Bivariate Bicycle Code: Quantum LDPC

Updated 1 February 2026

Bivariate Bicycle Code is a quantum LDPC code built from two circulant matrices over bivariate group algebras, enabling high encoding rates and fault tolerance.
It leverages algebraic constructions with polynomials in F2[x,y]/(x^l-1, y^m-1) to yield practical stabilizer weights of 6 or 8 and flexible modular designs.
Advanced decoding techniques such as BP-OSD and machine-learning based methods, along with connectivity reduction, enhance its performance in near-term quantum processors.

A bivariate bicycle code is a quantum low-density parity-check (qLDPC) code constructed via the CSS formalism from two circulant matrices derived from polynomials in a bivariate group algebra. These codes generalize the surface code and standard hypergraph-product codes, providing higher encoding rates and improved logical error performance at moderate code lengths. Their combinatorial and algebraic structure allows flexible code design for fault-tolerant quantum architectures, including both periodic and open-boundary implementations. The canonical BB code is specified by a pair of polynomials in the ring $\mathbb F_2[x,y]/(x^\ell-1, y^m-1)$ , with typical stabilizer weights of six or eight per check, and code parameters $[[n=2\ell m,\, k,\, d]]$ . Modern research includes explicit constructions, covering and pruning techniques, connectivity reductions, and advanced decoding strategies.

1. Algebraic Construction and Ring-Theoretic Foundation

Let $\ell, m \in \mathbb{N}$ and $R = \mathbb{F}_2[x,y]/(x^\ell - 1, y^m - 1)$ , the group algebra of $\mathbb{Z}_\ell \times \mathbb{Z}_m$ . Given two polynomials $A(x,y), B(x,y) \in R$ , one defines $n = \ell m$ and constructs two $n \times n$ circulant matrices (blocks) from the coefficient vectors of $A$ and $B$ . The CSS quantum code uses parity-check matrices: $H_X = [\,A\,|\,B\,], \quad H_Z = [\,B^T\,|\,A^T\,]$ where $T$ denotes transpose (index reversal modulo $\ell$ or $m$ ).

The commutation condition $H_X H_Z^T = A B^T + B A^T = 0$ follows from the commutativity of $x$ and $y$ in the group algebra. Stabilizer generators have support of $\operatorname{wt}(A) + \operatorname{wt}(B)$ per row, typically weight $6$ or $8$ for practical codes. The code encodes $k$ logical qubits and achieves minimum distance $d$ determined via codeword enumeration or cyclic-code bounds (Wang et al., 2024, Postema et al., 24 Feb 2025, Symons et al., 17 Nov 2025).

2. Dimension, Distance, and Coprime Constructions

In the special coprime case $\gcd(\ell, m) = 1$ , the bivariate ring $R$ reduces to $\mathbb{F}_2[z]/(z^n-1)$ via the mapping $z = x y$ , simplifying the analysis. If $A(x,y) = a(z)$ and $B(x,y) = b(z)$ , set $g(z) = \gcd(a(z), b(z), z^n - 1)$ . Then, as proven in several works (Postema et al., 24 Feb 2025, Rowshan, 3 Jan 2026): $k = 2 \deg g(z)$ Dimension is strictly tied to the degree of the shared gcd polynomial and the rates of stabilizer redundancy, establishing the equality $R = k/n = \gamma$ , with $\gamma$ the density of redundant stabilizers per type (Rowshan, 3 Jan 2026). The minimum distance $d$ is lower-bounded by the distances of associated cyclic codes from BCH-type bounds, though explicit closed-form expressions for $d$ generally require computational search.

The coprime subclass allows a predetermined rate: choosing a factor polynomial $f(x,y)$ and defining $a(x,y)$ and $b(x,y)$ as coprime divisors of $f$ specifies the code dimension in advance and accelerates code discovery (Wang et al., 2024).

3. Tanner Graphs, Covering Codes, and Pruned Variants

The Tanner graph of a BB code is bipartite, connecting qubits (split into two blocks of $\mathbb{Z}_\ell\times\mathbb{Z}_m$ ) to checks via the supports of $A$ and $B$ . Covering constructions yield an infinite family of new BB codes: for an $h$ -fold cover, lattice dimensions and polynomials are extended according to algebraic consistency modulo base parameters, and logical operators are projected/lifted by induced chain-maps (Symons et al., 17 Nov 2025). Cover codes satisfy $n_h = h n$ , $k_h \ge k$ , and $d \le d_h \le h d$ for odd $h$ .

For open-boundary implementations, "pruning" deletes rows/columns from the circulant blocks associated with $A$ and $B$ . For hypergraph-product instances, this halves the number of logical qubits while retaining locality and code distance: $\widetilde{\mathcal Q} = [\ell m + (\ell - r_A)(m - r_B),\, r_A r_B,\, \min(d_A, d_B)]$ where $r_A$ , $r_B$ are the degrees of univariate check polynomials (Eberhardt et al., 2024).

4. Circuit Implementation, Connectivity, and Morphing

Standard BB syndrome-extraction circuits require weight-6 connectivity per qubit. The morphing circuit principle reduces connectivity: splitting stabilizers into contracting sets using group homomorphisms enables measurement with only degree-5 connections (Shaw et al., 2024). This is accomplished via alternating rounds of contraction, measurement, and restoration with Clifford unitaries, preserving code properties. The method generalizes to arbitrary two-block group algebra codes and, for weight- $w$ codes, enables reduction to degree $w-1$ connectivity where suitable homomorphisms exist.

Degree-5 BB codes maintain thresholds and logical error rates compared to conventional degree-6 instances, supporting biplanar layouts and enabling efficient logical input/output with surface codes via lattice surgery.

5. Decoding Algorithms and Performance Benchmarks

Quantum LDPC BB codes necessitate efficient decoding. Belief-propagation (BP) plus ordered-statistics decoding (OSD) sets the current benchmark, but new decoders, such as Multiple-Bases BP-List Decoding (MBBP-LD), further improve error rates. For example, on the $[[144,12,12]]$ code, MBBP-LD reduces logical error rates by up to $40\%$ versus BP-OSD while retaining linear-time complexity (Rabeti et al., 4 Nov 2025).

Machine learning decoders using staged, transformer-based architectures (with code-aware self-attention) deliver orders-of-magnitude runtime gains and significant logical error suppression, e.g., nearly $5\times$ improvement for $[[72,12,6]]$ at $p = 0.1\%$ over BP-OSD (Blue et al., 17 Apr 2025). Thresholds for BB codes under BP-OSD and ML decoding are typically $\sim0.7\%$ for moderate block lengths.

Numerical simulations in the context of bilayer architectures and LOCC routing have demonstrated BB codes matching or surpassing surface codes in logical error rates per physical qubit, leveraging masking schedules and entanglement swapping to minimize overhead (Berthusen et al., 2024).

6. Logical Operators, Clifford Gates, and Bias Tailoring

The group-algebraic structure of BB codes yields explicit bases of logical operators. In principal cases, logical $Z$ ’s and $X$ ’s correspond to horizontal and vertical "pure" forms built from annihilator polynomials, admitting commutation structures analogous to toric codes (Eberhardt et al., 2024). Fold-transversal Clifford gates including SWAP-type, Hadamard-type, and CZ-type gates are implementable by constant-depth, transversal circuits leveraging symmetries (automorphisms, dualities) between $A$ and $B$ .

Self-dual BB codes admit weight-8 stabilizers and support transversal CNOT, Hadamard, and $S$ gates, outperforming surface and color codes in encoding rate and $k d^2 / n$ metric for moderate $n$ (Liang et al., 6 Oct 2025). Clifford-deformed variants on bipartite hexagonal lattices ("Romanesco codes") are bias-tailored, achieving O $(n)$ effective distance under noise bias, and are constructed from cellular automaton LDPC codes related by reflection (Leroux et al., 30 May 2025).

7. Modular Architectures, Resource Estimates, and Practical Impact

Modular fault-tolerant architectures built on BB codes ("bicycle architecture") demonstrate order-of-magnitude reductions in physical qubits required for executing large logical circuits compared to surface code platforms. Protocols leverage low-overhead qLDPC surgery for Clifford and $T$ gates, Pauli-based circuit compilation, and logical measurement via ancillary "logical processing units." For example, the $[[144,12,12]]$ and $[[288,12,18]]$ codes ("gross" and "two-gross") deliver logical error suppression scaling as $O(p^{d/2})$ , with much lower prefactors than topological codes (Yoder et al., 3 Jun 2025).

Research emphasizes the flexibility of BB constructions, including covering sequences, boundary pruning, connectivity reduction, and tailored decoding, which collectively enable scalable, resource-efficient, and experimentally viable quantum error-correcting architectures. While asymptotic scaling remains suboptimal, moderate-length BB codes are competitive with or superior to surface codes in realistic device regimes (Postema et al., 24 Feb 2025).

Representative Examples and Parameters

Code	Length $n$	Logical $k$	Distance $d$	Check Weight	Notable Properties
[[72,12,6]]	72	12	6	6	Classical ML decoding, degree-5 morphing circuits
[[144,12,12]]	144	12	12	6	Cover code, modular architecture, high threshold
[[288,12,18]]	288	12	18	6	Two-gross code, logical measurement via LPU
[[40,6,6]]	40	6	6	8	Self-dual, transversal Clifford gates
[[120,8,12]]	120	8	12	8	Weight-8 self-dual, higher metric than color codes
[[244,4,12]]	244	4	12	8	Bias-tailored, Romanesco construction

Summary of Key Techniques and Open Research

Bivariate bicycle codes integrate algebraic, combinatorial, and architectural innovations for qLDPC design. They employ circulant block matrices with polynomial structure in group algebras, support code covering and pruning for arbitrary boundaries, and enable low-connectivity circuits via morphing. Advanced list and machine learning decoders extend their operational utility. Self-dual and bias-tailored variants enhance gate compatibility and noise resilience. While their asymptotic scaling is “bad,” codes of moderate length are leading candidates for efficient, high-rate quantum error correction in near-term quantum computation. Continued research explores covering code hierarchies, single-shot decoder robustness, boundary engineering, and integration into modular quantum processors.

Principal References: (Wang et al., 2024, Postema et al., 24 Feb 2025, Symons et al., 17 Nov 2025, Berthusen et al., 2024, Yoder et al., 3 Jun 2025, Eberhardt et al., 2024, Blue et al., 17 Apr 2025, Liang et al., 6 Oct 2025, Rowshan, 3 Jan 2026, Leroux et al., 30 May 2025, Eberhardt et al., 2024, Shaw et al., 2024).