Generalized matching decoders for 2D topological translationally-invariant codes

Motivation

• Graph-matching decoders are motivated by the toric code’s efficient MWPM decoding and by the known equivalence between 2D TTI codes and multiple copies of the toric code.This equivalence makes efficient decoding for broader TTI families a natural and practically relevant question.Source: §1 · pp. 3–4

Contributions

• The proposed decoders correct errors up to a constant fraction of the code distance and achieve non-zero thresholds under i.i.d. bit-flip and phase-flip code-capacity noise.The fraction is determined by parity-check locality.Source: §1 · p. 4
• On the 12 × 6 distance-12 gross code, the optimized graph-matching decoder outperforms BP at high logical error rates but performs slightly worse than BP-OSD [PC08].The study also considers the distance-24 code with the same polynomial on a 24 × 24 lattice.Source: §1 · p. 4
• The method coarse-grains a TTI code into effective toric-code excitations and removes them with graph matching to obtain a correction for the original code.It relies on the polynomial representation of 2D TTI codes and their equivalence to multiple copies of the toric code.Source: §1 · p. 4
• For practically relevant BB codes, the decoder’s competitive performance and favorable computational complexity support graph matching as a viable decoding approach and motivate further optimization.The authors suggest such improvements may eventually outperform BP-OSD and be more feasible for larger codes.Source: §1 · p. 4

2 Overview

• General TTI codes can produce multi-defect hyperedges, making direct decoding a minimum-weight hypergraph matching problem that is generally NP-hard; coarse-graining instead exposes toric-code-like pairings.The decoupling theorem provides the conceptual basis by transforming topologically ordered 2D TTI codes into finitely many toric-code copies and trivial qubits.Source: §2 · pp. 5–7

2.1 Decoder 1: the layer-decoupling decoder

• The layer-decoupling decoder algebraically projects the measured syndrome into r toric-code sector syndromes, decodes each by matching, and lifts the sector corrections back through the inverse transformation.The induced sector noise can be correlated, so edge weights are constructed from phase-flip probabilities associated with qubits connected by the constant-depth CNOT circuit.Source: §2 · p. 7, §2.1 · pp. 8–9

2.2 Decoder 2: the cell-matching decoder

• The cell-matching decoder locally flushes violated checks into fixed basis subcells, represents residual violation classes by r-bit labels, and matches each type on a coarse lattice.It then combines local flushing operators with boundary edge-generator Paulis to produce a physical correction.Source: §2.2 · p. 9

2.3 Main Results

• Both decoders have non-zero code-capacity thresholds, exponentially decreasing logical error below a constant fraction of the toric-code threshold, and MWPM-scale asymptotic complexity.The guarantees apply to code families defined by fixed polynomials on torus sizes divisible by the precomputable coarse-graining parameter b.Source: §2.3 · pp. 10–11
• For BB codes, the adapted matching decoder improves over BP and approaches BP-OSD, with a gross-code pseudothreshold of ≈5.2% versus ≈5.0% for BP and ≈5.5% for BP-OSD.The evaluation uses code-capacity noise and includes gross, two-gross, and a 1152-qubit BB instance.Source: Figure 1 · p. 5, §2.3 · p. 11

3.1 Notation

• A vector’s Hamming weight counts its nonzero coordinates, with [n] denoting the positive integers up to n.All F2-vector spaces are assumed finite-dimensional, so Hamming weights are finite.Source: §3.1 · p. 12

3.2 Quantum codes

• A quantum CSS code is a pair of binary subspaces Q_X and Q_Z satisfying Q_X^⊥ ⊆ Q_Z, with dimension k = dim(Q_Z) − dim(Q_X).The code’s locality is the maximum row or column weight of its X- and Z-stabilizer parity-check matrices, and its binary symplectic matrix combines these checks.Source: §3.2 · pp. 12–13

3.3 Chain complexes

• A chain complex is a sequence of F2-vector spaces connected by boundary maps whose consecutive composition is zero; its cochain complex transposes those maps.Homology and cohomology are defined through cycles, boundaries, cocycles, and coboundaries, with systolic and cosystolic distances measuring minimum nontrivial weights.Source: §3.3 · p. 13
• Quantum CSS codes correspond to 3-term chain complexes whose X stabilizers, qubits, and Z stabilizers occupy consecutive spaces, with CSS orthogonality equivalent to a trivial boundary-of-a-boundary.X and Z logical operators are represented by 1-cohomology and 1-homology classes, and their distances are the corresponding 1-cosystolic and 1-systolic distances.Source: §3.3 · pp. 13–14

3.4.1 Laurent polynomials

• Laurent polynomials over F2 form the ring R = F2[x^±1, y^±1], with monomials x^i y^j providing a basis and negative powers enabling lattice translations in both directions.The degree is the maximum i + j among nonzero terms, and only finitely many coefficients are nonzero.Source: §3.4 · p. 14, §3.4.1 · p. 14

3.4.2 2D TTI codes

• A 2D TTI code is defined on a periodic two-dimensional lattice with local translationally invariant stabilizer checks, represented compactly by bivariate Laurent polynomials.Each site carries a constant number q of qubits, monomials label sites and checks, and multiplication by x or y translates horizontally or vertically.Source: §3.4.2 · pp. 14–15
• Polynomial terms act as finite repeating stencils, while the antipode reverses translation direction and the dagger operation transposes the parity-check matrix while applying that antipode.The formulation covers any 2D TTI code with a constant number of qubits per site, including bivariate bicycle codes; the q = 2 example yields the toric-code parity-check matrix.Source: §3.4.2 · pp. 15–16, Figure 6 · p. 15

4.1 Matching decoder framework

• The decoder virtually decouples the 2D TTI code into toric-code copies and trivial product states, applies matching decoders, then lifts the corrections back.The finite-size procedure maps measured syndromes and noise weights to sectors, runs Match on each toric-code graph, and applies the inverse decoupling map.Source: §4.1 · pp. 17–18
• The decoupling process yields algebraic row and column operations whose invertible matrices transform the coarse-grained parity-check matrix into a block-diagonal form.The matrices U and V encode valid symplectic operations over the coarse-grained Laurent-polynomial ring, enabling the sector decomposition.Source: §4.1 · pp. 17–18

4.2 Morphism of chain complexes

• A bijective chain isomorphism maps the original code to the direct sum of toric-code sectors and a trivial product-state complex, whose projections produce matching-decodable graph complexes.The isomorphism also maps original syndromes and Pauli error configurations to their decoupled counterparts.Source: §4.2 · pp. 19–20

4.3 Mapping the noise model for the TCs

• The noise mapping sums error probabilities from physical qubits disentangled into each toric-code qubit, conservatively treating induced sector noise as independent to weight graph edges.The mapped probabilities improve edge weighting, while translational invariance may require increasing the coarse-graining parameter.Source: §4.3 · pp. 20–21

4.4 Mapping the syndrome configuration for the TCs

• The syndrome mapping uses the chain isomorphisms and sector projections to convert the original code syndrome into toric-code and product-state syndromes.Because errors are random, the actual syndrome need not remain translationally invariant, despite translationally invariant code structure.Source: §4.4 · p. 21

4.5 Error correction operators and lifting

• The lifted correction is valid for the original code if and only if every toric-code-sector correction and the product-state correction is valid.Corrections are obtained with MWPM or Union-Find on sector graphs, while the product state uses a local correction without matching.Source: §4.5 · p. 22
• Corrections obtained independently in the toric-code sectors are lifted to the original code using the inverse chain isomorphisms, with the lifting identities following from Lemma C.1.The lifting step combines sector corrections into a valid correction for the original code.Source: §4.5 · p. 23, §4.6 · p. 24

4.6 Layer-decoupling decoding algorithm for finite-size 2D TTI codes

• For finite-size 2D TTI codes with lattice dimensions divisible by b, the decoder uses toric-code sectors with d_min=d_TC=min{L_x/b,L_y/b} and provides adversarial and local-stochastic error-correction guarantees.The adversarial guarantee applies when L_x,L_y are integer multiples of b and at least 3b; the local-stochastic guarantee requires w_max·p_max below the corresponding toric-code matching threshold.Source: §4.6 · p. 24
• If lattice dimensions are not divisible by b, the logical-qubit structure can differ substantially, requiring alternative coarse-graining parameters and matching strings, as demonstrated for gross codes in Section 6.The standard finite-size algorithm applies directly only when L_x and L_y are integer multiples of b.Source: §4.6 · p. 25
• The decoder’s effective distance is reduced by a factor b, requires each toric-code sector to have distance at least 3, and requires w_max·p below the toric-code threshold, approximately 10.3% for MWPM bit- and phase-flip decoding.Choosing decoupling circuits or symplectic transformations that minimize w_max can improve practical performance by reducing induced correlated errors.Source: §4.6 · p. 25
• The finite-size decoder has overall complexity dominated by matching on the toric-code sectors, while decoupling, syndrome mapping, and lifting contribute linear-time processing in L_xL_y.For fixed code families, the number of sectors and coarse-graining parameter are constants; unit-cell operations can be parallelized.Source: §4.6 · pp. 25–26

5 The cell-matching decoder for 2D TTI Codes

• The decoder flushes each unit-cell syndrome into a fixed basis subcell, interprets the residual coker H class as excitation types, and matches each excitation set on a coarse torus.Local physical syndromes flush to no excitation, while non-physical restrictions produce nontrivial basis-subcell configurations; global linearity ensures every excitation set has even cardinality for matching.Source: §5 · p. 26, §5.2 · pp. 27–28, §5.3 · p. 29

5.1 Unit cell construction

• Grouping sites into b×b unit cells makes excitations periodic, while a localized c×c basis subcell represents the independent excitation types of coker H.The basis subcell may contain excitation patterns involving several toric-code copies, and the construction assumes b divides both lattice dimensions.Source: §5.1 · p. 27

5.2 Flushing of excitations

• The flushing map independently transports all excitations within each unit cell into its basis subcell, annihilating locally physical syndromes and retaining nontrivial coker H classes.This provides a local representative of the residual syndrome class and prevents spurious basis-subcell excitations when the restricted syndrome is physical.Source: §5.2 · pp. 27–28

5.3 Mapping to TC copies

• Coarse edge generators toggle one excitation type at neighboring unit cells, allowing matched coarse corrections to lift to physical Pauli operators on the original code.If every residual coarse cycle is contractible, the lifted correction is a stabilizer; therefore any decoder-induced logical operator must arise from a nontrivial homology class in at least one coarse toric sector.Source: §5.3 · pp. 29–31

5.4 Correctness and complexity analysis of the cell-matching decoder

• A physical error of weight |E| induces, in each excitation sector, a minimum-weight coarse 1-chain with at most |E| horizontal and |E| vertical edges.The bound follows from the single-qubit geometry: errors affect at most four neighboring unit cells, yielding only local coarse-edge patterns before linearity extends the result to general errors.Source: §5.4 · p. 32
• Theorem 5.12 establishes that cell matching corrects bounded-weight adversarial errors and succeeds under local stochastic noise when induced coarse-edge rates lie below the toric-code matching threshold.Each physical single-qubit error induces at most one coarse edge per sector and affects only O(1) sectors; logical-failure probability then decays exponentially in d_min.Source: §5.4 · pp. 33–34
• The correctness proof runs MWPM independently on each excitation-type sector, lifts the resulting chains, and uses contractibility to show the total correction differs from the error by stabilizers.A logical failure would require a nontrivial homology class in at least one sector, which the argument excludes.Source: §5.4 · pp. 33–34
• For constant excitation count and coarse-graining parameter, runtime is dominated by matching and scales as O(MATCHING(L_xL_y)); local flushing and syndrome processing can be parallelized.The decoder’s performance can improve by choosing an anyon basis that enables intra-cell matching and increases effective toric distance for some logical qubits.Source: §5.4 · pp. 34–35

5.5 Cell-matching decoder threshold for the color code

• The hexagonal color code achieves a threshold of 7.31 ± 0.05% with the basic cell-matching decoder, below but comparable to existing optimized matching-based thresholds.The decoder flushes locally, independently applies MWPM to triangular-lattice toric graphs, and lifts corrections back to the color code.Source: Figure 7 · p. 35, §5.5 · p. 35

6 Adapting the cell-matching decoder for small and intermediate code sizes

• The adaptations target small and intermediate lattices, are presented using BB code examples, and apply to general small and intermediate 2D TTI codes.The algorithms lack rigorous performance guarantees because their construction relies on heuristics.Source: §6 · p. 36

6.1.1 Identifying equivalence class of syndromes

• A chosen coker H basis identifies each violated check’s equivalence class by solving a unique linear system, enabling lattice checks to be decomposed into translated canonical syndrome representatives.Basis elements with smaller translation scales ||u_i||_T < b are preferred because they increase the effective distance of the resulting matching graphs.Source: §6.1.1 · pp. 37, 39

6.1.2 Defining the matching graphs

• For each basis component, the decoder constructs a complete graph on syndrome checks containing that component; every graph has an even vertex count and therefore admits perfect matching.The graphs are intended to pair checks within the same error cluster, generalizing the toric-code matching objective.Source: §6.1.2 · pp. 39, 41, 42
• Edge weights use translation-invariant move sets and shortest-path searches to balance path length against remaining syndrome, providing a computable heuristic rather than an exact error-cluster lower bound.The tunable parameter λ controls the relative importance of minimizing |P| and |s_remain|; Dial’s algorithm costs O(ℓm) per start point when move costs are integer.Source: §6.1.2 · pp. 43–44

6.1.3 Re-matching and syndrome shifts

• The re-matching and syndrome-shift optimizations address MWPM’s overcounting of overlapping pairwise paths when multiple matched pairs belong to one error cluster.Independent edge costs can substantially overlap and make the summed matching cost exceed the true joint cost of explaining the cluster.Source: §6.1.3 · p. 45
• The decoder considers all b × b syndrome shifts, identifies candidate clusters with depth-limited DFS, and re-matches when matching consistency F_S exceeds F_limit by modifying internal and boundary edge weights.The DFS has at most 6w_fixed operations per matching edge, although smaller clusters generally reduce the actual cost.Source: §6.1.3 · p. 45

6.2 Intermediate code size machinery

• The intermediate-size decoder selects the syndrome shift minimizing the compatibility graph’s largest connected component, optionally redecoding that component, and applies short strings between basis-pattern cells.Short-string chains use the winding direction of the matched path; this avoids choosing among many candidate DFS clusters, whose incorrect selection can cause failure.Source: §6.2 · pp. 46–48

6.3 Small code size machinery

• For small codes, the decoder collects DFS subsets with F_S = 1 across shifts, selects disjoint subsets by multiplicity, and uses staged residual decoding and subset re-decoding to find a covering correction.It returns a below-e_limit correction when possible, otherwise the lowest-weight covering candidate, or declares failure if none exists; this works because small codes contain few total clusters.Source: §6.3 · pp. 48–49

6.4 Numerical results for BB codes

• Matching performs best for the gross code, with logical failure rates relatively close to BP-OSD; the small-size decoder is stronger on the gross than two-gross code, while the intermediate decoder improves relative to BP on the 24 × 24 lattice.The small-code limitation is attributed to increasing expected cluster counts, whereas short-string corrections avoid explicitly identifying every cluster in the intermediate regime.Source: §6.4 · pp. 50–51, Figure 13 · p. 51
• The current matching implementation lags BP-OSD in speed because shift enumeration, subset-removal redecoding, cluster-search overhead, and Python implementation impose substantial constant costs.These costs are readily parallelized, but runtime optimization is left to future work.Source: §6.4 · pp. 51–52

Conclusions

• The proposed layer-decoupling and cell-matching decoders exploit TTI–toric-code equivalence, with analytical guarantees and competitive performance against BP-based decoders [MAB+25].The cell-matching decoder locally flushes excitations within coarse-grained cells to respect noise locality and avoid correlated errors.Source: §7 · p. 52

Future directions

• Future work could optimize matching edge weights using BP-derived soft information [HBK+23] and extend decoding to phenomenological or circuit-level measurement errors.Hybrid BP–graph-matching decoders may better capture correlations between excitations in different toric-code copies and could integrate with color-code circuit-level decoders [CKYZ20] [GJ23] [LLB25].Source: §7 · p. 52
• Open-boundary lattices require accounting for modified excitation constraints, boundary-condensing excitations, and matching pairs between excitations and appropriate boundaries.These adaptations matter because boundaries can affect local excitation flushing and are emphasized in proposed experimentally feasible layouts [EPS24] [LEC25] [SCB+25].Source: §7 · p. 52
• Applying graph matching to quantum product codes remains challenging because their syndromes may lack the point-like, low-dimensional structure of toric-code excitations.A possible route is generalizing higher-dimensional color-code decoders, though the needed constant-depth mapping to multiple higher-dimensional toric codes remains unclear [Haa16].Source: §7 · p. 52

• BB codes are defined from classical cyclic codes on two ℓ×𝑚 lattices and include the toric and color codes as special cases.The work focuses on BB codes embeddable in a 2D toric layout, while noting broader constructions and applications in prior work [BCG+24] [YSR+25] [ES24] [LEC25] [LLSC25].Source: A Bivariate bicycle codes · p. 58
• For α≡−α, β≡−β, and α≥β≥0, a BB code BB(α, β, a, b) is defined by two polynomials and parity-check matrices H_X=(a(x,y)|b(x,y)) and H_Z=(b*(x,y)|a*(x,y)).Source: A Bivariate bicycle codes · p. 58
• Algebraically, BB codes use the ring R = F2[x^±1, y^±1]/⟨x^ℓ−1, y^m−1⟩, whose elements label qubits and define stabilizer checks.This labeling enables computational algebraic geometry tools such as Gröbner bases to analyze concrete code parameters.Source: A Bivariate bicycle codes · p. 59
• The X- and Z-stabilizer groups are principal R-modules generated by (a, b) and (b*, a*), respectively.Specifically, S_X = R·(a(x, y), b(x, y)) and S_Z = R·(b*(x, y), a*(x, y)).Source: A Bivariate bicycle codes · p. 59
• Error configurations and their syndromes are represented by polynomial pairs, with X and Z syndromes lying in polynomial ideals generated by the BB defining polynomials.Logical operators are characterized as syzygy modules modulo the corresponding stabilizer modules, yielding quotient R-modules for both logical types.Source: A Bivariate bicycle codes · pp. 59–60

Decoupling

• The approach decouples 2D TTI codes into independent copies of the toric code and a trivial product state.This decoupling simplifies the analysis of two-dimensional topological codes by treating them as equivalent up to Clifford gates.Source: B An algorithmic approach to… · pp. 60–61

Background

• Building on coarse-graining and the decoupling framework of Haah [Haa16] and Bombín, the section reformulates these ideas in modern quantum-error-correction language.The presentation also relates its terminology to the original works, whose decoupling notion plays a central role in understanding 2D topological codes.Source: B An algorithmic approach to… · p. 61

Algorithm and example

• The section introduces Clifford operations, coarse-graining, and valid symplectic transformations before presenting a modified decoupling algorithm and an illustrative example.The organization proceeds from foundational operations to the algorithm and then demonstrates the decoupling process in action.Source: B An algorithmic approach to… · p. 61

Clifford Operations and Symplectic Group

• Clifford operations correspond to elements of Sp(2n, F2), whose matrices preserve the symplectic form governing Pauli commutation relations.The symplectic group is a subgroup of GL(2n, F2).Source: §B.1 · p. 61
• CNOT, Hadamard, and Phase gates act as elementary column operations on a CSS code’s binary symplectic matrix, thereby permuting its Pauli operators.CNOT adds specified columns across the matrix partitions, Hadamard swaps paired columns, and Phase adds one paired column to another.Source: §B.1 · p. 61

Motivation

• Coarse-graining groups several lattice cells into one super-cell, restoring a visible interaction pattern under translations by a super-cell for locally longer-range codes.For the toric code, a single cell suffices, whereas longer-range interactions require grouping cells to expose the code’s translational symmetry.Source: §B.2 · p. 62

Algebraic construction

• The construction replaces R=F2[x^±1,y^±1] with R′=F2[x^±b,y^±b], where b sets the super-cell size and each coarse-grained site contains bq qubits.Larger b produces a larger and more strongly coarse-grained super-cell; the parity-check matrix grows by a factor of b over R′.Source: §B.2 · p. 62

Polynomial representation

• Because arbitrary polynomials are formed from generator matrices using sums, products, and inverses, the coarse-grained representation preserves the code’s polynomial interactions over R′.This faithful representation makes the code-Hamiltonian interactions on the coarse-grained lattice more explicit.Source: §B.2 · pp. 62–63

Generator representation

• The basis B contains the b^2 monomials x^i y^j with 0≤i,j<b, enabling matrix representations of the generators x and y over R′.The generator matrices implement ordered shifts of this basis, with boundary crossings represented by x′ and y′.Source: §B.2 · pp. 62–63

Local disentanglement

• Symplectic column transformations implement local Clifford operations that aim to disentangle the coarse-grained code into independent toric codes and a trivial product state.These operations act on qubits within each coarse-grained super-cell.Source: §B.3 · p. 63

Column transformations

• Paired column swaps physically exchange qubits within every super-cell while preserving the symplectic representation.Swapping columns i and j requires simultaneously swapping their paired columns i+n_c and j+n_c.Source: §B.3 · p. 63
• Multiplying paired columns by inverse monomials translates the corresponding qubits across the coarse-grained lattice, and paired column additions implement CNOT gates between super-cell qubits.A monomial scaling by x'^j y'^k is paired with x'^−j y'^−k; monomial-multiple additions decompose into scaling, addition, and rescaling.Source: §B.3 · pp. 63–64

Stabilizer relabeling

• Valid row swaps, monomial multiplications, and additions relabel stabilizer checks without changing the code’s stabilizer group.These row operations respectively exchange check labels, translate their labels across super-cells, and replace one check by its sum with another.Source: §B.3 · p. 64

B.4 Setting up for the Decoupling Algorithm

• The decoupling algorithm is formulated through sufficient conditions, expressed in standard quantum error-correction language and connected to Haah [Haa16] and Bombín.These works present the conditions in different abstract languages.Source: §B.4 · p. 64

B.4.1 Topological Order Condition

• B.4.1 Topological Order Condition: Topological order requires ker H = im H† over R = F2[x±1, y±1], equivalently trivial first homology.The associated chain complex is exact, so no finite-support Pauli product forms a homologically nontrivial cycle on the infinite lattice.Source: §B.4.1 · pp. 64–65
• B.4.1 Topological Order Condition: This condition ensures macroscopically large code distance and is preserved when coarse-graining replaces R by R′ = F2[x±b, y±b].Some self-dual BB codes can nevertheless have strong finite-size parameters and performance without satisfying the condition [XZB+25].Source: §B.4.1 · p. 65

B.4.2 Torsion of the Cokernel

• B.4.2 Torsion of the Cokernel: coker H is the module of excitation classes, identifying configurations equivalent when their difference is generated by a finite-support Pauli error.Such equivalent excitations can be fused to vacuum in the anyonic picture.Source: §B.4.2 · p. 65
• B.4.2 Torsion of the Cokernel: The torsion submodule T(coker H) consists of point-like topological excitation classes annihilated by a nonzero polynomial in R.A finite superposition of translated excitation patterns can then be created by a finite-support Pauli error, allowing the defect to move and fuse with vacuum.Source: §B.4.2 · pp. 65–66
• B.4.2 Torsion of the Cokernel: In the toric code, a single-qubit Z operator translates an X-type excitation by one lattice step, and strings can move it to annihilate another defect.The finite set of anyon types coincides with the finite abelian group of torsion classes, which is invariant under Clifford transformations, stabilizer relabelings, and translations.Source: §B.4.2 · p. 66

B.4.3 Annihilator and Mobility

• B.4.3 Annihilator and Mobility: The annihilator ideal ann(coker H) contains polynomials that annihilate every excitation class and therefore encodes which translations kill all torsion classes.For the toric code, it is generated by x−1 and y−1.Source: §B.4.3 · p. 66
• B.4.3 Annihilator and Mobility: For the toric code, the annihilator ideal’s generators x−1 and y−1 correspond to one-step translations of any point charge along the x or y direction.These are one-dimensional string operators whose commutation with local checks is confined to their endpoints [Bom14].Source: §B.4.3 · p. 66

Sufficient conditions

• If ker HΩ = im H† over R = F2[x±1, y±1], a transformed matrix H′ has im H† = im H′† = ker H′Ω and ker H′† = 0 [Haa16].H′ has size 2t × 4t over a coarse-grained ring R′ = F2[x±b, y±b].Source: §B.4.4 · p. 67
• The coarse-grained structure exposes super-cell translations, allowing point-like excitations to move between adjacent super-cells and revealing a toric-code excitation map up to Clifford equivalence.The annihilator ideal is generated by x^b − 1 and y^b − 1, corresponding to finite-support Pauli translations.Source: §B.4.4 · p. 67

Decoupling theorem

• Under the stated condition, the code becomes a tensor product of finitely many toric-code copies and a product state after finitely many Clifford-operation layers [Haa16].The constructive procedure recursively applies local disentanglement and stabilizer relabeling to the coarse-grained matrix H′.Source: §B.4.4 · p. 67
• The number of independent toric-code copies equals 1/2 dim_F2 𝒯(coker H′), reflecting the two point-like excitation types, e and m.Translationally invariant qubits are selected across super-cells to extract each toric-code copy; the remaining checks and qubits form a product state.Source: §B.4.4 · p. 67

Chain isomorphism

• Lemma C.1 establishes that φ=(φ_2,φ_1,φ_0) defines a chain map from 𝒞 to 𝒞̃.The proof derives the required commutativity relations from the defining relations of the decoupling outputs U and V.Source: C Layer-decoupling decoder lemmas and… · p. 68

Projection chain maps

• Lemma C.2 shows that the projections π^(i) and π_A are chain maps.This follows because the projections remove the other summands in a direct sum of chain complexes, while respecting the definitions of the projections and differentials.Source: C Layer-decoupling decoder lemmas and… · p. 68

Combined chain-map identities

• Corollary C.3 combines the decoupling isomorphism and projection maps into the composite chain-map identities in equations (167) and (168).The identities relate the original differentials, transformed differentials, and projected components through φ and π_A.Source: C Layer-decoupling decoder lemmas and… · p. 68

Rectangular partition

• The method partitions monomial labels into disjoint ℓ′ × m′ rectangles and assigns associated qubits and X stabilizers to the coarse lattice cells.For the 24×24 gross code, the construction gives ℓ′ = 3 = m′.Source: D Computing short strings · pp. 68–69

Transfer matrices

• For each syndrome basis excitation, local error representatives define a horizontal transfer matrix A_x, with an analogous vertical matrix A_y that may differ from A_x.The resulting representatives give short strings between syndrome patterns in adjacent rectangles.Source: D Computing short strings · p. 69
• Additional short strings are obtained from eigenvectors of the transfer matrices, including fixed-subspace vectors that reproduce identical patterns between separated rectangles.The same construction applies in the vertical direction.Source: D Computing short strings · p. 69

Basis verification

• Candidate quotient-basis elements are checked by representing the ring as an F2-vector space, generating the ideal from monomial translates, and appending candidate columns to the resulting matrix.The candidates form a basis precisely when they span the quotient and are linearly independent modulo the ideal.Source: D Computing short strings · p. 69