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Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read By deleting selected qubits and stabilizers from bivariate bicycle codes, one obtains a code on an open-boundary lattice with the same distance and locality but half the logical qubits.

desk verdict Useful methods paper with a sound univariate pruning result, exploratory bivariate examples, and a locality proof that needs spelling out. read the letter →

arxiv 2412.04181 v1 pith:LH6JBUBY submitted 2024-12-05 quant-ph

Jens Niklas Eberhardt , Francisco Revson F. Pereira , Vincent Steffan This is my paper · ORCID
classification quant-ph MSC 81P70 PACS 03.67.Pp
keywords quantumerrorcorrectionbivariatebicyclecodesqLDPCopenboundaryconditionspruninghypergraphproductfold-transversalgatescyclic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Bivariate bicycle codes are quantum low-density parity-check codes whose local checks live on a 2D lattice, but the lattice has periodic boundary conditions, which complicates hardware implementation. This paper introduces a pruning operation—deleting selected qubits and stabilizers while keeping the remaining checks commuting—and proves that every bivariate bicycle code that is a hypergraph product of two classical cyclic codes can be pruned onto a lattice with open boundary conditions. For $A(x)$ dividing $x^\ell - 1$ and $B(y)$ dividing $y^m - 1$, the pruned code has the same distance, roughly the same physical qubit count, and exactly half the logical qubits, with parameters $[[\ell m + (\ell - r_A)(m - r_B), r_A r_B, \min(d_A,d_B)]]$. The paper also gives examples of genuinely bivariate (non-hypergraph-product) codes that can be pruned, including the honeycomb color code, and shows that some fold-transversal gates survive the pruning. The importance is that this moves low-overhead bivariate bicycle codes one step closer to planar hardware.

What carries the argument

The key object is the pruning operation on stabilizer codes: choose subsets of qubits and of stabilizer generators, restrict the surviving generators to the surviving qubits, and require the restricted X- and Z-checks to commute. For cyclic codes, the construction uses reduced parity-check matrices: $A_{\mathrm{red}}$ deletes the first $r_A$ rows of $A(S_\ell)$, and $B_{\mathrm{red}}$ deletes the last $r_B$ columns of $B(S_m)$. The pruned code is then the hypergraph product $C_{A_{\mathrm{red}}} \boxtimes C^T_{B_{\mathrm{red}}}$, which is shown to be a literal row-and-column deletion of the original bivariate bicycle code's check matrices. This explicit deletion description is what guarantees the open-boundary layout: the deleted rows and columns are exactly the qubits and stabilizers whose checks would wrap around the periodic boundary, and since hypergraph products of cyclic codes automatically satisfy the CSS commutativity condition, the remaining code is valid and local.

What would settle it

Take a small example, say $\ell = m = 6$ with $A(x) = 1 + x + x^2$ and $B(y) = 1 + y + y^2$, construct $A_{\mathrm{red}}$ and $B_{\mathrm{red}}$ as in Lemma 4, and explicitly form the hypergraph product; check whether its X- and Z-check matrices satisfy $H_X H_Z^T = 0$ and whether its distance equals $\min(d_A,d_B)$. Any failure—non-commuting checks, or a lower distance—would disprove the theorem.

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Extended reading notes

Core claim

The central discovery is Theorem 1: if $Q$ is a bivariate bicycle code built from univariate polynomials $A(x)$ and $B(y)$ with $A(x) \mid (x^\ell - 1)$ and $B(y) \mid (y^m - 1)$, then there exists a pruned bivariate bicycle code that is as local as $Q$ on a lattice with open boundary conditions. Concretely, the pruned code is the hypergraph product $C_{A_{\mathrm{red}}} \boxtimes C^T_{B_{\mathrm{red}}}$, where $A_{\mathrm{red}}$ and $B_{\mathrm{red}}$ are the parity-check matrices of the original cyclic codes with redundant rows (respectively columns) deleted. Proposition 1 gives its parameters as $[[\ell m + (\ell - r_A)(m - r_B), r_A r_B, \min(d_A,d_B)]]$, so the distance is unchanged, the physical overhead is nearly unchanged, and the logical dimension drops from $2 r_A r_B$ to $r_A r_B$. The same construction, applied when the divisibility condition fails by first replacing $A$ and $B$ with their greatest common divisors with $x^\ell - 1$ and $y^m - 1$, yields pruned codes with identical parameters. For the non-hypergraph-product regime, explicit prunings of the honeycomb color code and of a code with parameters $[[30,2,4]]$ and $[[66,2,6]]$ are exhibited.

Load-bearing premise

Everything rests on the claim in Lemma 4 that the reduced hypergraph product code is literally obtained from the original code by deleting specified rows and columns of the check matrices, so that commutativity and parameters follow without any additional adjustment.

Editorial extensions

If this is right

  • Every hypergraph-product bivariate bicycle code satisfying the divisibility condition can be converted to an open-boundary code with the same distance and locality, at the cost of half its logical qubits.
  • The conversion preserves fold-transversal gates: the ZX-duality that implements the logical S/CZ-type gates restricts to the pruned code, so fault-tolerant Clifford operations remain available.
  • When $A(x)$ does not divide $x^\ell - 1$, one can first replace $A$ by $\gcd(A, x^\ell - 1)$ and $B$ by $\gcd(B, y^m - 1)$; the resulting code has the same parameters and then satisfies the divisibility condition, so pruning still applies.
  • The explicit examples—the honeycomb color code and the codes with parameters $[[30,2,4]]$ and $[[66,2,6]]$—show that pruning is not restricted to hypergraph products, suggesting a more general mechanism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the row-and-column deletion argument can be extended to all bivariate bicycle codes, the high-rate codes from the main motivating family could run on planar hardware; the examples here suggest the obstruction is practical rather than structural.
  • The halving of the logical dimension may reflect a general trade-off: changing a periodic-boundary local code into an open-boundary local code with the same check weights seems to force the loss of roughly half the logical qubits.
  • The computer search over convex lattice regions used for the examples could be systematized into an algorithm for finding open-boundary prunings of arbitrary bivariate bicycle codes.
  • Combining pruning with the boundary gauge operators studied in the works on extracting topological orders of Pauli stabilizer codes might yield open-boundary codes with better parameters than direct pruning alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper introduces a notion of "pruning" quantum stabilizer codes—deleting selected qubits and stabilizer generators while preserving commutativity—as a way to convert bivariate bicycle codes from periodic to open boundary conditions. For hypergraph products of cyclic codes, i.e., bivariate bicycle codes specified by univariate polynomials A(x) and B(y) with A(x)|(x^l-1) and B(y)|(y^m-1), the paper states Theorem 1 and Proposition 1: there exists a pruned code with parameters [[lm+(l-r_A)(m-r_B), r_A r_B, min(d_A,d_B)]] that is "as local as" the parent code on a lattice with open boundary conditions, halving the number of logical qubits while preserving distance and roughly preserving the physical qubit count. The paper also gives explicit examples of pruning non-hypergraph bivariate bicycle codes, including a honeycomb color code family, and discusses how fold-transversal gates can be carried over to the pruned codes.

Significance. Conditional on the proof being completed, the construction is appealing: it offers a concrete route from periodic bivariate bicycle codes to open-boundary codes with preserved distance and locality, which is directly relevant to hardware implementations. The algebraic part is clean and uses standard tools: Lemma 1 and Proposition 1 follow from cyclic-code theory and the Tillich-Zemor hypergraph product formula, and the parameter computation is transparent. The paper is also honest about its exploratory scope, presenting the non-hypergraph cases as examples rather than as a general theorem. Its main weakness is that the geometric claim underlying Theorem 1—the open-boundary locality of the pruned Tanner graph—is not actually proved, and the gate-compatibility proposition is asserted rather than demonstrated. These gaps are local in the sense that they may be fixable, but they are load-bearing for the paper's central claims.

major comments (2)
  1. [Section 3, Lemma 4 and following paragraph] The proof that C_Ared ⊠ C^T_Bred is "as local as" Q on a lattice with open boundary conditions is missing. The one-sentence proof of Lemma 4 only says that rows and columns of the check matrices are removed; it does not specify which deleted rows and columns correspond to which removed qubits and stabilizers, nor does it give an explicit embedding of the remaining Tanner graph into an open-boundary lattice. The following claim, "by construction as local ... but does not utilize the periodic boundary conditions," therefore does not follow from the stated argument. Concretely, for l=6 and A(x)=1+x+x^2, r_A=2, deleting the first two rows of A(S_6) leaves row 4 with support {4,5,0} modulo 6; if the open boundary is placed between positions 5 and 0, that stabilizer has diameter 5 in the open layout, not r_A+1. A different cut or a different choice of which r_A consecutive rows to delete may repair this, but the paper neither specifies nor analyzes such a choice. Since open-boundary locality is the central novelty of Theorem 1, an explicit placement rule and a proof that every remaining stabilizer has bounded support in the open layout are required.
  2. [Section 4, Proposition 2] The compatibility of the ZX-duality pi with pruning is asserted without proof. The statement that the restriction to the remaining physical qubits "still yields a unitary on the logical level" is not formalized: when pruning removes one member of a two-qubit orbit, one must specify how the fold-transversal physical gate is modified, and then prove that the modified operator preserves the stabilizer group of the pruned code and induces the claimed logical action on the surviving logical qubits. The appeal to Figures 4b and 5c is not a substitute for this argument. A concise proof for the general hypergraph-product case of Lemma 4 should be supplied.
minor comments (3)
  1. [Section 3, proof of Proposition 1] The sentence "Plugging this into Lemma 2" should refer to Theorem 2, since the hypergraph product parameter formula is Theorem 2, not Lemma 2.
  2. [Section 2.4 and Definition 1] The symbol l is used both for the number of stabilizer generators in Definition 1 and for the horizontal lattice size in Section 2.4; this overloaded notation should be disambiguated.
  3. [Example 2] The statement that the pruned examples were constructed by computer search over certain convex regions is not accompanied by any details of the search or by a verification method; please add enough information for the reader to reproduce these claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pruning construction and parameter formula derive from standard external results, with no fitted inputs and no load-bearing self-citation.

full rationale

The paper's central claim, Theorem 1 and Proposition 1, is derived from independent classical facts: Lemma 1 about cyclic code parity-check matrices, Lemma 2 about gcds and kernels, Lemma 3 about transpose cyclic codes, and the external hypergraph product parameter formula Theorem 2 from Tillich and Zemor. The pruning construction in Lemma 4 is an explicit algebraic manipulation: it deletes specific rows and columns from the check matrices and then verifies that the remaining matrices define the hypergraph product of the reduced classical codes. This is a direct construction, not a hidden reuse of the claimed conclusion. No parameter is fitted to data and then renamed as a prediction; the logical dimension rA*rB and distance min(dA,dB) follow from the stated hypotheses A(x)|(x^l - 1) and B(y)|(y^m - 1) together with the standard hypergraph product formula. The only self-citations are [ES24] and [BE21], and they are used for background or for the separate fold-transversal gate discussion, not to force the pruning theorem. The paper also cites external works [QWV23, BB24, TZ14] for the fold-transversal and hypergraph product facts. Regardless of whether the locality claim for the open-boundary embedding is fully proved, that would be a correctness or completeness concern, not circularity: the claimed reduction does not rely on its own conclusion being assumed. The analysis is therefore self-contained against external mathematical results, and no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper relies on standard CSS and cyclic code theory, plus the hypergraph product parameter theorem. No free parameters are fitted: the construction is deterministic given the polynomials and lattice sizes. No new physical entities are postulated; the pruned code is a mathematical construction, not a new physical object. The examples use specific polynomial choices, which are design decisions, not fitted constants.

assumptions (4)
  • standard math CSS code orthogonality condition H_X H_Z^T = 0 is preserved under column and row deletions that respect the pruned stabilizer set.
    Used in Definition 1 and throughout Section 3 to justify that pruning yields a valid stabilizer code.
  • standard math For a cyclic code with check polynomial h of degree r, any n-r consecutive rows of h(S_n) are linearly independent and span all rows (Lemma 1).
    Used in Lemma 4 and Proposition 1 to compute the parameters of the reduced matrices Ared and Bred.
  • standard math Tillich-Zemor hypergraph product parameter formula: C1 (tensor product) C2 has parameters [[n1 n2 + l1 l2, k1 k2 + k^T_1 k^T_2, min(d1,d2,d^T_1,d^T_2)]].
    Used in Proposition 1 to derive the pruned code parameters from the classical codes CAred and C^T_Bred.
  • domain assumption Bivariate bicycle code construction as defined in [KP13] and reviewed in Section 2.4, including the check matrix form H_X = (A(x,y)|B(x,y)), H_Z = (B(x,y)^T|A(x,y)^T).
    The entire paper operates within this code family, and the pruning analysis assumes the specific matrix representation (6).

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Pith. "Pith review of Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions." pith.science (2026). https://pith.science/paper/LH6JBUBY

@misc{pith2026241204181,
  author       = {Pith},
  title        = {Pith review of: Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LH6JBUBY}},
  note         = {Machine review of arXiv:2412.04181}
}
read the original abstract

Quantum low-density parity-check codes are promising candidates for quantum error correcting codes as they might offer more resource-efficient alternatives to surface code architectures. In particular, bivariate bicycle codes have recently gained attention due to their 2D-local structure, high encoding rate, and promising performance under simulation. In this work, we will explore how one can transform bivariate bicycle codes defined on lattices with periodic boundary conditions to codes with the same locality properties on a 2D lattice with open boundary conditions. For this, we introduce the concept of pruning quantum codes. We explain how pruning bivariate bicycle codes is always possible when the codes are hypergraph products of two classical cyclic codes. We also indicate that this might be possible for more general bivariate bicycle codes by constructing explicit examples. Finally, we investigate fault-tolerant quantum computation using the constructed pruned codes by describing fold-transversal gates.

Figures

Figures reproduced from arXiv: 2412.04181 by the authors.

Figure 1
Figure 1. The X-type and Z-type stabilizers of the bivariate bicyclic code associated with the polyno￾mials A(x, y) = 1+x+y 2 , B(x, y) = 1+y+x 2 . The white qubits are the horizontal qubits, the black ones the vertical qubits. The colored qubits are the support of the marked stabilizer. The other polynomial terms sketch how the X- and Z-type stabilizers look in general. For the X-type stabilizer depicted in (a), the terms of… view at source ↗
Figure 2
Figure 2. Examples for Tanner graphs. The variable vertices are white, the check vertices black. In (a), we display two possible Tanner graphs for the repetition code that encodes 1 logical bit into 3 physical bits via 0 7→ 000, 1 7→ 111. The first one is the cyclic code with parity check matrix h(S3) where h = 1+x. For the second one, we discard the first (redundant) check. In (b), we display the Tanner graph of the cyclic c… view at source ↗
Figure 3
Figure 3. For the surface code, the qubits are placed on the edges of a lattice with two rough and two smooth boundaries as shown in (a). For each plaquette of the lattice P, there is a stabilizer Q i∈P Zi . For each vertex v of the lattice, there is a stabilizer Q v∈i Xi . In (b), we show the corresponding Tanner graph with X-check vertices in red and Z-check vertices in blue. Finally, in (c), we see how the surface code ari… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The bivariate bicycle code Q associated with A(x) = 1 + x + x 2 and B(y) = 1 + y + y 2 is the hypergraph product of the classical cyclic code associated with 1 + x + x 2 and its transpose. We depict the edges in the Tanner graph for two specific stabilizer generators i…
Figure 5
Figure 5. Figure 5: The (6.6.6)-honeycomb color code as a bivariate bicycle code. In (a), we depict the stabilizer generators for the polynomials A(x, y) = 1 + x + xy and B(x, y) = 1 + y + xy. By tiling the plane as in (b), we see that the stabilizer generators induce a (6.6.6)- honeycomb…
Figure 6
Figure 6. Figure 6: Pruning the code associated with the polynomials A(x, y) = 1 + x + y −1 + xy and B(x, y) = 1 + y + xy + x −1y −1 . In 6a, we depict examples of X- and Z-type stabilizers where we write ¯x = x −1 and ¯y = y −1 for better visibility. In 6b and 6c, we depict pruned versio…
Figure 7
Figure 7. Figure 7: In (a), we depict the ZX duality π. For any symmetric bivariate bicycle code, it induces a logical operation by applying S, S † , and CZ on the physical qubits as depicted in (b). In (c), we depict a basis of logical operators for the bivariate bicycle code associated …

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Forward citations

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    quant-ph 2026-02 accept novelty 7.0 of 10

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    Bivariate bicycle quantum codes form infinite families via graph covers: the [[144,12,12]] gross code is a double cover of [[72,12,6]], with logical-operator lifting and parameter bounds.

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