We present new upper and lower bounds on the minimum distance of certain generalized bicycle (GB) codes beyond the reach of techniques for classical codes capable of even capturing the true minimum distance for some cases. These bounds are then applied to illustrate the existence and analyze two highly degenerate GB code families with parameters $[[d^2+1,2,d]]$ for odd $d \geq 3$ and $[[d^2,2,d]]$ for even $d \geq 4$, both having the property that each check qubit is connected to exactly four data qubits similar to surface codes. For the odd-distance family, we analyze the structure of low-weight logical Pauli operators and demonstrate the existence of a fault-tolerant logical CNOT gate between the two logical qubits, achievable through a simple relabeling of data qubits. We further construct a syndrome extraction pattern for both families that does not imply minimum distance reduction arising from extraction circuit faults that propagate from the check qubits to the data qubits. Finally, we numerically evaluate their logical error rates under a code capacity depolarizing noise model using the belief propagation ordered statistics decoding (BP-OSD) and minimum-weight perfect-matching (MWPM) decoders, yielding thresholds of approximately $14-16\%$ for the odd and even families, very similar to those of rotated surface codes.
We present new upper and lower bounds on the minimum distance of certain generalized bicycle (GB) codes beyond the reach of techniques for classical codes capable of even capturing the true minimum distance for some cases. These bounds are then applied to illustrate the existence and analyze two highly degenerate GB code families with parameters [[d2+1,2,d]] for odd d≥3 and [[d2,2,d]] for even d≥4, both having the property that each check qubit is connected to exactly four data qubits similar to surface codes. For the odd-distance family, we analyze the structure of low-weight logical Pauli operators and demonstrate the existence of a fault-tolerant logical CNOT gate between the two logical qubits, achievable through a simple relabeling of data qubits. We further construct a syndrome extraction pattern for both families that does not imply minimum distance reduction arising from extraction circuit faults that propagate from the check qubits to the data qubits. Finally, we numerically evaluate their logical error rates under a code capacity depolarizing noise model using the belief propagation ordered statistics decoding (BP-OSD) and minimum-weight perfect-matching (MWPM) decoders, yielding thresholds of approximately 14−16% for the odd and even families, very similar to those of rotated surface codes.
Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters [[2n2, 2,n]], [[4r2, 2, 2r]], and [[(2t+1)2+1, 2, 2t+1]]. These families match the performance of Kitaev’s toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent (2, 2)-GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.
Hook errors are a major challenge in implementing logical operations with the surface code, because they can reduce the fault distance below the code distance. This motivates syndrome-extraction circuits that suppress hook-error effects for the stabilizer layouts that appear during logical operations. However, the existing methods either increase circuit depth or require simultaneous execution of measurements and CNOT gates, both of which introduce additional overheads and degrade the threshold. We propose the ZX interleaving syndrome extraction, which preserves the full fault distance d for any surface-code layout with regular stabilizer tiles at minimum depth, i.e., four layers of CNOT gates, without requiring additional circuit depth or simultaneous execution of measurements and CNOT gates. The key idea is to interleave the Z and X stabilizer tiles so that hook-error edges in the decoding graph are shortened and effectively eliminated. Numerical simulations under uniform depolarizing noise for memory and lattice-surgery experiments confirm that the proposed method achieves a full fault distance of d, whereas the best existing minimum-depth approach achieves d−1. Since the full fault distance is achievable for any regular tiling layout of the surface code, the proposed method may serve as an indispensable technique for practical fault-tolerant quantum computation.
Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.
The discovery of new quantum error-correcting codes that encode several logical qubits into relatively few physical qubits motivates the development of efficient and accurate methods of decoding these systems. Here, we adopt the minimum-weight perfect matching algorithm, a subroutine invaluable to decoding topological codes, to decode bivariate bicycle codes. Using the equivalence of bivariate bicycle codes to copies of the toric code, we propose a method we call the `cylinder trick' to rapidly find a correction using matching on code symmetries. We benchmark our decoder on the gross code family, cyclic hypergraph-product codes, generalized toric codes, and recently proposed directional codes under code capacity and phenomenological noise models, demonstrating the general applicability of our protocol. For a subset of these codes, we find that our decoder can be significantly improved by augmenting matching with strategies including belief propagation and `over-matching', thus achieving performance competitive with state-of-the-art approaches.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group G and two subsets of G whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find [[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>n.
Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of 0.2%, approximately matching that of the [[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
Automorphisms of quantum codes, when they exist, offer a pathway toward fault-tolerant gate implementation via qubit relabeling. Although useful, the conditions under which automorphisms appear in a given code remain poorly understood. In this paper, we develop an algebraic framework for systematically analyzing and engineering automorphisms in Generalized Bicycle (GB) codes. Central to our approach is the derivation of a three-space dependency between the polynomial ring space, the parity check matrix space, and the F22ℓ qubit space, similar to the structure found in the study of classical cyclic codes. By expressing GB codes as a pair of cyclic submodules of Rℓ2, where Rℓ≅F2[x]/⟨xℓ−1⟩, we reduce the search for code automorphisms to a deterministic algebraic problem, deriving necessary and sufficient conditions for the existence of block-separable automorphisms built from cyclic shifts, ring automorphisms and block-swaps. We connect these conditions to the fold-transversal gate framework, providing explicit criteria for the existence of H-, S-, and CX-type fold-transversal gates. We further discuss structured bases for logical operators in order to determine the logical action of a given automorphism. Finally, we introduce the Maximal Cube Root (MCR) code family, a family of GB codes constructed around the principle of maximizing automorphism flexibility and fold-CX gates. We demonstrate a collection of k=2 MCR codes up to d=13 generating the 2-qubit Clifford group via automorphism and fold-transversal gates, with stabilizer weight ranging from 8 to 16, and k>2 MCR codes with a minimum of 20 distinct logical gates achievable from automorphisms. This serves as a first demonstration of inverse design: using these methods to build codes around a rich automorphism structure from the ground up.