Vine Codes: Low-Overhead Quantum LDPC Codes on a Planar Square Grid

Abstract: The surface code is a promising route towards large-scale quantum computing, requiring only nearest-neighbour gates amenable to superconducting hardware. However, surface codes incur large qubit overheads. Novel quantum low-density parity check (qLDPC) codes promise to reduce overheads but require long-range connections that are difficult to achieve on superconducting platforms. Here, we introduce "Vine Codes" - qLDPC codes that are implementable on a planar square grid through nearest-neighbour, two-qubit gates native to superconducting platforms (iSWAP and CZ). Our approach generalises "Directional Codes" recently introduced by Gehér et. al. (2025) which are constrained to a torus. In contrast, vine codes have open boundary conditions constructed with the aid of routing qubits. We perform extensive numeric searches and find promising candidate vine codes, e.g. [[121,4,6]], [[221,6,7]], and [[234,9,6]] codes. We verify the circuit distances and show that data and measure qubits required can be reduced by up to ~28% relative to the surface code at a circuit distance of 7. Even including routing qubits, vine codes require fewer total qubits than the surface code (e.g. ~18% reduction at circuit distance 10) and benefits are expected to increase at higher distances. We perform circuit-level noise simulations to demonstrate that under a realistic noise model and at a near-term noise rate of $10^{-3}$, vine codes can perform better than the surface code while using fewer qubits. We give an exhaustive list of all unique vine codes up to stabiliser-weight 9. We additionally introduce "Flip-Vine Codes" which possess single-qubit transversal Clifford gates useful for fault-tolerant logic and magic state cultivation. We furthermore construct examples of generalised open boundaries for vine codes that go beyond the familiar X/Z boundaries of the surface and tile codes.

• The paper presents vine codes, a novel quantum LDPC family that achieves up to a 28% reduction in qubit overhead compared to surface codes while using only local, planar interactions.
• The paper introduces flip-vine codes, which enable transversal Clifford gates and simplify fault-tolerant logic through dynamic measure qubit switching.
• The paper demonstrates via circuit-level noise simulations that vine codes can outperform surface codes in logical error rates under realistic noise conditions.

Vine Codes: Low-Overhead Quantum LDPC Codes on a Planar Square Grid

Introduction

This work introduces "vine codes," a new family of quantum low-density parity check (qLDPC) codes engineered for compatibility with planar, square-grid qubit architectures. The motivation for vine codes arises from the limitations of the surface code—despite its high fault-tolerance threshold and straightforward implementation using only nearest-neighbor interactions, it entails significant qubit overhead. Conversely, high-performance quantum LDPC codes can dramatically reduce overhead but typically require complex, non-local connectivity, posing considerable challenges for deployment on hardware such as superconducting platforms. Vine codes bridge this gap by achieving both low overheads and strict two-dimensional, nearest-neighbor requirements, and by leveraging native gates (iSWAP, CZ).

Construction of Vine Codes

Vine codes are defined on a regular square lattice where each measure qubit interacts with four proximal data qubits. The core design principle is the use of a "step sequence," which dictates the set and order of entangling operations executed in the syndrome extraction circuit. The step alphabet comprises both CXSWAP and CX gates; CXSWAP is related to an iSWAP operation followed by a CNOT, while CX is equivalent to the CZ gate (up to single-qubit Clifford corrections).

The code’s stabilizer generators trace characteristic "vine"-shaped support patterns, a hybrid of extended pathways (via CXSWAPs) and local branches (via CXs), which allow for large stabilizer support sizes within a localized interaction framework. Code construction employs layouts with alternating X- and Z-measure qubits. A sequence of entangling operations, dictated by the step sequence, produces a dynamic relocation of measure qubits essential for syndrome acquisition while maintaining commutativity and topological order criteria.

Vine codes crucially generalize the directional codes previously constrained to toroidal, periodic boundary conditions. By introducing specific boundary constructions—either Pauli type (analogous to surface code X/Z boundaries) or generalized color code-like boundaries—and by incorporating "routing" qubits at boundaries, vine codes maintain locality and reduce overheads even with open boundaries.

Flip-Vine Codes and Transversal Clifford Gates

A noteworthy variant, the flip-vine code, exploits a dynamical switching of measure qubit type (X ↔ Z) between consecutive QEC rounds. This weakly self-dual structure allows for transversal implementation of logical single-qubit Clifford gates (notably Hadamard and, with suitable bipartitions, S gates), greatly simplifying fault-tolerant logic and enabling application to magic state cultivation. Flip-vine codes achieve this self-duality while further reducing the number of active measure qubits by designating subsets as routing-only, preserving code commutativity and logical properties.

Boundary Constructions

The authors detail explicit protocols to construct both Pauli and generalized (colored) boundaries. The boundary geometry is parameterized using parallelogram vectors, and the stabilizer supports are truncated to conform to the planar patch, ensuring no distance-reducing hook errors arise. The generalized boundary construction is framed in terms of anyon condensation theory—by projecting syndromes associated with certain tunneling operators, the code is set into equivalence with the color code, allowing for colored boundary terminations and the possibility of further fault-tolerant logical gate implementations. All stabilizer measurements near boundaries are engineered to avoid circuit depth increases and maximize qubit reuse.

Overhead and Performance Benchmarks

Systematic numerical searches yield a library of valid vine codes up to stabilizer weight $w = 9$. The authors present several codes with parameters such as $[[121,4,6]]$, $[[221,6,7]]$, $[[234,9,6]]$, and $[[437,6,10]]$, where $n$ is the number of physical qubits, $k$ the number of logical qubits, and $d$ the minimum circuit distance. For a fair comparison, total overhead calculations include routing qubits; nevertheless, vine codes show:

• Up to 28% reduction in data+measure qubits at distance $d=7$ vs. surface code.
• At code distance $d=10$, an ~18% total qubit reduction (including routing) relative to surface code, with further advantages projected as $[[121,4,6]]$0 increases.
• Efficient scaling observed in polynomial fits of overhead vs. distance, with asymptotic improvements predicted for all geometries permitting $[[121,4,6]]$1 (where $[[121,4,6]]$2 parameterizes quadratic scaling in qubit count).

Circuit-level noise simulations at a physically realistic $[[121,4,6]]$3 demonstrate that vine codes can outperform the surface code in logical error rate per round for comparable distances (particularly for Z memory), even with fewer total qubits. The observed performance variations between X and Z logicals arise from code patch anisotropies and entropic differences in logical operator distribution.

Implications and Future Directions

Vine codes provide a compelling architectural option for planar, fixed-layout quantum processors—especially superconducting hardware—where hardware connectivity cannot be readily altered. Compared to surface codes, vine codes deliver a clear quantitative reduction in required physical qubits and maintain or improve logical error rates under realistic noise models.

The introduction of flip-vine codes enables transversal Clifford operations, facilitating fault-tolerant circuit synthesis and resource-efficient magic state distillation pathways. From a theoretical perspective, the generality of the step sequence and boundary constructions provides a framework for exploring more exotic code geometries, alternative lattice tilings, and dynamical ("Floquet") code families.

Potential developments include:

• Generalizing boundary constructions for arbitrary patch geometries, including triangles and pentagons.
• Integration with advanced decoding techniques and overhead-reduction strategies (e.g., yoking, leakage reduction, and bias tailoring).
• Adapting vine/flip-vine codes for topological lattice surgery and multi-logical gate operations.
• Exploring embedding on lattices with increased valency (hex, pentagon tilings) and non-Euclidean (hyperbolic) geometries for potential further increases in code rate and scalability.

Conclusion

Vine codes fundamentally extend the paradigm of planar, local-connectivity qLDPC codes by providing high-rate, high-distance logical encoding with strict nearest-neighbor gate requirements, low routing overhead, and compatibility with hardware-native operations. This work offers both practical implementation recipes and a broad theoretical foundation for future research on scalable quantum error correction and enables significant resource reductions for next-generation quantum processors.