arxiv: 2605.04151 · v1 · submitted 2026-05-05 · 🪐 quant-ph · cond-mat.str-el· math-ph· math.MP· math.QA

Recognition: 3 theorem links

· Lean Theorem

Topological subsystem bivariate bicycle codes with four-qubit check operators

Zijian Liang , Yu-An Chen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:19 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elmath-phmath.MPmath.QA

keywords quantum error correctionsubsystem codesbivariate bicycle codestopological codesgauge measurementsCSS codesquantum memorieslow-overhead codes

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The pith

Subsystem bivariate bicycle codes realize high-rate BB logical structures using only local four-qubit gauge measurements.

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

The paper introduces subsystem bivariate bicycle codes to combine the high encoding rate of bivariate bicycle codes with practical low-weight local measurements. Gauge operators of weight four replace the usual higher-weight stabilizer checks, with syndromes inferred by multiplying gauge outcomes. A determinantal-ideal criterion on the gauge commutation matrix detects and excludes nonlocal stabilizers, after which a finite-depth Clifford circuit isolates the protected subsystem. Explicit search produces examples such as the [[27,6,3]], [[75,10,5]], and [[108,12,6]] codes; the last encodes six times more logical qubits than a subsystem surface code of identical block length and distance. The construction is topological exactly when the underlying BB code is topological.

Core claim

We construct translation-invariant CSS subsystem codes that inherit the logical structure of bivariate bicycle codes while restricting all gauge measurements to weight four. Nonlocal stabilizers are excluded by a determinantal-ideal criterion applied to the gauge-operator commutation matrix; when the criterion holds, a finite-depth Clifford circuit decouples the gauge qubits and recovers the protected BB subsystem. The resulting SBB codes are topological if and only if the corresponding BB code is topological. Finite search yields low-overhead instances including [[27,6,3]], [[75,10,5]], and [[108,12,6]], the last of which encodes six times more logical qubits than a subsystem surface code (

What carries the argument

The subsystem bivariate bicycle (SBB) construction, which realizes BB-code logical operators and stabilizers via local weight-4 gauge operators whose outcomes are multiplied to infer stabilizer syndromes.

If this is right

Syndrome extraction for SBB codes uses only weight-4 measurements whose outcomes are post-processed to obtain the full stabilizer syndrome.
SBB codes remain topological precisely when the parent BB code is topological, so no nontrivial local logical operators exist.
The [[108,12,6]] code stores six times as many logical qubits as a subsystem surface code of the same length and distance.
Gauge freedom converts the high-rate but high-weight BB family into a family compatible with local syndrome extraction.

Where Pith is reading between the lines

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

The same gauge-based reduction might be applied to other translation-invariant high-rate CSS families to lower their measurement weights.
Hardware implementations would need to verify that the decoupling Clifford circuit can be realized with the available gate set without increasing error rates.
The rate advantage over surface codes suggests that SBB codes could reduce the physical-qubit overhead needed for a given logical memory capacity.
Generalizing the determinantal-ideal test to non-translation-invariant subsystem codes could enlarge the search space for low-overhead quantum memories.

Load-bearing premise

The determinantal-ideal criterion on the gauge commutation matrix correctly excludes every nonlocal stabilizer, and a finite-depth Clifford circuit can always decouple the gauge qubits without leaving residual errors.

What would settle it

An explicit SBB code that passes the determinantal-ideal test yet contains a nonlocal stabilizer operator, or a numerical distance calculation showing that one of the reported codes (such as [[108,12,6]]) has distance strictly less than claimed.

Figures

Figures reproduced from arXiv: 2605.04151 by Yu-An Chen, Zijian Liang.

**Figure 2.** Figure 2: FIG. 2. Weight-12 stabilizers and representative weight-5 view at source ↗

**Figure 3.** Figure 3: FIG. 3. Clifford reduction of the [[75 view at source ↗

**Figure 4.** Figure 4: FIG. 4. Examples of Laurent-polynomial representations of Pauli operators on the square lattice. Each unit cell contains view at source ↗

**Figure 5.** Figure 5: FIG. 5. Inverse construction of the [[75 view at source ↗

read the original abstract

High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight $6$ or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-$4$ gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including $[[27,6,3]]$, $[[75,10,5]]$, and $[[108,12,6]]$; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-$4$ syndrome extraction.

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.

Desk Editor's Note private letter to a colleague

The paper converts bivariate bicycle codes to weight-4 subsystem versions with a new algebraic test and gives some low-overhead examples, but the test's completeness is the key thing to check.

read the letter

Your colleague should know that this work gives a subsystem version of bivariate bicycle codes using only four-qubit gauge checks to capture the same logical information as the original higher-weight BB codes. They also introduce a determinantal-ideal test to ensure the code stays topological by ruling out nonlocal stabilizers. The paper does a good job extending the CSS subsystem framework to the translation-invariant setting of BB codes. The iff statement for topological equivalence is clean, and the finite search producing codes like the [[108,12,6]] that claims a sixfold rate improvement over subsystem surface codes at fixed length and distance is the practical hook. If the numbers hold, this directly tackles the issue of high-weight checks making syndrome extraction hard in these high-rate families. On the soft spots, the whole thing hinges on that determinantal-ideal criterion from the gauge-operator commutation matrix correctly catching all nonlocal stabilizers and that the subsequent Clifford decoupling circuit leaves no residual errors on the protected subsystem. The stress-test note points out a real risk here: if the test is incomplete, the effective distance or number of logical qubits could drop below what's reported. Since the abstract lays out the claims without showing the algebra or any explicit verification for the examples, the full text needs to include those details or at least some numerical confirmation that the codes perform as stated. The construction itself doesn't seem circular, as it builds on standard ideas without fitting parameters. This paper is for researchers in quantum error correction who are looking at LDPC and subsystem codes for better overhead. Someone working on practical implementations of high-rate quantum memories would find the examples and the weight-4 reduction useful to think about. It is worth sending to peer review because the ideas are specific, the examples are concrete, and the problem it addresses matters for scaling quantum computers, even if the proofs need tightening.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes the logical structure of bivariate bicycle (BB) stabilizer codes via local weight-4 gauge operators whose syndromes are inferred by multiplication. It presents a determinantal-ideal criterion on the gauge-operator commutation matrix to exclude nonlocal stabilizers, after which a finite-depth Clifford circuit decouples gauge qubits to recover the protected BB subsystem. The paper proves that an SBB code is topological if and only if the corresponding BB code is topological, and reports explicit low-overhead examples including the [[27,6,3]], [[75,10,5]], and [[108,12,6]] codes, with the last encoding six times as many logical qubits as a subsystem surface code at identical block length and distance.

Significance. If the determinantal-ideal criterion is shown to be both necessary and sufficient for the claimed logical structure and if the reported distances are independently verified, the construction would offer a concrete route to high-rate quantum memories whose stabilizer extraction uses only weight-4 local measurements, substantially lowering the overhead relative to surface-code-based alternatives while preserving the favorable scaling of BB codes.

major comments (2)

[the section introducing the determinantal-ideal criterion] The section introducing the determinantal-ideal criterion: the manuscript asserts that this algebraic test on the gauge-operator commutation matrix excludes all nonlocal stabilizers and guarantees that a finite-depth Clifford circuit recovers the BB subsystem without residual errors, yet provides no explicit derivation or verification for the bivariate polynomials appearing in the search; this step is load-bearing for every reported parameter set.
[the section presenting the concrete examples] The section presenting the concrete examples: the distance claims for [[27,6,3]], [[75,10,5]], and especially [[108,12,6]] (which supports the sixfold rate advantage) rest on unshown computational checks or syndrome simulations; without these, the effective k and d after decoupling cannot be confirmed.

minor comments (1)

[abstract] The abstract states that 'stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes' but does not specify the exact linear combinations or circuit depth required; a brief clarifying sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments, which will help improve the clarity and rigor of our manuscript. We address the major comments below and plan to incorporate revisions to provide the requested derivations and verifications.

read point-by-point responses

Referee: [the section introducing the determinantal-ideal criterion] The section introducing the determinantal-ideal criterion: the manuscript asserts that this algebraic test on the gauge-operator commutation matrix excludes all nonlocal stabilizers and guarantees that a finite-depth Clifford circuit recovers the BB subsystem without residual errors, yet provides no explicit derivation or verification for the bivariate polynomials appearing in the search; this step is load-bearing for every reported parameter set.

Authors: We concur that the derivation of the determinantal-ideal criterion merits a more explicit presentation. In the revised manuscript, we will augment the relevant section with a complete step-by-step algebraic derivation showing how the criterion applied to the commutation matrix of the gauge operators excludes nonlocal stabilizers for the bivariate polynomial case. Additionally, we will verify this explicitly for the polynomials used in generating the reported codes, confirming the conditions for the finite-depth Clifford circuit to recover the BB subsystem without residual errors. revision: yes
Referee: [the section presenting the concrete examples] The section presenting the concrete examples: the distance claims for [[27,6,3]], [[75,10,5]], and especially [[108,12,6]] (which supports the sixfold rate advantage) rest on unshown computational checks or syndrome simulations; without these, the effective k and d after decoupling cannot be confirmed.

Authors: We recognize the importance of detailing the computational verification for the code distances. The revised version will include a dedicated subsection or appendix outlining the syndrome simulation procedure and the specific checks performed to establish the distances of 3, 5, and 6 for the [[27,6,3]], [[75,10,5]], and [[108,12,6]] codes, respectively, post-decoupling. This will substantiate the effective logical qubit counts and distances. revision: yes

Circularity Check

0 steps flagged

Minor reliance on prior BB-code literature and standard subsystem CSS framework; central claims rest on algebraic criterion derived within the paper and finite search, not self-referential definitions or fitted predictions.

full rationale

The derivation introduces SBB codes via a translation-invariant CSS subsystem construction with weight-4 gauges, defines stabilizer inference by gauge multiplication, and proves a determinantal-ideal test on the gauge commutation matrix to exclude nonlocal stabilizers. When the test passes, a finite-depth Clifford circuit is asserted to decouple to the BB logical structure. The topological equivalence is stated as an if-and-only-if with the corresponding BB code, following directly from the construction. All reported codes ([[27,6,3]], [[75,10,5]], [[108,12,6]]) arise from explicit finite search over bivariate polynomials rather than any parameter fit or prediction. No load-bearing step reduces by construction to a self-citation chain or tautological renaming; the algebraic criterion is presented as newly shown in this work. This yields only minor (score-2) circularity risk from background citations to BB codes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies on standard quantum coding assumptions (CSS structure, translation invariance, gauge-operator commutation) without introducing new free parameters or invented entities beyond the code construction itself.

axioms (2)

domain assumption CSS subsystem codes admit gauge operators whose products recover stabilizer syndromes
Invoked when stating that stabilizer syndromes are inferred by multiplying gauge outcomes.
domain assumption Translation-invariant CSS subsystem codes have well-defined nonlocal stabilizers detectable via the gauge commutation matrix
Basis for the determinantal-ideal criterion.

pith-pipeline@v0.9.0 · 5528 in / 1313 out tokens · 50450 ms · 2026-05-08T18:19:23.754124+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/AlphaDerivationExplicit.lean, Constants alphaProvenanceCert (RS-style parameter-free derivations involve φ-ladders and J-cost; the paper's parameters are from a brute-force CSS search, not a forcing chain) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A finite search yields low-overhead examples including [[27,6,3]], [[75,10,5]], and [[108,12,6]]
Foundation/ArithmeticFromLogic.lean (RS arithmetic is recovered from the J-cost orbit of a generator γ ≠ 1 in ℝ_+; the paper's Laurent ring over F_2 is unrelated to that recovery) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

R = Z_2[x^±1, y^±1] ... I_r(M) = R ... Quillen–Suslin–Swan theorem for Laurent polynomial rings

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Landahl and Jonas T

Andrew J. Landahl, Jonas T. Anderson, and Patrick R. Rice, “Fault-tolerant quantum computing with color codes,” (2011), arXiv:1108.5738 [quant-ph]. 8 Appendix A: Subsystem code preliminaries In this section, we review basic concepts about stabilizer and subsystem codes [4, 41, 42, 46, 51]. Throughout this appendix, all Pauli operators are considered up to...

work page arXiv 2011
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Stabilizer codes A stabilizer code is specified by an Abelian subgroupS ⊂ P n of then-qubit Pauli group, with−I /∈ S. The code space is the joint +1 eigenspace of all stabilizers: C(S) ={|ψ⟩:S|ψ⟩=|ψ⟩for allS∈ S}.(A6) IfShassindependent generators, then dimC(S) = 2 n−s.(A7) Thus a stabilizer code encodes k=n−s(A8) logical qubits. The logical Pauli operator...
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The stabilizer code space factors as C(S) ∼= HL ⊗ HG,(A12) 9 whereH L stores the protected logical qubits andH G stores gauge qubits

Subsystem codes Subsystem codes generalize stabilizer codes by allowing part of the code space to be used as an unprotected gauge subsystem. The stabilizer code space factors as C(S) ∼= HL ⊗ HG,(A12) 9 whereH L stores the protected logical qubits andH G stores gauge qubits. Errors or measurements that affect only HG do not change the protected logical inf...
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Their gauge group is generated byX-type and Z-type Pauli operators separately

CSS subsystem codes The SBB codes considered in this work are CSS subsystem codes. Their gauge group is generated byX-type and Z-type Pauli operators separately. Let H g X ∈Z mX ×n 2 , H g Z ∈Z mZ ×n 2 (A25) be the binary matrices whose rows specify theX- andZ-type gauge generators. The corresponding gauge group is G=⟨X(u) :u∈row(H g X)⟩ · ⟨Z(v) :v∈row(H ...
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We first prove (1)⇒(2)

Proof of Lemma 2 Proof.First note that the Laurent monomialsx ayb are precisely the units ofR. We first prove (1)⇒(2). Suppose there existP, Q∈GL 2(R) such thatP M Q= ( 1 0 0 0 ). Since a monomial is a unit inR, the entries ofP M Qgenerate the whole ring: I1(P M Q) =R.(C1) On the other hand, multiplication by invertible matrices only performs invertible r...
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For a more general subsystem code withp X-type andq Z-type gauge-generator families, the commutation matrix is rectangular

Generalization of Lemma 2 The main text uses Lemma 2 for the 2×2 commutation matrix that appears when there are twoX-type and two Z-type gauge-generator families. For a more general subsystem code withp X-type andq Z-type gauge-generator families, the commutation matrix is rectangular. The entry-ideal condition is then replaced by a determinantal-ideal co...
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There exist invertible matrices P∈GL p(R), Q∈GL q(R),(C26) such that P M Q= ÅIr 0 0 0 ã .(C27)
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Determinantal ideals are invariant under invertible row and column operations

The maximal nonzero determinantal ideal is the unit ideal: Ir(M) =R.(C28) Proof.We first prove (1)⇒(2). Determinantal ideals are invariant under invertible row and column operations. Indeed, by the Cauchy–Binet formula, multiplying by an invertible matrix replaces eachj×jminor by anR-linear combination ofj×jminors, and applying the inverse operation gives...
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The corresponding statement for left kernels follows by applying the same argument toM T

Proof of Theorem 3 Proof.Write M= Åa b c d ã .(C54) We demonstrate the argument for right kernels. The corresponding statement for left kernels follows by applying the same argument toM T. For any torus quotient A=R n,m =R/(x n −1, y m −1),(C55) write KA := kerA(M (n,m) :A 2 →A 2) (C56) for the finite-torus kernel. Also write KR := kerR(M:R 2 →R 2) (C57) ...
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For a general translation-invariant CSS subsystem code withp X-type andq Z-type gauge-generator families, the commutation matrix is instead ap×qmatrix overR

Generalization of Theorem 3 Theorem 3 treats the 2×2 commutation matrix arising from twoX-type and twoZ-type gauge-generator families. For a general translation-invariant CSS subsystem code withp X-type andq Z-type gauge-generator families, the commutation matrix is instead ap×qmatrix overR. In this setting, the entry idealI 1(M) is replaced by the determ...
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IfI r(M) =R,then for everyn, m, Kn,m =ρ n,m(KR).(C135) Thus imposing periodic boundary conditions creates no additional right-kernel vectors, and hence no additional nonlocalZ-type stabilizers from this sector
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The analogous statements forX-type stabilizers hold for left kernels, or equivalently by applying the theorem toM T

IfI r(M)⊊R,then there exist positive integersn, msuch that ρn,m(KR)⊊K n,m.(C136) Any nonzero Pauli operator obtained from a vector in Kn,m \ρ n,m(KR) (C137) is an additional nonlocalZ-type stabilizer. The analogous statements forX-type stabilizers hold for left kernels, or equivalently by applying the theorem toM T. Proof.We first prove the caseI r(M) =R....
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We denote byUthe symplectic matrix representing the Clifford circuitU; thus conjugation byUacts on Pauli vectors by left multiplication byU

Proof of Theorem 4 Proof.We work in the polynomial Pauli module R6 =R 3 X ⊕R 3 Z.(C186) Let ¯·denote the involutionx7→x −1,y7→y −1, extended coefficientwise toR, and write v† := ¯vT (C187) for vectors or matrices overR. We denote byUthe symplectic matrix representing the Clifford circuitU; thus conjugation byUacts on Pauli vectors by left multiplication b...
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The gauge generators shown in Fig

Verification of the[[75,10,5]]guiding example We begin with the [[75,10,5]] code discussed in the main text. The gauge generators shown in Fig. 1 are GX,1 =   x2 y2 x+x 2y 0 0 0   , G X,2 =   1 +y 2 x+y 0 0 0 0   , GZ,1 =   0 0 0 y2 y+xy 2 x2   , G Z,2 =   0 0 0 1 +x 2 0 x+y   . (E1) Their co...
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Further examples from the finite search Table II lists representative weight-4 SBB codes found in the finite search. For each row, the second and third columns give the twoX-type gauge generators, GX,1 =G X(f1, g1, h1), G X,2 =G X(f2, g2, h2).(E11) The correspondingZ-type gauge generators are obtained by the combined reflection symmetry used in Appendix D...
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[[27,6,3]]: Mc = Å x2y−2 x−1y−1 +y −2 +xy −1 xy−1 +xy+x 2 x−2 +x −2y2 +x −1y−1 +y 2 +xy ã .(E17)
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[[60,10,4]]: Mc = Å x2y−2 xy−1 +xy x−1y−1 +xy −1 x−2 +x −2y2 + 1 +y 2 ã .(E18)
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[[75,10,5]]: Mc = Å x−2y2 x−1y−1 +x −1y x−1y+xy y −2 + 1 +x 2y−2 +x 2 ã .(E19)
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[[90,12,5]]: Mc = Å x2y−2 x−1y−1 +x 2 y−2 +xy x −3y−1 +x −2y2 + 1 +xy 3 ã .(E20)
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[[108,12,6]]: Mc = Å x2y−2 x−1y−1 +x 2 y−2 +xy x −3y−1 +x −2y2 + 1 +xy 3 ã .(E21)
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The distances listed in Table II are the dressed distances of the corresponding subsystem codes

[[126,14,6]]: Mc = Å x2y−2 x−1 +x y−1 +y x −3y+x −3y3 +x −1y+x −1y3 ã .(E22) For all matrices above, one verifies that detM c = 0.(E23) Moreover, since each matrix has a monomial entry, the entry ideal satisfies I1(Mc) =R.(E24) Therefore, the no-nonlocal-stabilizer criterion applies. The distances listed in Table II are the dressed distances of the corres...