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arxiv: 2605.18961 · v1 · pith:2GK2IIAOnew · submitted 2026-05-18 · 🪐 quant-ph · cs.IT· math-ph· math.IT· math.MP

4D and 5D Layer Codes through Color Routing

Andrew C. Yuan , Nou\'edyn Baspin This is my paper

Pith reviewed 2026-05-20 10:58 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath-phmath.ITmath.MP
keywords quantum error correctionLDPC codeshigher-dimensional codesCSS codescolor routingBPT boundslayer codesenergy barrier
0
0 comments X

The pith

Embedding qLDPC codes via color routing produces 4D and 5D layer codes that saturate the BPT bounds exactly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an explicit construction for quantum error-correcting codes in four and five dimensions by embedding lower-dimensional quantum LDPC codes into a layered structure. Color routing arranges the check layers and line defects so that the resulting codes inherit good scaling properties from the input. A sympathetic reader would care because the output codes meet the theoretical upper limits on distance and size in those dimensions while remaining modular enough for use in three-dimensional physical hardware.

Core claim

We introduce an explicit CSS code construction that generalizes layer codes to D=4 and D=5. From an input [[n,k,d]] qLDPC code with energy barrier Δ the method yields a D-dimensional layer code with parameters [[Θ(n^{D/(D-2)}), k, Θ(d n^{1/(D-2)})]] and energy barrier Ω(Δ). When the input codes are good, the construction saturates the D=4,5 BPT bounds exactly. The higher-dimensional codes are modular and therefore suited to architectures built from modular network patches despite the physical limitation to three dimensions.

What carries the argument

Color routing, the mechanism that resolves the structure of check layers and line defects during the embedding of qLDPC codes into the higher-dimensional lattice.

If this is right

  • The resulting codes achieve distance scaling Θ(d n^{1/(D-2)}) for D=4 and D=5.
  • Energy barrier of the output code scales as Ω(Δ) from the input.
  • The codes remain explicit CSS codes that preserve the original logical dimension k.
  • The construction is modular and therefore compatible with network-patch architectures in three physical dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same routing technique might extend to dimensions higher than five if the layer structure can be generalized further.
  • Modularity of the codes suggests they could support distributed error-correction protocols across separate hardware modules.
  • The embedding approach offers a systematic way to lift good codes from lower to higher dimensions while controlling defect weights.

Load-bearing premise

The color-routing embedding must preserve energy-barrier scaling and produce the claimed code parameters without introducing new low-weight errors or violating the CSS structure.

What would settle it

Build the explicit 4D code from a small known good qLDPC code and measure whether its minimum distance equals the predicted Θ(d n^{1/2}) scaling and whether any unexpected low-weight logical operators appear.

Figures

Figures reproduced from arXiv: 2605.18961 by Andrew C. Yuan, Nou\'edyn Baspin.

Figure 1
Figure 1. Figure 1: FIG. 1. 3D Layer Code Example. Depicts the 3D layer code [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Queue and Congestion. (a) depicts the projection of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Color Routes. (a) denotes a fixed arrangement of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Shor Code. The grey circles denote qubits. The blue [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Contracting Cycles. (a), (b) denotes the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Shor Defects. (a)-(g) denotes the defects (green lines) [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Check Routing. Each color denotes the AB route [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Check Layer. (a) The vertical ˆz [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Defects. (a) The red AB route denotes that of an [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Color Route on [ [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Contracting Cycles on [ [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Defects on [ [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Syndrome Cleaning. (a) is adopted from Fig. [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
read the original abstract

We introduce and explicit Calderbank-Shor-Steane (CSS) code construction that generalizes the Layer codes to $D=4,5$ dimensions. Much like its predecessor, the present construction is based on embedding quantum low-density parity check (qLDPC) codes; from an $[[n,k,d]]$ code with energy barrier $\Delta$, we obtain a $D=4,5$ dimensional Layer code with parameters $[[\Theta(n^{D/(D-2)}), k, \Theta(dn^{1/(D-2)})]]$ and energy barrier $\Omega(\Delta)$. Using good qLDPC codes as input, our construction saturates the $D=4,5$ dimensional BPT bounds exactly. The higher dimensional Layer Codes are modular, and thus well suited to architectures composed of modular network patches, despite our physical limitation to three dimensions. We overcome the hurdles encountered by previous generalization attempts through the use of \textit{color routing}, allowing us to resolve the structure of the check layers and line defects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces an explicit CSS construction that generalizes layer codes to four and five dimensions by embedding an input [[n, k, d]] qLDPC code with energy barrier Δ via color routing. The resulting D=4,5 layer code has parameters [[Θ(n^{D/(D-2)}), k, Θ(d n^{1/(D-2)})]] and energy barrier Ω(Δ). When the input is a good qLDPC code, the construction is claimed to saturate the BPT bounds exactly while remaining modular and compatible with three-dimensional hardware.

Significance. If the color-routing embedding is shown to preserve distance scaling and CSS structure without introducing new low-weight logical operators, the result would supply the first explicit modular construction achieving optimal scaling in D=4 and D=5. This would be a concrete advance for fault-tolerant architectures built from network patches.

major comments (2)
  1. [Construction via color routing] The central saturation claim rests on the assertion that color routing produces no logical operators of weight below Θ(d n^{1/(D-2)}). The manuscript describes routing to resolve check layers and line defects but supplies neither an exhaustive enumeration of post-routing operators nor a rigorous upper bound on their weights; this gap is load-bearing for the exact BPT saturation statement.
  2. [Parameter derivation] The output parameters are stated directly in terms of the input qLDPC parameters, yet the derivation of the precise exponents D/(D-2) and 1/(D-2) is not accompanied by an explicit counting argument or small-case verification showing how the embedding multiplies the number of physical qubits and logical operators.
minor comments (1)
  1. The notation for the higher-dimensional code parameters would benefit from an explicit small example (e.g., taking a small input qLDPC code and listing the resulting qubit and check counts after routing).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the presentation and proofs.

read point-by-point responses

  1. Referee: [Construction via color routing] The central saturation claim rests on the assertion that color routing produces no logical operators of weight below Θ(d n^{1/(D-2)}). The manuscript describes routing to resolve check layers and line defects but supplies neither an exhaustive enumeration of post-routing operators nor a rigorous upper bound on their weights; this gap is load-bearing for the exact BPT saturation statement.

    Authors: We agree that the current manuscript would benefit from a more explicit and rigorous argument establishing the distance lower bound after color routing. While the construction is designed such that routing resolves check layers and line defects without creating new low-weight logical operators (by leveraging the input code's distance and the CSS structure), we will add a dedicated subsection in the revised version that enumerates the possible operator types post-routing and provides a formal weight bound of Ω(d n^{1/(D-2)}) derived from the input qLDPC distance and the modular embedding. This will directly support the BPT saturation claim. revision: yes

  2. Referee: [Parameter derivation] The output parameters are stated directly in terms of the input qLDPC parameters, yet the derivation of the precise exponents D/(D-2) and 1/(D-2) is not accompanied by an explicit counting argument or small-case verification showing how the embedding multiplies the number of physical qubits and logical operators.

    Authors: The exponents arise from the D-dimensional layer embedding geometry: the physical qubit count scales as the product of the input block size n with a (D-2)-dimensional factor from the routing and layering, yielding n^{D/(D-2)}, while the distance scales with the additional n^{1/(D-2)} factor from the line defects and check layers. We will include an explicit counting argument in the revised manuscript (with a small illustrative example for D=4 using a simple input code) to make this derivation transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit construction derives parameters from inputs without reduction to fits or self-citations

full rationale

The paper presents an explicit CSS construction that embeds an input [[n,k,d]] qLDPC code with barrier Δ into a D=4,5 layer code via color routing, directly yielding the stated parameters [[Θ(n^{D/(D-2)}), k, Θ(d n^{1/(D-2)})]] and barrier Ω(Δ). This scaling follows from the embedding geometry and is not a fitted prediction or self-definition; the saturation of BPT bounds is claimed as a consequence when the input is a good qLDPC code. No load-bearing self-citation chain, ansatz smuggling, or renaming of known results appears in the derivation. The argument for preserving distance and CSS structure rests on the color-routing mechanism itself rather than reducing to prior fitted values or unverified uniqueness theorems. The derivation is therefore self-contained as a new modular construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the existence of good qLDPC input codes and on the unproven claim that color routing correctly organizes the higher-dimensional checks without introducing new errors.

axioms (1)
  • domain assumption Good qLDPC codes with energy barrier Δ exist and can be embedded while preserving the barrier scaling.
    The output parameters and energy barrier Ω(Δ) are stated to follow directly from any such input code.

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

  1. [1]

    Check Routing Despite being motivated by the 2D greedy routing al- gorithm in Lemma III.16, an arbitrary checkx∈ Xcan have weight>2 and thus a more general scheme must be considered. Specifically, Definition IV.1(AB Graph).For every qubitq= (a, b), let theAB graphAB(q) denote the union of the row and column that passes throughq, i.e., AB(q) = (a×[L])∪([L]...

    work page

  2. [2]

    Contracting Cycles Note that AB(x) may have nontrivial cycles so that ηAB(x)⊗R Z can have nontrivial internal logicals. Therefore, in this section, we will attempt to make the cycles contractible by introducing additional faces along some coordinate in the ˆzaxis, similar to Example III.8 and Lemma III.19. However, we first need the following Lemma Theore...

    work page

  3. [3]

    Gluing Maps Similar to the conventional layer codes, we define the gluing mapsg QX , gZQ in [Yua26] as follows, which are also depicted by the vertical dashed lines in Fig. 8. gQX |x;i, q, k •⟩=|i, q, k •⟩1{i=η(x), q∼x}(56) gZQ|i•, q, k⟩= X z∼q |i•, q, k;z⟩1{k=η(z)}(57) wherei • denotes eitheriori + fori∈[χ X], and similarly fork •, and thusg QX , gZQ act...

    work page

  4. [4]

    However, 14 (a)η(x)×η(z) =η X ×η Z ∈ˆx׈z (b)η(x)×η(z) =η X ×η Z ∈ˆx׈z FIG

    Defect Map Similar to the conventional layer codes, the defect mapsp ZX will correspond to strings between pairs of qubitsq 1, q2 ∈x∧zfor adjacentx, zchecks. However, 14 (a)η(x)×η(z) =η X ×η Z ∈ˆx׈z (b)η(x)×η(z) =η X ×η Z ∈ˆx׈z FIG. 9. Defects. (a) The red AB route denotes that of an x-check in Fig. 7. In the case where there exists az-check such thatx∧...

    work page

  5. [5]

    Note that the max degree isO(wq)

    Check Routing Definition V.1(Induced Routing onQ).LetG Q = (Q,P Q) denote theinduced graphonQwith vertices consisting of qubitsQand edges (also referred aspack- ets)p= (q, q ′) if there exists anx-check (orz-check) such thatq, q ′ ∼x(orq, q ′ ∼z). Note that the max degree isO(wq). Order each packetparbitrarily. Let theinduced routing problemonQbe such tha...

    work page

  6. [6]

    (Source and Destination) γ00(p) =γ 0(p) (81) γ11(p) =γ ∞(p) (82)

    work page

  7. [7]

    (One Plane at a Time) Continguous points in the sequence share the same plane, i.e.,γ 00, γ01 share the same[2]-coordinate, and similarly, forγ 10, γ11, whileγ 01, γ10 share the same3-coordinate

    work page

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    (Density) For anys∈F 2 2 andq∈[L] 3, |{p:γ s(p) =q}|=O(wq) (83) Proof.Write γ0(p) =a 0 ×b 0 ×c 0, γ ∞(p) =a ∞ ×b ∞ ×c ∞,(84) Then the color route is given by γ00(p) =γ 0(p) =a 0 ×b 0 ×c 0 γ10(p) =a 0 ×b 0 ×c 1 γ01(p) =a ∞ ×b ∞ ×c 1 γ11(p) =γ ∞(p) =a ∞ ×b ∞ ×c ∞ 17 For an intermediate positionc 1 ∈[L] 2 dependent onp that we define as follows. Consider a (...

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    The filled planes Λ(x) are similarly defined

    Contracting Cycles Let us denote the planes induced by filling in the star graphsλ(q) via Λ(q), which are naturally 3-term cell complexes. The filled planes Λ(x) are similarly defined. Specifically, 18 Definition V.6(Star Planes).Letq∈ Q= [L] 3. Then define thestar planeofqas Λ(q) = [ i<j q|[1,i) ×[L]×q| (i,j) ×[L]×q| (j,3] (101) so that|Λ(q)|=O(L 2) in t...

    work page

  10. [10]

    Gluing Maps Similar to the conventional layer codes, we define the gluing mapsg QX , gZQ as follows (analogue of the vertical dashed lines in Fig. 8). gQX |x;i, q, k •⟩=|i, q, k •⟩1{i=η(x), q∼x}(126) gZQ|i•, q, k⟩= X z∼q |i•, q, k;z⟩1{k=η(z)}(127) wherei • denotes eitheriori + fori∈[L X], and similarly fork •, so thatg QX , gZQ actlocallyin 5D

    work page

  11. [11]

    However, due to the 5D hypercube structure (3D grid structure of qubits), the strings are somewhat more complicated, even more so than the 4D case

    Defect Map Similar to the 3D Layer Codes, the defect mapsp ZX will correspond to strings between pairs of qubitsq 1, q2 ∈ x∧zfor adjacentx, zchecks. However, due to the 5D hypercube structure (3D grid structure of qubits), the strings are somewhat more complicated, even more so than the 4D case. Fortunately, our construction of the check layers exactly ac...

    work page

  12. [12]

    LetA=X→Q→Zbe the input CSS code withX, Q, Zdenoting theX-checks, qubits, andZ- checks

    Review Following [Yua26], though with somewhat different no- tation, let us define the conventional 3D layer codes as follows. LetA=X→Q→Zbe the input CSS code withX, Q, Zdenoting theX-checks, qubits, andZ- checks. LetX,Q,Zdenote the collection of correspond- ing basis elementsx, q, z, all of which are arranged on a 1D linear array so thatX= [|X |] and sim...

    work page

  13. [13]

    Then ∆X(C) = Ω 1 wX qZ ∆X(A) (A11) ∆Z(C) = Ω 1 wZqX ∆Z(A) (A12) Proof.The statement follows from Theorem A.5 and Re- mark 9

    Result Theorem A.1(Energy Barrier).LetCdenote the 3D layer code as defined in Eq.(A2)-(A9)withX- andZ- type (1-)energy barrier∆ X(C),∆ Z(C), and similarly for the input codeA. Then ∆X(C) = Ω 1 wX qZ ∆X(A) (A11) ∆Z(C) = Ω 1 wZqX ∆Z(A) (A12) Proof.The statement follows from Theorem A.5 and Re- mark 9

    work page

  14. [14]

    Lemma A.2(Xlayer error).Letσ x ∈x⊗R ⊤ Q ⊗R ⊤ Z be a 0-chain (syndrome)

    Proof of Energy Barrier Since the construction is symmetric inX, Z, we shall be concerned with theX-type energy barrier ∆(C), while theZ-type is similarly derived. Lemma A.2(Xlayer error).Letσ x ∈x⊗R ⊤ Q ⊗R ⊤ Z be a 0-chain (syndrome). Then there exists a mapσ x 7→ˆex such that the 1-chainˆe x is such that∂ xˆex =σ x and ∆1(ˆex)≤(1 +w X)|σx|(A13) and |gqX...

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  15. [15]

    Let us now check that the embedded column code is =A

    Logicals Preserved Proof of Theorem IV.9.By Lemma IV.8, we see that the gluing maps and the defect map are compatible and thus by the framework in [Yua26], the output codeCis well- defined. Let us now check that the embedded column code is =A. SinceC(x) is a cell-complex with a (unique) con- nected componentC 0(x), we see thatH 0(C(x)) ∼= F2 has basis [∥x...

    work page

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    Distance Preserved To prove that the distance of the 4D Layer Codes is preserved in some manner, the procedure is similar to that of the∞-dimensional Layer Codes [YBW26] (which also applies to the conventional 3D Layer Codes [WB24, Yua26]). Specifically, we will utilize Appendix (A) of [YBW26] to compute the relative expansion coefficient of the check lay...

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    Energy Barrier Preserved To prove that the energy barrier of the 4D Layer Codes is preserved in some manner, the procedure is similar to that of the conventional 3D Layer codes [WB24] (see Appendix A for a detailed exposition), though additional care must be taken due to the contracting layers, e.g., ηω(x) in Fig. 8b. In particular, since the algorithm La...

    work page

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    Logicals Preserved Proof of Theorem V.16.The proof is similar to that of Theorem IV.9 and thus we will only discuss the differ- ences. By Theorem V.12 and [Yua26], the output code and input code has isomorphic logicals, i.e., H1(C) ∼= H1(A) (C1) Note that the number of qubits ofCper edge follows from Remark 14 and the fact that an edge is either along the...

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    Distance Preserved To prove that the distance of the 5D Layer Codes is preserved in some manner, the procedure is similar to that of the∞-dimensional Layer Codes [YBW26] (which also applies to the 3D case in[WB24, Yua26] and 4D case in Theorem IV.10.). Specifically, we will utilize Appendix (A) of [YBW26] to compute the relative expansion coeffi- cient of...

    work page

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    In particular, since the 5D Layer Codes are symmetric inX, Z-checks, we shall restrict our attention to theX-type energy barrier

    Energy Barrier Preserved To prove that the energy barrier of the 5D Layer Codes is preserved in some manner, the procedure is similar to that of the 3D case [WB24] (see Appendix A for a de- tailed exposition) and 4D case in Theorem IV.11, though similar to the 4D case, additional care must be taken due to the contracting layersηΛω(x). In particular, since...

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    Check Layers a. Check Routing Definition D.1(Induced Routing onQ).LetG Q = (Q,P Q) denote theinduced graphonQwith vertices consisting of qubitsQand edges (also referred aspack- ets)p=qq ′ if there exists anx-check (orz-check) such thatq, q ′ ∼x(orq, q ′ ∼z). Note that the max degree is O(wq). Order each packetpbased on the lexicographic order ofQ= [L] D. ...

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    (Source and Destination) γ¯0(p) =γ 0(p) (D2) γ¯1(p) =γ ∞(p) (D3)

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    (One Plane at a Time) Ifs| [i] =s ′|[i], then γs(p)|[2i] =γ s′(p)|[2i] (D4) If, in addition,δ(s) =δ(s ′) =i+ 1, then γs(p), γs′(p)have the same coordinates except pos- sibly for the2i+ 1- and2i+ 2- coordinate

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