There has been significant recent interest in near-term qLDPC codes as high-performance alternatives to surface and color codes. One such class of codes is 2-dimensional translationally invariant (TI) codes, such as bivariate bicycle codes, which share similar properties to topological codes. Fundamental objects in the study of such codes are the materialised symmetries, which can be used for the construction of matching-based decoders. These decoders generalise the minimum-weight perfect matching decoder for toric codes and provide similar asymptotic performance guarantees that heuristic decoders such as BP and its variants lack. Despite this, the mathematical structure of symmetries along with their properties under translation and restriction to finite-sized lattices has not been well-studied. We describe a decomposition of symmetry spaces of 2D CSS TI codes on infinite lattices into translation-invariant subspaces which allow us to write an explicit basis of symmetries in a plane-wave-like form. We then describe how to adapt this infinite lattice basis so that any corresponding rectangular periodic lattice basis can be obtained by restriction to basis elements compatible with the periodic boundaries. This allows us to describe how the symmetry space varies with different rectangular dimensions. We illustrate via examples that it is often straightforward in practice to determine a basis of symmetries on twisted boundaries as well, without needing to recompute the decomposition. We comment on the applications of this framework to matching-based decoders and provide examples of symmetries of such codes as the gross code.
There has been significant recent interest in near-term qLDPC codes as high-performance alternatives to surface and color codes. One such class of codes is 2-dimensional translationally invariant (TI) codes, such as bivariate bicycle codes, which share similar properties to topological codes. Fundamental objects in the study of such codes are the materialised symmetries, which can be used for the construction of matching-based decoders. These decoders generalise the minimum-weight perfect matching decoder for toric codes and provide similar asymptotic performance guarantees that heuristic decoders such as BP and its variants lack. Despite this, the mathematical structure of symmetries along with their properties under translation and restriction to finite-sized lattices has not been well-studied. We describe a decomposition of symmetry spaces of 2D CSS TI codes on infinite lattices into translation-invariant subspaces which allow us to write an explicit basis of symmetries in a plane-wave-like form. We then describe how to adapt this infinite lattice basis so that any corresponding rectangular periodic lattice basis can be obtained by restriction to basis elements compatible with the periodic boundaries. This allows us to describe how the symmetry space varies with different rectangular dimensions. We illustrate via examples that it is often straightforward in practice to determine a basis of symmetries on twisted boundaries as well, without needing to recompute the decomposition. We comment on the applications of this framework to matching-based decoders and provide examples of symmetries of such codes as the gross code.
Two-dimensional topological translationally-invariant (TTI) quantum codes, such as the toric code (TC) and bivariate bicycle (BB) codes, are promising candidates for fault-tolerant quantum computation. For such codes to be practically relevant, their decoders must successfully correct the most likely errors while remaining computationally efficient. For the TC, graph-matching decoders satisfy both requirements and, additionally, admit provable performance guarantees. Given the equivalence between TTI codes and (multiple copies of) the TC, one may then ask whether TTI codes also admit analogous graph-matching decoders. In this work, we develop a graph-matching approach to decoding general TTI codes. Intuitively, our approach coarse-grains the TTI code to obtain an effective description of the syndrome in terms of TC excitations, which can then be removed using graph-matching techniques. We prove that our decoders correct errors of weight up to a constant fraction of the code distance and achieve nonzero code-capacity and circuit-level thresholds in the specified noise models. We further numerically study a variant optimized for practically relevant BB codes and observe performance comparable to that of belief propagation with ordered statistics decoding (BP-OSD). We also study the code-capacity threshold for the hexagonal color code as a proof of concept. Our results indicate that graph-matching decoders are a viable approach to decoding BB codes and other TTI codes.
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.
In recent years, there have been many studies on local stabilizer codes. Under the assumption of translation and scale invariance Yoshida classified such codes. His result implies that translation invariant 2D color codes are equivalent to copies of toric codes. Independently, Bombin, Duclos-Cianci, and Poulin showed that a local translation invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev's toric code. In this paper we focus on 2D topological color codes and relax the assumption of translation invariance. Using a linear algebraic framework we show that any 2D color code (without boundaries) is locally equivalent to two copies of a related surface code. This surface code is induced by color code. We study the application of this equivalence to the decoding of 2D color codes over the bit flip channel as well as the quantum erasure channel. We report the performance of the color code on the square octagonal lattice over the quantum erasure channel. Further, we provide explicit circuits that perform the transformation between 2D color codes and surface codes. Our circuits do not require any additional ancilla qubits.
We study finite-length qudit quantum low-density parity-check (LDPC) codes from translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. Recent qubit work [PRX Quantum 6, 020357 (2025)] showed that, within the bivariate-bicycle viewpoint, twisting generalized toric patterns can significantly improve the finite-size distance of qubit codes. Building on this insight, we extend the construction and the search to qudit codes over finite fields. Using algebraic methods based on Laurent-polynomial ideals and Gröbner-basis computations over Fq, we compute the number of logical qudits on any twisted torus and identify compact codes with favorable rate--distance tradeoffs. Overall, for the finite sizes explored, twisted-torus qudit constructions typically achieve larger estimated distances than their untwisted counterparts and, at the same number of physical and logical qudits, larger estimated distances than previously reported twisted qubit instances. The best new codes are tabulated.
Quantum low-density parity-check (qLDPC) codes promise low-overhead fault-tolerant quantum memory, but many leading constructions are naturally defined on periodic lattices, whereas scalable quantum hardware favors planar architectures with open boundaries. In this work, we develop a general operator-algebraic framework and computational algorithms for constructing all gapped boundaries and defects of two-dimensional topological Pauli stabilizer codes. Starting from a truncated bulk code, our method identifies boundary gauge operators, boundary anyons, and their string operators, and then constructs microscopic boundary Hamiltonians through anyon condensation and topological-order completion, thereby eliminating local logical operators near the boundary. We establish a one-to-one bulk-boundary correspondence between bulk anyons and excitations of the anomalous boundary subsystem code, enabling fusion rules, topological spins, and braiding statistics to be extracted directly from the boundary operator algebra. For translation-invariant codes, Laurent-polynomial representations reduce these tasks to finite matrix computations based on Hermite and Smith normal forms, enabling automated constructions for Z d qudits of both prime and composite dimension. We demonstrate the framework on toric, color, double-semion, six-semion, and anomalous three-fermion codes, and apply it to bivariate bicycle codes, revealing large logical dimensions and unusually long translation periods for anyon strings. By systematically converting bulk topological data into explicit boundary and defect Hamiltonians, our framework can serve as a foundation for constructing low-overhead planar qLDPC codes with open boundaries and for engineering fault-tolerant logical operations.
Directional codes, recently introduced by Gehér--Byfield--Ruban \cite{Geher2025Directional}, constitute a hardware-motivated family of quantum low-density parity-check (qLDPC) codes. These codes are defined by stabilizers measured by ancilla qubits executing a fixed \emph{direction word} (route) on square- or hex-grid connectivity. In this work, we develop a comprehensive \emph{word-first} analysis framework for route-generated, translation-invariant CSS codes on rectangular tori. Under this framework, a direction word W deterministically induces a finite support pattern P(W), from which we analytically derive: (i)~a closed-form route-to-support map; (ii)~the odd-multiplicity difference lattice L(W) that classifies commutation-compatible X/Z layouts; and (iii)~conservative finite-torus admissibility criteria. Furthermore, we provide: (iv)~a rigorous word equivalence and canonicalization theory (incorporating dihedral lattice symmetries, reversal/inversion, and cyclic shifts) to enable symmetry-quotiented searches; (v)~an ``inverse problem'' criterion to determine when a translation-invariant support pattern is realizable by a single route, including reconstruction and non-realizability certificates; and (vi)~a quasi-cyclic (group-algebra) reduction for row-periodic layouts that explains the sensitivity of code dimension k to boundary conditions. As a case study, we analyze the word $W=\texttt{NE$^2NE^2$N}$ end-to-end. We provide explicit stabilizer dependencies, commuting-operator motifs, and an exact criterion for dimension collapse on thin rectangles: for (Lx,Ly)=(2d,d) with row alternation, we find k=4 if 6∣d, and k=0 otherwise.