Three Sheets, One Architecture

Three Sheets, One Architecture:

Inter-Sheet Joint Pauli Measurements 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 joint Pauli measurements

between co-located CSS code blocks. The construc-

tion 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 − 1)

, 𝐿, 𝐿 − 1]]

as a planar variant), uniform weight-4 stabilizers, and

𝐾=4

active per-qubit connectivity. Three triad sheets share the FCC lattice geometry, en-

coding

6𝐿

logical qubits at distance

𝐿

, deployable as a monolithic 2D chip (small

𝐿

a three-layer stacked architecture, or native 3D hardware. Second, we introduce a

fault-tolerant surgery protocol using local FCC triangle measurements to implement

joint Pauli measurements between logical qubits in dierent sheets.

Our quantitatively characterized claims are:



Static memory baseline.

Single-sheet logical memory simulation at

𝐿 ∈

{4, 6}

on a custom Stim circuit gives an FSS crossing near

≈ 1.0%

(Section 7.4),

consistent with surface-code-like behavior under circuit-level depolarizing noise

with MWPM decoding. This is reported as a baseline, not the central charac-

terized contribution.



Joint-

𝑍𝑍

surgery (ZZ-merge): FSS-crossing threshold estimate

1.07% ± 0.05%

for the toric variant and

0.76% ± 0.05%

for the boundary-

aware planar variant, from pairwise crossings at

𝐿 ∈ {4, 6, 8}

(Sections 7, 9.2).

This is the central characterized FT primitive.



Joint-

𝑋𝑋

surgery (XX-merge): FSS-crossing threshold estimate

≈

1.0% ± 0.1%

on the toric variant (Section 9.3). The planar boundary-aware

variant has a structural blocker for the dual primitive under the standard

boundary choice.

As a synthesis claim, we verify the three-sheet Horsman [5] CNOT logical truth table

𝐿 ∈ {4, 6, 8}

on the toric variant (

𝑝 = 0

, deterministic on all four computational-

basis inputs, DEM builds cleanly), and observe a

4.2𝜎

distance improvement at

𝑝 = 10

−3

from

𝐿 = 4

𝐿 = 6

under MWPM. However, a direct diagnostic shows

the protocol as constructed is

not yet fault-tolerant in the standard sense

: increasing

the per-merge depth

𝑑

each

does not reduce LER (at

𝐿 = 4

𝑑

each

= 8

gives

higher

LER

than

𝑑

each

= 4

), because the Pauli-frame correction enters the observable through

specic gauge measurements that are not themselves repeated

𝑑

times. Reaching

FT-thresholded status for the full CNOT requires either FT-repeated gauge mea-

surements or a gauge-aware decoder; both are concrete follow-up directions.

The architecture’s

deployment niche

is therefore: high-density FT memory plus

FT joint-Pauli measurements at

𝐾=4

, with the full universal Cliord layer iden-

tied as a near-term protocol renement rather than a structural blocker. The

density advantage is concrete and consistent:

∼ 16

physical qubits per logical at

𝐿 = 4

in a three-sheet deployment vs

∼ 31

for distance-4 rotated surface code (a

∼ 1.9×

improvement, essentially at in

𝐿

). Application classes that consume only

memory and joint-Pauli primitives — quantum networking nodes, large-scale logical

benchmarking, NISQ-to-FT bridge demonstrations — are immediately addressable

on existing or near-term

𝐾=4

hardware (Google Willow, IQM, OQC). Compared to

surface code lattice surgery (conned within a single patch), the sheet code provides

2𝐿

logicals per sheet at the same distance and connectivity; compared to bivariate

bicycle codes (inter-block gates between physically separated

𝐾=6

modules), the

sheet code achieves inter-sheet operations 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

[2, 4] pairs

𝐾=4

planar hard-

ware compatibility with mature fault-tolerant protocols, including lattice surgery [5, 6]

for logical gates within a single 2D substrate. The

bivariate bicycle (BB) code

family [10]

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 [11].

This paper introduces a third option that occupies a previously unlled niche: a CSS

code at

𝐾=4

connectivity (matching the surface code) with

native inter-sheet joint Pauli

measurements

between co-located CSS code blocks — a fault-tolerant primitive that the

surface code provides only within a single patch’s lattice surgery zone. This is a dierent

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 sepa-

rated

𝐾=6

modules and where modular logical operation protocols are an area of active

architectural development). Full FT logical CNOT composition between sheets is a syn-

thesis of two such joint-Pauli primitives that we verify at the truth-table level but identify

as not yet fault-tolerantly characterized in this work (Section 10); the primary character-

ized contribution is the joint-Pauli primitive itself. 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

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 dierent

sheets, with the merged code preserving distance

𝑑 = 𝐿

The two ingredients work in tandem: the sheet code provides ecient storage at

𝐾=4

;

the triangle surgery primitive provides fault-tolerant inter-sheet joint Pauli measurements

at the same envelope. The result lls a gap that neither surface code nor bivariate bicycle

codes address:

𝐾=4

planar connectivity with inter-sheet joint Pauli measurements as

a native primitive. Composing these primitives into full FT logical gates (Cliord cir-

cuits, magic-state distillation, etc.) requires the further protocol or hardware renements

identied in Section 10.

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

𝑎

, 𝑣

𝑏

, 𝑣

𝑐

}

partition naturally across

𝑆

𝑥𝑦

𝑆

𝑥𝑧

𝑆

𝑦𝑧

. A single weight-

3 Pauli measurement on a triangle simultaneously couples data qubits across all three

sheets.

Summary of results.

(i) sheet code with parameters

[[𝐿

, 2𝐿, 𝐿]]

on a torus,

[[𝐿(𝐿 −

, 𝐿, 𝐿 −1]]

on a plane, uniform weight-4 stabilizers,

𝐾=4

data-qubit connectivity (Sec-

tion 2); (ii) three-sheet hardware architecture via time-multiplexed syndrome extraction

(Section 3); (iii) triangle algebra spanning

6𝐿−3

of the

6𝐿

cross-sheet Z-logicals (Sec-

tion 4); (iv) fault-tolerant surgery protocol with

𝐾=4

verication and

𝑂(𝐿)

gate overhead

(Section 5); (v) distance preservation under merge for both Z- and X-sides (Section 6); (vi)

threshold simulation via custom Stim at

𝐿 = 4, 6, 8

giving

𝑝

𝑍𝑍

= 1.07% ± 0.05%

(toric)

and

𝑝

planar

= 0.76% ± 0.05%

(boundary-aware planar) (Section 7); (vii) comparison with

state of the art (Section 8); (viii) CSS-dual XX-merge primitive demonstrated with com-

parable threshold

≈ 1.0%

𝐿 = 4, 6, 8

on the toric variant (Section 9.3); (ix) three-sheet

Horsman CNOT logical truth table veried at

𝐿 ∈ {4, 6, 8}

but

not yet fault-tolerantly

characterized

(Section 10).

Claim status matrix.

To make the epistemic status of each claim transparent up front:

Claim Status Evidence / Section

[[𝐿

, 2𝐿, 𝐿]]

code parameters (toric);

[[𝐿(𝐿 − 1)

, 𝐿, 𝐿 − 1]]

(planar)

Proven Theorem 2, §2

𝐾=4

active per-qubit connectivity Veried by con-

struction

§5, gate schedule

Z- and X-distance preservation under

merge

Proven (general

𝐿

) + compu-

tational checks

(

𝐿 = 4, 6, 8

)

Theorems 5, 6

Static memory FSS-crossing baseline

≈ 1.0%

(toric, single sheet)

Simulation,

𝐿 ∈

{4, 6}

, MWPM

§7.4

ZZ-merge FSS-crossing threshold

1.07%±0.05%

(toric) /

0.76%±0.05%

(planar boundary-aware)

Simulation,

𝐿 ∈ {4, 6, 8}

MWPM, FSS

data-collapse t

available

§7, §9.2

XX-merge FSS-crossing threshold

≈

1.0% ± 0.1%

(

toric only

; planar XX

has structural blocker)

Simulation,

𝐿 ∈ {4, 6, 8}

MWPM

§9.3

Three-sheet Horsman CNOT logical

truth table at

𝑝 = 0

Veried de-

terministic at

𝐿 ∈ {4, 6, 8}

𝑑

each

∈ {2, 3, 4}

𝐿 = 4

§10

CNOT distance suppression at

𝑝 =

−3

Observed:

4.2𝜎

improvement

𝐿 = 4 → 𝐿 = 6

§10, Fig. 6

CNOT fault-tolerant threshold

Not extracted

;

𝑑

each

-scaling

fails (non-FT)

§10, follow-up direction

Three-layer stacked architecture with

vertical inter-sheet couplers

Specied struc-

turally; per-

coupler delity

not modeled

§3

Magic-state distillation, non-Cliord

layer

Suggested by

octahedral

symmetry; no

protocol con-

structed

§10.8

2 The FCC Sheet Code

2.1 The Triad Decomposition

The FCC lattice has

𝐾 = 12

nearest-neighbor vectors, partitioning naturally into three

orthogonal sheets of 4:

𝑆

𝑥𝑦

: (±1, ±1, 0)

𝑆

𝑥𝑧

: (±1, 0, ±1)

(1)

𝑆

𝑦𝑧

: (0, ±1, ±1)

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

𝐿

(even), each sheet contains

𝐿

edges. Restricted to a single sheet, each FCC vertex has

𝐾=4

incident edges.

2.2 The Sheet Code Stabilizers

Denition 1

(FCC sheet code)

Fix one triad sheet

𝑆

(say

𝑆

𝑥𝑦

). 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

𝑋

𝐻

𝑇

𝑍

= 0

over

GF(2)

satised 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 verication:

𝐿 𝑛 rank(𝐻

𝑍

) rank(𝐻

𝑋

) 𝑘

4 64 28 28 8

6 216 102 102 12

8 512 248 248 16

In each case

rank(𝐻

𝑍

) = rank(𝐻

𝑋

) = (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 single-edge Pauli that anticommutes with

the appropriate opposite-type sheet stabilizers (a Z-edge with the X-stabilizers of the

sheet; an X-edge with the Z-stabilizers) and is therefore detected. 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

𝑥𝑦

, edges have

𝑑𝑧 = 0

, so each

𝑆

𝑥𝑦

edge

has a well-dened

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

𝑥𝑧

(layered by

𝑦

) and

𝑆

𝑦𝑧

(layered by

𝑥

Theorem 2

(Layer decomposition)

The FCC sheet code on sheet

𝑆

𝑥𝑦

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 deciency equals 1 (one product redundancy among

𝐿

vertex stabilizers).

Proof.

Edge partition.

Each

𝑆

𝑥𝑦

edge

(𝑣

, 𝑣

)

has

𝑧(𝑣

) = 𝑧(𝑣

)

since the displacement

vector

𝑣

−𝑣

∈ {(±1, ±1, 0)}

has

𝑑𝑧 = 0

. Dene

layer(𝑒) = 𝑧(𝑣

)

. The map

layer : 𝑆

𝑥𝑦

→

{0, 1, . . . , 𝐿 − 1}

partitions the

𝐿

edges of

𝑆

𝑥𝑦

into

𝐿

disjoint sets of

𝐿

edges each.

Stabilizer partition.

A vertex Z-stabilizer at vertex

𝑣

acts on the 4 sheet-

𝑆

𝑥𝑦

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

𝑥𝑦

, since the 4 edges of an oct void restricted to

𝑆

𝑥𝑦

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

vertex Z-stabilizers, sat-

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

rank(𝐻

layer

𝑍

) = 𝐿

/2 − 1

. Across

𝐿

layers,

rank(𝐻

sheet

𝑍

) = 𝐿 · (𝐿

/2 − 1) = (𝐿

− 2𝐿)/2

The same argument applies to

𝐻

sheet

𝑋

Code parameters.

𝑘 = 𝑛 − rank(𝐻

𝑍

) − rank(𝐻

𝑋

) = 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

verication of the rank: the formula

rank(𝐻

𝑍

) = rank(𝐻

𝑋

) = (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)

, 1, 𝐿−1]]

via standard boundary

engineering [4]. The resulting

planar sheet code

has parameters

[[𝐿(𝐿 − 1)

, 𝐿, 𝐿 − 1]] (

planar boundaries

(2)

The distance drops by one due to the standard rotated-surface-code boundary truncation,

and the 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

𝐿 ≤ 8

(

≤ 1,500

data qubits total), a monolithic planar processor hosts all three sheets

via time-multiplexed syndrome extraction: data qubits occupy xed positions; per-sheet

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

plers recongure between rounds to activate one sheet at a time (

𝑆

𝑥𝑦

, 𝑆

𝑥𝑧

, 𝑆

𝑦𝑧

in successive

rounds).

Wire-length scaling:

on a monolithic 2D embedding of the

Θ(𝐿

)

-qubit 3D

lattice, average nearest-neighbor distance scales as

Θ(𝐿

1/2

)

— straightforward at

𝐿 = 4

(192 qubits), requires active calibration at

𝐿 = 8

(1,536 qubits), impractical at

𝐿 ≥ 12

without coupler-reach upgrades.

Idle penalty:

while one sheet is measured, the other

two idle; with round time

𝑡

, full cycle is

3𝑡

and each data qubit idles

2𝑡

per cycle, captured

in the Section 7 noise model.

3.2 Option B: Three-Layer Stacked Architecture (Recommended

Deployment)

For

𝐿 ≥ 8

, we recommend three planar

𝐾=4

processors, one per triad sheet, bonded with

through-silicon-vias or inter-layer capacitive couplers for triangle-mediated cross-sheet

operations.

Each chip is a standard planar

𝐾=4

device. Layer hosting

𝑆

𝑥𝑦

carries the

𝐿

data qubits

𝐿

stacked rotated

𝐿 × 𝐿

lattices (Theorem 2). 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. Inter-layer couplers activate only during surgery operations

and remain inactive otherwise. The chip-internal

𝐾=4

connectivity is unaected; the

inter-layer couplers add

𝐾 = 1

per data qubit per active triangle, preserving the

𝐾=4

eective constraint during surgery (Section 5). Three-layer stacked QEC architectures

appear in recent hardware roadmaps [7, 10]; IBM’s stacked-die approach for bivariate

bicycle uses analogous couplers. The sheet code’s three-chip architecture has lower per-

chip connectivity (

𝐾=4

vs.

𝐾=6

) but simpler inter-chip coupling (only triangle ancillas

need vertical bonds).

Vertical crosstalk.

Only triangle ancillas (a minority,

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 static memory is unaected by inter-layer phenomena. Cou-

plers activate via control electronics; o-state isolation via detuning is platform-dependent

(superconducting TSVs reach

∼ 40

–

dB [7]; neutral-atom and ion-trap platforms can

achieve higher via mechanical separation). Platform-specic crosstalk budgeting is iden-

tied as future work (Section 10.8).

3.3 Option C: Native FCC Hardware

For maximally ecient 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 oers it.

3.4 Hardware Footprint

Across all options:

3𝐿

data qubits (one per FCC edge, partitioned by sheet),

3 × 𝐿

vertex Z-ancillas,

3 × 𝐿

octahedral void X-ancillas, and

𝐿

transient triangle ancillas

per active surgery primitive (reusable). Total:

6𝐿

recurring qubits with a

𝐾=4

active

syndrome-extraction schedule per data qubit, plus

𝑂(𝐿)

transient ancillas during surgery.

3.5 Comment on the “Cross-Block” Terminology

The three triad sheets occupy the same FCC lattice geometry (time-multiplexed in option

A, stacked in option B, co-located in option C), not spatially separated modules in the

bivariate bicycle sense. Throughout this paper we use the phrase

inter-sheet operations

for

any operation between logical qubits in distinct sheets implemented by triangle surgery.

These decompose into two strictly distinguished categories. (1)

Inter-sheet joint Pauli

measurements

— the ZZ-merge and XX-merge primitives of Sections 7 and 9.3 — are

fault-tolerant primitives with FSS-crossing threshold estimates. (2)

Composite logical

Cliord gates

(e.g., the three-sheet Horsman CNOT obtained by sequencing two joint

Pauli measurements) are constructed in Section 10 as correct logical gates at

𝑝 = 0

but

are

not yet fault-tolerantly characterized

; calling these “logical gates” is correct but does

not imply they are FT-thresholded in this work. The sheets are logically distinct CSS

code blocks (each independently encoding

2𝐿

logical qubits with independent stabilizer

groups and decoders) but 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

𝑎

, 𝑣

𝑏

, 𝑣

𝑐

, the three edge-vectors

𝑣

𝑏

−𝑣

𝑎

𝑣

𝑐

− 𝑣

𝑎

𝑣

𝑐

− 𝑣

𝑏

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. See Figure 1.

4.2 Triangle Operators

Denition 2

(Triangle operator)

For an FCC triangle

𝑇

with edges

𝑒

𝑥𝑦

∈ 𝑆

𝑥𝑦

𝑒

𝑥𝑧

∈ 𝑆

𝑥𝑧

𝑒

𝑦𝑧

∈ 𝑆

𝑦𝑧

, dene the Z-triangle operator

𝒵

𝑇

= 𝑍

𝑒

𝑥𝑦

⊗ 𝑍

𝑒

𝑥𝑧

⊗ 𝑍

𝑒

𝑦𝑧

and similarly

𝒳

𝑇

= 𝑋

𝑒

𝑥𝑦

⊗ 𝑋

𝑒

𝑥𝑧

⊗ 𝑋

𝑒

𝑦𝑧

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

𝑋

the cross-sheet X-stabilizer matrix. Dene the space of valid triangle

products

𝒱 = {𝑚 · ℬ : 𝑚 ∈ GF(2)

|𝑇 |

, 𝐻

𝑋

(𝑚 · ℬ)

𝑇

= 0}.

Then

dim(𝒱 mod row span(𝐻

𝑍

)) = 6𝐿 − 3

, and every operator in this space has support

on exactly two sheets.

Verication:

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

𝑖

, the projection of triangle-reachable

cross-sheet logicals onto the Z-logical space of

𝑆

𝑖

has dimension

𝐿

for each partner sheet

𝑆

𝑗

(

𝑗 = 𝑖

). The union of projections via both partners covers the full

2𝐿

-dimensional

Z-logical space of

𝑆

𝑖

Verication at

𝐿 = 4

Each sheet pair reaches a

-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 [5, 6], these reachability

results suggest that arbitrary logical-pair interactions can in principle be mediated using

ancilla logicals and logical-basis routing; a complete routing-depth construction (and the

associated FT-thresholded characterization, since the composed CNOT itself is not yet

FT-thresholded in this work, Section 10) is left to future work.

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).

Figure 2 illustrates the canonical

𝐿 = 4

four-triangle primitive.

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 ag qubits are needed:

the per-sheet code distance suces for fault-tolerant triangle measurements. For

𝐿 = 4

a ag-qubit protocol [9] catches the worst-case weight-2 propagation.

5.2

𝐾=4

Connectivity Verication

Two senses of

𝐾=4

The connectivity claim made by this paper, here and throughout,

is about

active connectivity per syndrome round

: each data qubit participates in at most

4 two-qubit gates per round of syndrome extraction, including during surgery. This is the

constraint that matters for hardware compatibility (gate scheduling, crosstalk, parallel

CNOT capacity, calibration overhead) and is what makes the architecture compatible

with platforms such as Google Willow and IQM Star/Garnet whose native processors are

designed around

𝐾=4

active connectivity. The

physical coupler layout

of a real chip may

include additional couplers (e.g., diagonal ones used only during specic phases or never

used at all in this protocol); this is a separate hardware-implementation question, and we

do not claim the physical layout must literally have exactly 4 couplers per data qubit.

Throughout this paper, “

𝐾=4

” should be read as active per-qubit connectivity during

any single syndrome round.

Each data qubit therefore participates in at most 4 two-qubit gates per syndrome round,

with no exceptions during surgery. During surgery, some gate slots recongure from

per-sheet ancillas to triangle ancillas. At

𝐿 = 4

with the canonical 4-triangle primitive:

the two

𝑥𝑦

-sheet edges shared between pairs of surgery triangles see 2 triangle-coupler

gates per round (retaining 2 per-sheet coupler gates each); the other 8 data qubits use 1

triangle-coupler gate with 3 per-sheet coupler gates retained.

The

𝐾 = 4

active envelope

is preserved throughout the surgery operation.

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

𝑥𝑧

and 4 edges in

𝑆

𝑦𝑧

, implementing

𝑍

𝐴

⊗ 𝑍

𝐵

where

𝐴

a logical of sheet

𝑥𝑧

and

𝐵

is a logical of sheet

𝑦𝑧

. Red stars indicate triangle ancilla

positions at each triangle’s centroid.

5.3 Gate Schedule

The triangle measurements’ CNOT gates schedule via graph coloring on the conict graph

𝐺

conﬂict

(nodes: surgery triangles; edges: shared data qubits). At

𝐿 = 4

, the 4-triangle

primitive’s conict 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

5𝐿

Time slots

–

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 impact.

Triangle measurements during the merge execute in parallel with

per-sheet syndrome extraction; in a time-multiplexed schedule, triangle CNOTs occupy

the same time slots as per-sheet CNOTs without extending wall-clock time. The merge

phase costs

𝐿

syndrome rounds at the same clock cycle as memory, for total surgery

overhead of

3𝐿

rounds vs.

𝐿

for memory. On superconducting transmons (100–400 ns

cycles [8]),

3𝐿

rounds at

𝐿 = 8

2.4

–

s additional wall-clock, well below typical

𝑇

, 𝑇

(

∼ 100

s); on neutral-atom platforms (

∼ 1

ms cycles) the

∼ 24

ms total requires sustained

coherence achievable in recent demonstrations. Idle errors during merge are captured by

the noise model of Section 7.

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 ips both

triangles’ outcomes simultaneously. Standard MWPM [13] applies directly to this graph;

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

triangle measurements [5, 6].

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

𝑥𝑦

, 𝑒

𝑥𝑧

, 𝑒

𝑦𝑧

)

, 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

𝑥𝑦

, two in

𝑆

𝑥𝑧

, and two in

𝑆

𝑦𝑧

Proof.

A triangle Z-operator

𝒵

𝑇

acts on three edges. Each edge

𝑒 ∈ 𝑆

𝑖

is contained in

exactly two octahedral voids of

𝑆

𝑖

, 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

𝒵

𝑇

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, the

𝐿

triangles of

the surgery primitive collectively anticommute with

3𝐿

distinct per-sheet X-stabilizers

(each broken by exactly two triangles, accounting for the

6𝐿

total triangle-stab anticom-

mutations of Lemma 2). These

3𝐿

X-stabilizer measurements become

gauge bits

during

the merge: their outcomes are correlated with the triangle outcomes but do not constrain

the merged code’s logical subspace. Veried 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 ip count across the

𝐿

triangles. This is zero by construction: the triangle product



𝑇

𝒵

𝑇

= 𝑍

𝐴

⊗𝑍

𝐵

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

3𝐿

initially-broken

X-stabilizer outcomes are reconstructed from the

𝐿

triangle measurement outcomes plus

surviving stabilizer constraints: each broken X-stabilizer’s eigenvalue is the modulo-2 sum

of (i) its pre-merge eigenvalue, (ii) the triangle measurements whose Z-operators overlap

it in odd parity, and (iii) 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 [5]: per-sheet boundary stabilizers are temporarily opened as gauges

and closed upon surgery completion. No per-sheet stabilizer is permanently modied;

the aected weight-4 stabilizers are still physically measured throughout the merge, but

their round-to-round detector constraints are gauge-dependent during the merge window

and are therefore omitted from the DEM during that window (see Section 7.1); they are

restored post-merge using the triangle-outcome records described above.

Single-fault error propagation across sheets.

A single fault on a triangle ancilla

mid-circuit can propagate to at most two data qubits in dierent sheets: for the canonical

CNOT schedule

𝑒

𝑥𝑦

→ 𝑒

𝑥𝑧

→ 𝑒

𝑦𝑧

, a

𝑍

error on the ancilla after the

𝑆

𝑥𝑧

CNOT and

before the

𝑆

𝑦𝑧

CNOT propagates to one data qubit each in

𝑆

𝑥𝑧

and

𝑆

𝑦𝑧

. This weight-2

cross-sheet error triggers detectors in two distinct per-sheet syndrome graphs; the decoder

graph must include edges spanning these per-sheet graphs (Section 7).

6 Distance Preservation

Theorem 5

(Merged code distance)

For any even

𝐿 ≥ 2

and any cross-sheet measure-

ment operator

Op = 𝑍

𝐴

⊗ 𝑍

𝐵

implemented by a weight-

2𝐿

triangle product, with

𝑍

𝐴

supported in sheet

𝑆

𝑖

and

𝑍

𝐵

in sheet

𝑆

𝑗

(

𝑖 = 𝑗

), the merged code formed by adding

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.

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

𝑋

). Furthermore,

is not in the row span of

𝐻

𝑍

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

Therefore appending

𝐻

𝑍

increases the rank by 1, and the merged code has

𝑘

merged

𝑘

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

modulo the original stabilizers.

Decompose any

𝑔

𝑔 = 𝑔

𝑥𝑦

⊕𝑔

𝑥𝑧

⊕𝑔

𝑦𝑧

where

𝑔

𝑠

denotes the restriction of

𝑔

to sheet

𝑆

𝑠

Similarly

decomposes as

Op = (Op)

𝑖

⊕ (Op)

𝑗

with

(Op)

𝑖

= 𝑍

𝐴

of weight

𝐿

𝑆

𝑖

and

(Op)

𝑗

= 𝑍

𝐵

of weight

𝐿

𝑆

𝑗

. Then

wt(𝑔) = wt(𝑔

𝑥𝑦

) + wt(𝑔

𝑥𝑧

) + wt(𝑔

𝑦𝑧

(3)

wt(𝑔 ⊕ Op) = wt(𝑔

𝑘

) + wt(𝑔

𝑖

⊕ 𝑍

𝐴

) + wt(𝑔

𝑗

⊕ 𝑍

𝐵

(4)

where

𝑘

is the third sheet (

𝑘 /∈ {𝑖, 𝑗}

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

𝑔

𝑠

on sheet

𝑆

𝑠

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:

(a)

𝑔

𝑘

= 0 (mod

stab

𝑆

𝑘

)

, i.e.,

𝑔

𝑘

is a non-trivial logical of

𝑆

𝑘

. Then

wt(𝑔

𝑘

) ≥ 𝐿

. Both

wt(𝑔)

and

wt(𝑔 ⊕ Op)

contain

wt(𝑔

𝑘

) ≥ 𝐿

as a summand, so the class minimum is

≥ 𝐿

(b)

𝑔

𝑘

is trivial in

𝑆

𝑘

(i.e.,

𝑔

𝑘

= 0

or a stabilizer), but

𝑔

𝑖

is a non-trivial logical of

𝑆

𝑖

Then

wt(𝑔

𝑖

) ≥ 𝐿

, so

wt(𝑔) ≥ 𝐿

. For

wt(𝑔 ⊕ Op)

, the contribution

wt(𝑔

𝑗

⊕ 𝑍

𝐵

)

either

≥ 𝐿

(if

𝑔

𝑗

and

𝑍

𝐵

are in distinct logical classes, or if

𝑔

𝑗

is a stabilizer leaving

the

𝑍

𝐵

contribution of weight

𝐿

) or zero (if

𝑔

𝑗

≡ 𝑍

𝐵

(mod

stab

𝑆

𝑗

)

). In the zero

case,

𝑔 ⊕Op

has contributions only from

𝑔

𝑖

⊕𝑍

𝐴

(in

𝑆

𝑖

, weight

≥ 𝐿

) and

𝑔

𝑘

(in

𝑆

𝑘

possibly weight 0). Therefore

wt(𝑔 ⊕ Op) ≥ 𝐿

𝑖

and

𝑗

(d) 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

𝐴

has representatives

{𝑍

𝐴

, 𝑍

𝐴

⊕ Op}

. Since

𝑍

𝐴

⊕ Op = 𝑍

𝐴

⊕ 𝑍

𝐴

⊕ 𝑍

𝐵

= 𝑍

𝐵

modulo per-sheet stabilizers, this class equals

{𝑍

𝐴

, 𝑍

𝐵

}

with representatives of weight

𝐿

each. Therefore the minimum is achieved.

Computational verication.

The proof above is independent of

𝐿

. To rule out edge

cases, we additionally veried 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

modied by the cross-sheet Z-measurement

Op = 𝑍

𝐴

⊗ 𝑍

𝐵

. The X-side requires a sep-

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

that commute with

, 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

, modulo the (unchanged) X-stabilizer group.

Proof.

Setup.

The original code has X-stabilizer matrix

𝐻

𝑋

and Z-logical group

ℒ

𝑍

X-logical group

ℒ

𝑋

. After surgery, the merged code has stabilizer matrices

𝐻

merged

𝑋

= 𝐻

𝑋

(unchanged) and

𝐻

merged

𝑍

= 𝐻

𝑍

∪ {Op}

. The merged X-logical group is the normalizer

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

stabilizers:

ℒ

merged

𝑋

= {𝑔 ∈ ℒ

𝑋

: [𝑔, Op] = 0}



⟨𝐻

𝑋

⟩.

The commutativity condition

[𝑔, Op] = 0

for

𝑔

an X-operator and

a Z-operator reduces

to:

𝑔

has even-parity overlap with the support of

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

𝐴

⊗𝑍

𝐵

where

𝐴

is a Z-logical of sheet

𝑆

𝑖

and

𝐵

is a Z-logical of sheet

𝑆

𝑗

. The X-logical

𝑋

𝐴

(the conjugate of

𝐴

𝑆

𝑖

) anticommutes

with

𝑍

𝐴

(per-sheet anticommutation) and commutes with

𝑍

𝐵

(disjoint sheet supports),

𝑋

𝐴

anticommutes with

. Symmetrically,

𝑋

𝐵

anticommutes with

. All other

generators of

ℒ

𝑋

commute with

: per-sheet

𝑋

-logicals in sheets

𝑆

𝑘

𝑘 = 𝑖, 𝑗

have

disjoint support from

and commute trivially; per-sheet

𝑋

-logicals in

𝑆

𝑖

𝑆

𝑗

other

than

𝑋

𝐴

𝑋

𝐵

are independent of

𝑋

𝐴

(resp.

𝑋

𝐵

) and the per-sheet anticommutation

structure ensures they commute with the relevant component of

The merged X-logical group has

6𝐿 − 2

commuting generators from the original

6𝐿

but the product

𝑋

𝐴

· 𝑋

𝐵

(sum of two anticommuting generators) commutes with

(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

Minimum weight.

The merged X-logical generators are of two types:

(a) Original per-sheet X-logicals that commute with

: weight

𝐿

each (per-sheet 2D

toric distance, Theorem 2).

(b) The new generator

𝑋

𝐴

·𝑋

𝐵

: weight

2𝐿

(disjoint supports across sheets

𝑆

𝑖

and

𝑆

𝑗

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 verication.

𝐿 = 4, 6

, all

6𝐿

X-logical generators have weight

𝐿

; exactly 2 anticommute with

(conrming the

6𝐿 − 2

count of single-generator

commuters), and the minimum weight among merged X-logical generators is exactly

𝐿

Combined with Theorem 5, the merged code distance is

𝐿

for both Z- and X-side errors.

7 Threshold Simulation

We characterize the surgery operation using a custom Stim circuit [12] with explicit tri-

angle measurements, decoded with MWPM via PyMatching [13], and report a nite-size-

scaling (FSS) crossing threshold estimate from pairwise crossings at

𝐿 ∈ {4, 6, 8}

. The

static memory FSS-crossing estimate is reported as a complementary baseline.

7.1 Custom Stim Circuit Construction

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

(

𝑆

𝑥𝑧

, 𝑆

𝑦𝑧

for the

𝐿 = 4

primitive) with per-sheet vertex Z- and oct-void X-stabilizer

measurements,

𝐿

triangle ancilla measurements during the merge phase, and auxiliary

𝑆

𝑥𝑦

data qubits used by the triangle measurements. Three phases of

𝑑

syndrome rounds

(pre-merge, merge, post-merge); the

3𝐿

X-stabilizers broken during merge (Lemma 2)

have detectors skipped per the gauge structure of Section 5.6. Key statistics:

𝐿

Total qubits Triangles Broken X-stabs DEM error mechanisms

4 324 4 12 (8 in

𝑆

𝑥𝑧

+ 𝑆

𝑦𝑧

, 4 in

𝑆

𝑥𝑦

) 23,946

6 750 6 18 (12 in

𝑆

𝑥𝑧

+ 𝑆

𝑦𝑧

, 6 in

𝑆

𝑥𝑦

) 70,434

8 2,568 8 24 (16 in

𝑆

𝑥𝑧

+ 𝑆

𝑦𝑧

, 8 in

𝑆

𝑥𝑦

) 539,215

The DEM decomposes via Stim’s

decompose_errors=True

(hyperedges into graph-like

edges where possible) at all three sizes; the resulting graphs are compatible with Py-

Matching’s MWPM decoder.

7.2 Surgery Operation Threshold

Terminology.

The numerical threshold estimates throughout this section and the rest

of the paper are

nite-size crossing estimates

extracted from pairwise crossings of logical

error rates at

𝐿 ∈ {4, 6, 8}

with MWPM decoding. We use the abbreviation FSS-crossing

estimate or simply “threshold estimate” for compactness, but the reader should not read

these as asymptotic (

𝐿 → ∞

) thresholds; they are tight empirical crossings at three

accessible code distances with quoted uncertainty drawn from the three pairwise crossings

and a formal FSS data-collapse t where available (Section 9.2 for the planar variant; the

toric FSS t gives

1.134% ± 0.033%

𝜈 = 1.50 ± 0.18

, statistically consistent with the

pairwise-crossing estimate). Extending to

𝐿 = 10

𝐿 = 12

would tighten the band but

is computationally expensive (Section 10.8).

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

𝑍

𝐴

⊗ 𝑍

𝐵

(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

𝐿 = 6

𝐿 = 4

𝐿 = 8

, and

𝐿 = 6

𝐿 = 8

give three independent estimates: