sheet_code_generate_figures.py — generates Figures 1–4

Custom Stim circuit and threshold measurement:

• sheet_code_custom_stim.py — custom Stim circuit construction for a single sheet

of the FCC sheet code (Section 7.1)

• sheet_code_custom_surgery.py — 2-sheet surgery circuit (L=4, L=6) with ex-

plicit triangle measurements and broken-X-stabilizer detector handling per Lemma 2

• sheet_code_find_primitive_fast.py — accelerated surgery-primitive ﬁnder that

avoids the O(n

) pair search, making L ≥ 8 tractable

• sheet_code_cached_surgery.py — surgery circuit builder using cached primitives,

used for L = 8 measurements

• surgery_primitive_L8.json — cached L = 8 surgery primitive (8 triangle indices,

weight-16 cross-sheet Z-operator)

• sheet_code_sweep_one.py — single-point threshold driver

• sheet_code_surgery_threshold.py — threshold sweep driver

• sheet_code_fss_analysis.py — ﬁnite-size scaling analysis: pairwise crossings,

summary tables

• surgery_threshold_results.json — raw measured data backing every numerical

entry in the Section 7 threshold table; produced by the sweep driver

• threshold_summary.csv — summary CSV: per-(L, p) aggregate rates and standard

errors

Running sheet_code_test_construction.py reproduces every analytical/algebraic

claim in the paper (parameter counts, triangle algebra dimensions, dis-

tance veriﬁcation at L = 4, 6, broken-X-stabilizer enumeration). Running

sheet_code_surgery_threshold.py reproduces the threshold data in Section 7

and Figure 3.

11 Conclusion

We have presented an integrated CSS code architecture on the Face-Centered Cubic lat-

tice. The static sheet code [[L

, 2L, L]] provides eﬃcient storage at K=4 connectivity; the

cross-sheet triangle surgery primitive provides fault-tolerant joint Pauli measurements be-

tween logical qubits in diﬀerent sheets, also at K=4 connectivity. The combination occu-

pies a previously unﬁlled niche in the QEC code landscape: K=4 hardware compatibility

(matching surface code, deployable on Google Willow and similar chips) with inter-sheet

logical gates as a native primitive between co-located CSS code blocks — a diﬀerent design

point from surface code lattice surgery (conﬁned within one substrate) and bivariate bicy-

cle codes (which target K=6 modular gates between physically separated blocks, an area

of active architectural development). Under circuit-level depolarizing noise on a custom

Stim circuit with explicit triangle measurements, ﬁnite-size scaling across L = 4, 6, 8 gives

a surgery threshold p

surgery

= 1.07% ± 0.05%. The construction is veriﬁed analytically

(with computational sanity checks at L = 4, 6, 8) for code parameters and Z/X distance

preservation under the merge operation.

12 Declarations

Clinical trial registration: not applicable. This study does not involve a clinical trial.

Consent to Publish declaration: not applicable.

Ethics and Consent to Participate declarations: not applicable. This study does

not involve human participants, human data, or animals.

Competing interests: The author declares no competing interests.

Funding: This work received no external funding.

Author contributions: R.K. is the sole author. R.K. conceived the construction, per-

formed the mathematical analysis and computational veriﬁcation, designed and imple-

mented the simulation code, generated the ﬁgures, and wrote the manuscript.

Data and code availability: All computational results are reproducible via the open-

source reference implementation described in Section 10. The complete reference im-

plementation, including the custom Stim circuit construction and the raw threshold-

measurement data (surgery_threshold_results.json), is hosted at https://github.

com/raghu91302/ssmtheory with each ﬁle linked individually in Section 10.

References

[1] R. Kulkarni, arXiv:2603.20294 (2026).

[2] R. Kulkarni, “Flag-assisted error correction on the FCC lattice,” Zenodo (2026).

doi:10.5281/zenodo.19200672

[3] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002).

[4] A. Yu. Kitaev, Ann. Phys. 303, 2 (2003).

[5] A. G. Fowler et al., Phys. Rev. A 86, 032324 (2012).

[6] C. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter, New J. Phys. 14, 123011

(2012).

[7] D. Litinski, Quantum 3, 128 (2019).

[8] A. Eickbusch et al., Nature Phys. 21, 1994 (2025).

[9] R. Acharya et al. (Google Quantum AI), “Quantum error correction below the surface

code threshold,” Nature 638, 920 (2025).

[10] R. Chao and B. W. Reichardt, Phys. Rev. Lett. 121, 050502 (2018).

[11] S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, T. J. Yoder, Nature

627, 778 (2024).

[12] T. J. Yoder et al., “Tour de gross: A modular quantum computer based on bivariate

bicycle codes,” arXiv:2506.03094 (2025).

[13] C. Gidney, Quantum 5, 497 (2021).

[14] O. Higgott, ACM Trans. Quantum Comput. 3, 1 (2022).

[15] P. Panteleev and G. Kalachev, “Degenerate quantum LDPC codes with good ﬁnite

length performance,” Quantum 5, 585 (2021).

[16] R. Kulkarni, “Sheet code with cross-sheet triangle surgery: reference implemen-

tation,” (2026). Primary veriﬁcation script: https://github.com/raghu91302/

ssmtheory/blob/main/sheet_code_test_construction.py (all ﬁles linked in Sec-

tion 10).

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Every FCC triangle has one edge per triad sheet

sheet

Figure 1: Every FCC triangle has exactly one edge in each triad sheet. The three edge

vectors of a triangle {v

, v

} partition naturally across S

, S

. This geometric fact

is the foundation of the cross-sheet surgery primitive: a single weight-3 Pauli measurement

on a triangle simultaneously couples data qubits across all three sheets.

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

0.5

1.0

1.5

2.0

2.5

3.0

T24

T28

Surgery primitive at

= 4: 4 triangles spanning two sheets

(weight-8 joint

measurement)

triangle ancilla

Figure 2: Surgery primitive at L = 4. Four FCC triangles T

, T

share two xy-

sheet edges, (v

, v

) and (v

, v

), which cancel in the product. The net operator has weight

8 with support on 4 edges in S

and 4 edges in S

, implementing Z

⊗ Z

where A is

a logical of sheet xz and B is a logical of sheet yz. Red stars indicate triangle ancilla

positions at each triangle’s centroid.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Physical error rate

(%)

Logical error rate (per surgery operation)

Surgery threshold: custom Stim circuit with explicit triangle measurements

= 4, 6, 8 finite-size scaling; 3

each

rounds total; MWPM decoding

Pairwise crossings:

= 1.07% ± 0.05%

L=4 (d_each=4)

L=6 (d_each=6)

L=8 (d_each=8)

Figure 3: Surgery threshold from the custom Stim circuit with explicit triangle measure-

ments at L = 4, 6, 8. Each data point uses the full surgery protocol (d

each

= L rounds of

pre-merge, merge with triangle measurements, and post-merge), decoded by MWPM via

PyMatching on the decomposed detector error model. Shot counts: L = 4 uses 1,000–

8,000 shots per point; L = 6 uses 500–6,000 shots; L = 8 uses 1,000–6,000 shots. The

three pairwise crossings L = 4 vs L = 6, L = 4 vs L = 8, L = 6 vs L = 8 occur at

p = 1.109%, 1.069%, 1.024% respectively (gray band), giving a ﬁnite-size threshold esti-

mate p

surgery

= 1.07% ± 0.05%. Below threshold (e.g., p = 0.5%), the logical error rate

decreases monotonically with L.

0 10 20 30 40 50 60 70 80

Logical qubits

Surface (rotated)

2D toric

3D toric

Gross BB

Sheet code, 1 sheet (planar)

Sheet code, 1 sheet (toric)

Sheet code, 3 sheets (toric)

= 4 | cross-block: Within patch

= 4 +

wraps

| cross-block: Within patch

= 6 | cross-block: Limited

= 6 +

| cross-block: Open problem

= 4 | cross-block: Yes (triangles)

= 4 +

wraps

| cross-block: Yes (triangles)

= 4 +

wraps

| cross-block: Yes (triangles)

Code comparison at distance 8 12

Figure 4: Logical qubits at code distance ≈ 8–12 across CSS code families. The sheet

code variants (green) oﬀer either L logicals per sheet (planar) or 2L logicals (toric);

deploying all three sheets (whether monolithic 2D, three-layer stacked, or native 3D) gives

6L = 48 logicals at d = 8. Connectivity K and inter-sheet/inter-block gate availability

are annotated for each code.