Bivariate Bicycle qLDPC Codes

• Bivariate Bicycle qLDPC Codes are defined via sparse bivariate polynomials over a quotient ring, yielding block-circulant, quasi-cyclic CSS stabilizer matrices.
• They feature a toric 2D periodicity with constant stabilizer weight and controlled nonlocal interactions, achieving higher encoding rates than comparable surface codes.
• The codes support diverse decoding strategies and hardware implementations, with explicit algebraic structures that facilitate logical operator construction and transversal gate analysis.

Bivariate bicycle qLDPC codes are CSS stabilizer codes built from sparse bivariate polynomials over the quotient ring $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1)$, or equivalently from two-block group-algebra constructions over $\mathbb{Z}/\ell \times \mathbb{Z}/m$. Their parity checks are block-circulant, quasi-cyclic, and sparse; in the widely studied instantiations derived from three monomials in each of $A$ and $B$, stabilizers have weight $6$, each data qubit has constant degree, and the Tanner graph exhibits 2D periodicity together with a controlled amount of long-range connectivity (Shaw et al., 27 Apr 2026, Eberhardt et al., 2024). The family has consequently become a focal point for three interconnected research programs: algebraic analysis of logical operators and symmetries, decoding on sparse but loopy Tanner graphs, and hardware architectures that either exploit or compensate for the nonlocal edges.

1. Algebraic framework and CSS structure

The standard BB construction starts from two sparse bivariate polynomials,

$$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$

over

$R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$

With the commuting shift operators

$x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$

each monomial $x^iy^j$ becomes a block-circulant shift on an $\ell\times m$ torus, and $\mathbb{Z}/\ell \times \mathbb{Z}/m$0 become $\mathbb{Z}/\ell \times \mathbb{Z}/m$1 binary matrices. The CSS parity-check matrices are then

$\mathbb{Z}/\ell \times \mathbb{Z}/m$2

with the commutation condition

$\mathbb{Z}/\ell \times \mathbb{Z}/m$3

equivalently $\mathbb{Z}/\ell \times \mathbb{Z}/m$4 in the bicyclic presentation (Shaw et al., 27 Apr 2026, Eberhardt et al., 2024).

The same family admits a two-block or group-algebra description. Writing $\mathbb{Z}/\ell \times \mathbb{Z}/m$5 for $\mathbb{Z}/\ell \times \mathbb{Z}/m$6, one forms a three-term complex

$\mathbb{Z}/\ell \times \mathbb{Z}/m$7

with differentials $\mathbb{Z}/\ell \times \mathbb{Z}/m$8 and $\mathbb{Z}/\ell \times \mathbb{Z}/m$9. In this language, physical qubits occupy horizontal and vertical edge sectors, and logical $A$0-operators are represented by homology classes

$A$1

This formulation is central to explicit logical-basis constructions and fold-transversal gates (Eberhardt et al., 2024).

In the instantiations studied in several implementation papers, $A$2 and $A$3 are each sums of three monomials in $A$4 or $A$5. Each row and column of $A$6 and $A$7 then has weight $A$8, so the stabilizers derived from $A$9 have weight $B$0 (Shaw et al., 27 Apr 2026). This sparse constant-weight structure is the reason BB codes are classed as qLDPC rather than merely quasi-cyclic stabilizer codes.

2. Tanner-graph geometry and representative code families

The Tanner graph of a BB code is bipartite between data qubits and check nodes. In the principal benchmark family, each data vertex has degree $B$1 and each check vertex has degree $B$2, and the quasi-cyclic structure creates a 2D toric arrangement with periodic boundary conditions. Long-range edges arise from monomials $B$3 that wrap support around the torus, so BB codes are not strictly planar nearest-neighbor codes even when they admit regular 2D embeddings (Shaw et al., 27 Apr 2026). This has two immediate consequences that recur throughout the literature: short cycles and trapping sets degrade iterative decoding, and hardware implementations must either realize or virtualize a limited set of nonlocal couplings (Rabeti et al., 13 May 2026, Berthusen et al., 2024).

A distinct but related geometric statement is that bicycle Tanner graphs have thickness $B$4, which has motivated bilayer and two-sided layouts in both neutral-atom and superconducting-style discussions (Poole et al., 2024). At the same time, several papers emphasize that BB codes retain nonzero rate and significantly higher encoding rate than the surface code at comparable blocklengths (Blue et al., 17 Apr 2025, Berthusen et al., 2024).

Code	Reported parameters	Typical context
$B$5	$B$6	circuit-level decoding, networked and erasure-aware studies
$B$7	$B$8	networked, fusion-based, and 2D-local studies
$B$9	$6$0	2D-local and photonic studies
$6$1	$6$2	main decoder and hardware benchmark
$6$3	$6$4	larger BB benchmark; modular “two-gross” instance
$6$5	B1 code	large-code decoder generality

Beyond these benchmark instances, symmetry-focused work constructs $6$6 and $6$7 BB codes with explicit logical bases and fold-transversal Clifford gates (Eberhardt et al., 2024). Implementation papers also single out the “gross” code $6$8 and “two-gross” code $6$9 as architectural primitives in modular schemes (Yoder et al., 3 Jun 2025).

A common misconception is to treat BB codes as surface-code variants with a few extra edges. The papers instead describe them as sparse CSS qLDPC codes with structured 2D periodicity and controlled nonlocality: they are toric in layout, but their algebraic definition and logical content are not reducible to topological nearest-neighbor stabilizer patterns (Shaw et al., 27 Apr 2026, Eberhardt et al., 2024).

3. Logical operators, homology, and transversal gate structure

The logical-operator theory of BB codes is unusually explicit for a qLDPC family. In the two-block formalism, logical $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$0-operators live in

$$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$1

For odd $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$2, the ring $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$3 is semisimple, and the codes are pure and principal: $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$4 decomposes as

$$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$5

where $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$6 and $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$7 generate $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$8 and $$A(x,y)=\sum_{i,j} a_{ij}x^iy^j,\qquad B(x,y)=\sum_{i,j} b_{ij}x^iy^j,$$9 (Eberhardt et al., 2024). This yields explicit horizontal and vertical logical sectors analogous to toric-code cycles, while preserving BB’s high-rate qLDPC structure.

For the symmetric examples $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$0 and $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$1, the papers construct “nice” logical bases and corresponding dual $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$2-bases. These bases make the action of translation automorphisms, axis exchange, and ZX-duality transparent. In the $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$3 case, the fold-transversal gates generated by $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$4, $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$5, $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$6, and $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$7 act as symplectic matrices over $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$8, and the resulting logical gate group modulo Paulis is

$R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1).$9

For $x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$0, the corresponding group is

$x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$1

(Eberhardt et al., 2024). These are Clifford-only constructions; the same work explicitly identifies the absence of a non-Clifford transversal gate as an open issue.

Self-dual extensions modify the BB family rather than the logical theory. Stacking a non-self-dual BB code with its transpose-related partner yields a self-dual double-layer code with

$x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$2

preserving LDPC sparsity while enabling transversal Clifford structure (Liu et al., 17 Feb 2026). Numerical calculations under a circuit-level noise model report pseudo-thresholds exceeding $x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$3 for odd-weight double-layer families and up to $x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$4 for even-weight stacked codes (Liu et al., 17 Feb 2026).

A further extension appears in the lifted-product setting, where 2D bivariate bicycle codes act as component codes of 3D bivariate tricycle codes. Under the algebraic conditions $x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$5 odd and $x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$6, the LP structure gives a one-way transversal CNOT from a 3D code to its 2D BB component, enabling teleportation-based code switching, while the 3D tricycle admits a depth-2 CCZ (Li et al., 8 Oct 2025). This suggests a division of labor in which BB layers provide low-overhead Clifford handling and related product codes provide non-Clifford resources.

4. Decoding on sparse but loopy graphs

Decoding BB codes is dominated by the tension between LDPC sparsity and abundant short cycles. In the standard CSS setting, $x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$7- and $x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$8-components are decoded independently, and many works analyze one binary component under a binary symmetric channel or a detector-space reduction of circuit-level noise. A representative normalized min-sum update takes the form

$x=S_\ell\otimes I_m,\qquad y=I_\ell\otimes S_m,$9

$x^iy^j$0

with $x^iy^j$1 and hard decision $x^iy^j$2 if $x^iy^j$3, else $x^iy^j$4 (Rabeti et al., 13 May 2026). The central obstacle is that BP is exact on trees, whereas BB Tanner graphs contain many short cycles and trapping sets.

The most explicit decoder tailored to this observation is the Multiple-Bases Belief-Propagation List Decoder. MBBP-LD constructs redundant parity-check “bases” by cycle-free subtree decompositions of the Tanner graph, augments

$x^iy^j$5

runs BP in parallel on each basis, and selects a candidate by the frequency-weighted score

$x^iy^j$6

For $x^iy^j$7, MBBP-LD achieves $x^iy^j$8–$x^iy^j$9 logical-error-rate reduction versus BP-OSD and $\ell\times m$0–$\ell\times m$1 versus BPGD across low-to-moderate $\ell\times m$2; for $\ell\times m$3, LER reductions reach up to $\ell\times m$4 compared to BP-OSD and up to $\ell\times m$5 compared to BPGD. On the larger B1 code $\ell\times m$6, it remains competitive with BPGD while maintaining BP-like latency under parallel implementation (Rabeti et al., 13 May 2026).

Circuit-level decoding has produced several other BB-specific benchmarks. A sliding-window BP decoder with guided decimation guessing attains a similar logical error rate as BP with ordered-statistics decoding and combination-sweep of order $\ell\times m$7; for a window size of three syndrome cycles, a multi-threaded CPU implementation achieves a worst-case decoding latency of $\ell\times m$8 ms per window for the $\ell\times m$9 code (Gong et al., 2024). Ambiguity Clustering, which clusters ambiguous post-BP regions and decodes them independently, is up to $\mathbb{Z}/\ell \times \mathbb{Z}/m$00 faster than BP-OSD with matched accuracy at $\mathbb{Z}/\ell \times \mathbb{Z}/m$01 circuit-level depolarising noise, and decodes the 144-qubit Gross code in $\mathbb{Z}/\ell \times \mathbb{Z}/m$02s per round of syndrome extraction on an M2 CPU (Wolanski et al., 2024). A recurrent transformer with code-aware attention goes further on $\mathbb{Z}/\ell \times \mathbb{Z}/m$03: at $\mathbb{Z}/\ell \times \mathbb{Z}/m$04, its logical error rate is almost $\mathbb{Z}/\ell \times \mathbb{Z}/m$05 times lower than BP-OSD, while maintaining more consistent runtime. On $\mathbb{Z}/\ell \times \mathbb{Z}/m$06, the same model is worse at low $\mathbb{Z}/\ell \times \mathbb{Z}/m$07, with logical error rate roughly $\mathbb{Z}/\ell \times \mathbb{Z}/m$08 higher than BP-OSD at $\mathbb{Z}/\ell \times \mathbb{Z}/m$09, but still retains a speed advantage of more than one order of magnitude (Blue et al., 17 Apr 2025).

Erasure-dominated settings lead to different decoders and different structural metrics. The Quantum Maxwell Erasure Decoder extends peeling with bounded symbolic guessing, runs in

$\mathbb{Z}/\ell \times \mathbb{Z}/m$10

and on tested BB instances approaches maximum-likelihood performance with small budgets; for the BB code $\mathbb{Z}/\ell \times \mathbb{Z}/m$11, budgets $\mathbb{Z}/\ell \times \mathbb{Z}/m$12 interpolate smoothly between peeling and ML and already recover ML-like performance in the observed range (Freire et al., 15 Jan 2026). In erasure-aware circuit models, BiBiEQ compiles BB memory circuits with explicit erasure checks. For the studied $\mathbb{Z}/\ell \times \mathbb{Z}/m$13, $\mathbb{Z}/\ell \times \mathbb{Z}/m$14, and $\mathbb{Z}/\ell \times \mathbb{Z}/m$15 families, the distance hop $\mathbb{Z}/\ell \times \mathbb{Z}/m$16 yields a drop in per-round LER by $\mathbb{Z}/\ell \times \mathbb{Z}/m$17–$\mathbb{Z}/\ell \times \mathbb{Z}/m$18 larger than the hop $\mathbb{Z}/\ell \times \mathbb{Z}/m$19, and the $\mathbb{Z}/\ell \times \mathbb{Z}/m$20EC schedule keeps the exact and approximate engines close enough that the approximate engine becomes a reliable proxy for faster sweeps (Bhave et al., 7 Feb 2026).

5. Implementation pathways

The geometry of BB codes has motivated a wide range of hardware mappings. In a strictly 2D-local setting, a bilayer architecture with LOCC-based routing measures light checks every round and heavy checks less frequently. Circuit-level simulations show that, in some parameter regimes, BB codes implemented with this protocol have logical error rates comparable to the surface code while using fewer physical qubits; for example, the $\mathbb{Z}/\ell \times \mathbb{Z}/m$21 BB code reaches $\mathbb{Z}/\ell \times \mathbb{Z}/m$22 with $\mathbb{Z}/\ell \times \mathbb{Z}/m$23 qubits, compared with $\mathbb{Z}/\ell \times \mathbb{Z}/m$24 surface-code patches using $\mathbb{Z}/\ell \times \mathbb{Z}/m$25 qubits and $\mathbb{Z}/\ell \times \mathbb{Z}/m$26 (Berthusen et al., 2024).

A related but distinct route is networked implementation. Here the combined Tanner graph

$\mathbb{Z}/\ell \times \mathbb{Z}/m$27

is bipartitioned by balanced min-cut so that cross-node stabilizer interactions are implemented by Bell-pair-assisted teleported CNOTs. For $\mathbb{Z}/\ell \times \mathbb{Z}/m$28, a typical bipartition has $\mathbb{Z}/\ell \times \mathbb{Z}/m$29 bridge edges; for $\mathbb{Z}/\ell \times \mathbb{Z}/m$30, $\mathbb{Z}/\ell \times \mathbb{Z}/m$31; and for $\mathbb{Z}/\ell \times \mathbb{Z}/m$32, $\mathbb{Z}/\ell \times \mathbb{Z}/m$33. Circuit-level simulations indicate that a $\mathbb{Z}/\ell \times \mathbb{Z}/m$34 threshold exists for local gate noise in the two-node networked setting, and that Bell pairs with fidelity $\mathbb{Z}/\ell \times \mathbb{Z}/m$35 are necessary for the networked implementation to be competitive with the monolithic architecture at practical $\mathbb{Z}/\ell \times \mathbb{Z}/m$36–$\mathbb{Z}/\ell \times \mathbb{Z}/m$37 (Shaw et al., 27 Apr 2026).

Neutral-atom and photonic realizations have pushed the implementation story further. A neutral-atom Rydberg proposal combines a dilation-and-folding remapping with optimized ancilla placement, reducing the maximum Euclidean communication distance for $\mathbb{Z}/\ell \times \mathbb{Z}/m$38 from $\mathbb{Z}/\ell \times \mathbb{Z}/m$39 or $\mathbb{Z}/\ell \times \mathbb{Z}/m$40 lattice spacings in earlier layouts to $\mathbb{Z}/\ell \times \mathbb{Z}/m$41, and achieves an optimized $\mathbb{Z}/\ell \times \mathbb{Z}/m$42 gate with fidelity $\mathbb{Z}/\ell \times \mathbb{Z}/m$43 at distances greater than $\mathbb{Z}/\ell \times \mathbb{Z}/m$44. For the $\mathbb{Z}/\ell \times \mathbb{Z}/m$45 code this yields a full error-correction cycle estimate of $\mathbb{Z}/\ell \times \mathbb{Z}/m$46 ms (Poole et al., 2024). In fusion-based photonic architectures with emitter-generated resource states, small BB codes attain Pauli-error-only pseudo-thresholds up to $\mathbb{Z}/\ell \times \mathbb{Z}/m$47 and erasure-only pseudo-thresholds up to $\mathbb{Z}/\ell \times \mathbb{Z}/m$48 for $\mathbb{Z}/\ell \times \mathbb{Z}/m$49; under modified repeat-until-success fusion, the photon-loss pseudo-threshold for $\mathbb{Z}/\ell \times \mathbb{Z}/m$50 saturates around $\mathbb{Z}/\ell \times \mathbb{Z}/m$51 at $\mathbb{Z}/\ell \times \mathbb{Z}/m$52 (Chen et al., 21 Sep 2025).

The modular “bicycle architecture” treats $\mathbb{Z}/\ell \times \mathbb{Z}/m$53 and $\mathbb{Z}/\ell \times \mathbb{Z}/m$54 as code modules with explicit logical instruction sets, adapters, and T factories. For these two specific BB codes, the architecture supplies fault-tolerant logical instruction sets, decoder assumptions, and a compilation strategy adapted to module constraints, and its end-to-end resource estimates show that an order of magnitude larger logical circuits can be implemented with a given number of physical qubits than on surface-code architectures (Yoder et al., 3 Jun 2025).

6. Boundary conditions, derived families, and structural limits

The original BB construction is toric, but open-boundary variants are now explicit. For the hypergraph-product or univariate specialization $\mathbb{Z}/\ell \times \mathbb{Z}/m$55, $\mathbb{Z}/\ell \times \mathbb{Z}/m$56, pruning removes qubits and checks that enforce periodic wraparound while preserving locality. If $\mathbb{Z}/\ell \times \mathbb{Z}/m$57 and $\mathbb{Z}/\ell \times \mathbb{Z}/m$58, the pruned code has parameters

$\mathbb{Z}/\ell \times \mathbb{Z}/m$59

and more specifically

$\mathbb{Z}/\ell \times \mathbb{Z}/m$60

while remaining “as local” as the toric parent (Eberhardt et al., 2024). The same paper also constructs explicit pruned examples beyond the univariate case, including color-code-related and inverse-monomial instances, and shows that certain fold-transversal phase-type gates descend to the pruned codes (Eberhardt et al., 2024).

A different frontier concerns measurement noise and single-shot correction. In the coprime BB subclass with $\mathbb{Z}/\ell \times \mathbb{Z}/m$61, trinomials $\mathbb{Z}/\ell \times \mathbb{Z}/m$62, and

$\mathbb{Z}/\ell \times \mathbb{Z}/m$63

the logical dimension is

$\mathbb{Z}/\ell \times \mathbb{Z}/m$64

and the same polynomial governs stabilizer redundancy and the classical syndrome codes required for single-shot decoding. Per stabilizer type,

$\mathbb{Z}/\ell \times \mathbb{Z}/m$65

The syndrome codes are cyclic codes generated by $\mathbb{Z}/\ell \times \mathbb{Z}/m$66, so BCH-like root arguments can raise the syndrome distance $\mathbb{Z}/\ell \times \mathbb{Z}/m$67, but the same algebra imposes the cap

$\mathbb{Z}/\ell \times \mathbb{Z}/m$68

This is the structural bottleneck identified in the coprime ansatz: high quantum rate fixes a corresponding upper bound on single-shot measurement tolerance (Rowshan, 3 Jan 2026).

Several derived families alter the noise model or even the Pauli structure. Romanesco codes are Clifford-deformed bivariate bicycle codes on a bipartite hexagonal lattice; they are non-CSS, self-dual, and each stabilizer is half $\mathbb{Z}/\ell \times \mathbb{Z}/m$69 and half $\mathbb{Z}/\ell \times \mathbb{Z}/m$70. Under strong noise bias, their effective distance approaches the distance of the underlying classical cellular-automaton input codes, and simulations show that a $\mathbb{Z}/\ell \times \mathbb{Z}/m$71 Romanesco code can outperform the $\mathbb{Z}/\ell \times \mathbb{Z}/m$72 BB CSS code over a wide bias range because its large-bias effective distance is governed by the classical distance $\mathbb{Z}/\ell \times \mathbb{Z}/m$73 rather than the CSS distance $\mathbb{Z}/\ell \times \mathbb{Z}/m$74 (Leroux et al., 30 May 2025). Twisted-torus generalizations over finite fields extend the bivariate-bicycle framework to qudits. In the finite-length search reported so far, twisted-torus qudit constructions typically achieve larger distances than untwisted counterparts and outperform previously reported twisted qubit instances; representative examples include $\mathbb{Z}/\ell \times \mathbb{Z}/m$75, $\mathbb{Z}/\ell \times \mathbb{Z}/m$76, and $\mathbb{Z}/\ell \times \mathbb{Z}/m$77 (Halla, 4 Feb 2026).

These developments clarify the present status of BB qLDPC codes. The family is no longer a single toric construction with a single decoder, but a broader algebraic ecosystem with explicit logical bases, several decoding paradigms, open-boundary and self-dual variants, bias-tailored deformations, and hardware-specific realizations. The enduring open questions are correspondingly structural rather than merely numerical: how to enlarge syndrome distance without sacrificing rate, how to preserve favorable gate symmetries under boundary modifications, and how to co-design code families, decoders, and architectures around the long-range but highly organized couplings that define the bivariate bicycle paradigm (Rowshan, 3 Jan 2026, Eberhardt et al., 2024).