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

Georgia M. Nixon, Campbell K. McLauchlan, Charles C. L. van Rest

arXiv:2606.20263v1quant-phcond-mat.str-el

TL;DR

Surface-code-compatible local connectivity is attractive for superconducting hardware, but surface codes have high qubit overhead and many qLDPC codes require non-local connections. This paper introduces planar vine codes with routing-assisted open boundaries and reports reduced qubit use alongside competitive circuit-level performance.

  • Problem

    Many qLDPC codes can reduce surface-code overhead but require non-local connectivity, which remains challenging for superconducting devices.

    Source: §I · p. 1

  • Method

    The authors construct vine qLDPC codes on planar square grids with routing-assisted open boundaries, and introduce flip-vine variants with transversal single-qubit Clifford gates.

    Source: §I · p. 2

  • Results

    Vine codes use up to 28% fewer data and measure qubits at circuit distance 7 and ~18% fewer total qubits at distance 10, while matching or beating surface-code performance near 10^-3.

    Source: §I · p. 2, §V · p. 13

  • Takeaways & Limitations

    The authors expect vine-code overhead gains over the surface code to improve at higher distances.

    Source: §V · p. 13

    The total-qubit comparison counts routing qubits, though some may be absorbed into corridors already needed for surface-code lattice surgery.

    Source: §V · p. 14

Abstract

from arXiv · show

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 Geh\'er 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.

I. INTRODUCTION

Surface codes suit superconducting hardware but carry substantial overhead, while many qLDPC codes rely on non-local connectivity; directional codes also retain a torus constraint [1, 48]. Vine codes extend this approach to planar square grids with open boundaries, and the paper reports lower overhead, noise-simulation results, and additional code constructions.

I. INTRODUCTION

  • Vine codes generalize directional codes with open boundaries built using routing qubits and syndrome extraction using both iSWAP and CZ gates.The construction uses a planar square grid with nearest-neighbour connectivity.Source: §I · p. 2
  • Flip-vine codes alter measure-qubit roles between X and Z stabilisers each round, yielding weak self-duality and single-qubit transversal Clifford gates.The paper identifies fault-tolerant logic and magic state cultivation as uses for these features.Source: §I · p. 2
  • At circuit distance 7, data and measure qubits fall by up to 28% versus the surface code; at distance 10, total qubits including routing fall by ~18%.Source: §I · p. 2
  • Routing-qubit minimisation techniques reduce routing requirements by up to ~66% in relevant examples compared with naive requirements.Source: §I · p. 3
  • At p = 10^-3, vine codes perform better or comparably to the surface code while using fewer qubits in circuit-based simulations.Source: §I · p. 3
  • Examples of generalised open boundaries extend beyond Pauli boundaries, and their stabilisers can be measured without increasing circuit depth.Source: §I · p. 3
  • The numerical search reports all valid vine codes up to stabiliser weight 9 and identifies candidate codes with lower overhead than the surface code.Source: §I · p. 3

II. VINE CODES

Vine codes are specified by a layout and a step sequence that determines syndrome-extraction gates; moving measure qubits create vine-shaped stabiliser supports. The section also describes validity constraints and overhead scaling across open-boundary geometries.

II. VINE CODES

  • A layout gives the initial positions of X- and Z-type measure qubits, while a step sequence specifies syndrome-extraction gates using directional CXSWAP or CX operations.Uppercase directional steps use CXSWAP; lowercase steps use CX without swaps.Source: §II · p. 3
  • CXSWAP moves let a measure qubit reach new data qubits, producing vine-shaped stabiliser supports with a main root and smaller offshoots.Source: §II · p. 3
  • In even syndrome-extraction rounds, the measure qubit follows the opposite route and returns to its initial position.Source: §II · p. 3
  • Valid step-sequence and layout combinations require even overlap of opposite-type stabilisers, non-interleaved measurement, topological order, and entanglement across the patch.Source: §II · pp. 3–4
  • For the k = 6 code, the plotted fits have C < 2k and A < k, leading the authors to expect vine-code efficacy over the surface code to improve at larger distances.Source: Figure 2 · p. 4
  • The nSeWWwSs code is shown across horizontal-parallelogram, square, and rotated open-boundary patches, each encoding k = 9 logical qubits.Source: Figure 3 · p. 4

A. Flip-vine codes for transversal Clifford gates

Flip-vine codes alternate the Pauli type measured by each active measure qubit across QEC rounds, making the X and Z checks share supports. With suitable boundaries, weak self-duality supports transversal Clifford operations.

A. Flip-vine codes for transversal Clifford gates

  • Weak self-duality in flip-vine codes means their X and Z parity-check matrices are equivalent.The supplied passage is truncated after this definition.Source: A. Flip-vine codes for transversal… · p. 4
  • In odd rounds active measure qubits measure one Pauli type, then measure the opposite type on the same support in even rounds as the step sequence reverses.The step sequence has even length, and the measure-qubit Pauli type flips each QEC round.Source: A. Flip-vine codes for transversal… · p. 5
  • Because active measure qubits support both Pauli types, approximately half the measure qubits are assigned to routing so the stabilizer count does not increase and checks commute.Open boundaries can make the active fraction approximate rather than exactly one half.Source: A. Flip-vine codes for transversal… · p. 5
  • Starting from an active measure qubit, the construction demotes measure qubits with odd-overlap checks to routing qubits to preserve commutation.The routing-qubit lattice is built systematically from seed positions and even sums of the relevant displacement vectors.Source: A. Flip-vine codes for transversal… · pp. 5–6
  • Routing qubits follow the vine using SWAPs without entangling data, while active measure qubits alternate X/Z checks and reverse the step sequence on even rounds.Data qubits follow the reverse step sequence; some layers entangle them, while others swap them with routing qubits.Source: A. Flip-vine codes for transversal… · p. 6
  • With suitable boundaries, flip-vine codes support a transversal Hadamard; a data-qubit bipartition satisfying the stated mod-4 condition also permits a transversal S gate.The paper notes that all codes it finds admit such a bipartition.Source: A. Flip-vine codes for transversal… · p. 6

III. BOUNDARY CONSTRUCTION

This section describes constructing finite-size vine-code instances with open boundaries, beginning with Pauli boundaries and then treating a more general approach.

III. BOUNDARY CONSTRUCTION

  • The section gives procedures for constructing explicit finite-size vine-code instances with open boundaries.It first introduces X/Z Pauli boundaries, then turns to a more general approach.Source: §III · p. 6

A. Pauli boundaries

Pauli boundaries are built by cutting a parallelogram patch from the infinite plane, truncating or removing stabilizers according to boundary type, and adding routing qubits where needed to preserve connectivity. Boundary geometry affects overhead, and removing unnecessary boundary SWAPs can reduce routing-qubit requirements.

A. Pauli boundaries

  • Pauli boundaries are expected to encode half as many logical qubits as the maximum possible with periodic boundary conditions.This is stated as an expectation, by analogy with tile and surface codes.Source: A. Pauli boundaries · p. 6
  • Pauli boundaries truncate X (Z) stabilizers at X (Z) boundaries and remove them at the opposite boundary type.The patch procedure also deletes outside data qubits, measure qubits for removed checks, and unsupported data qubits that could create low-weight logicals.Source: A. Pauli boundaries · p. 6
  • 82% of equivalent surface-code qubits are required by the rotated distance-10 patch, while the horizontal parallelogram requires 104%.The examples encode the same k = 6 and use the same vine-code sequence at distance 10.Source: Figure 5 · p. 7, A. Pauli boundaries · p. 8
  • The boundary procedure retains the largest Tanner-graph component and excludes from the exhaustive search some sequences that split into approximately equal disconnected regions.Such sequences fail to generate entanglement across the whole infinite-plane system, according to the passage.Source: A. Pauli boundaries · p. 7
  • Routing qubits preserve planar-grid connectivity when boundary construction discards data or measure qubits along a qubit’s trajectory.The circuit substitutes SWAPs for entangling gates when a trajectory encounters a routing qubit; routing qubits can be reset and used with CZSWAP gates.Source: A. Pauli boundaries · pp. 7–8
  • ∼66.2% fewer routing qubits are needed for the distance-10 rotated patch after unnecessary boundary SWAPs are removed, from 198 to 67.The figure compares rotated, square, horizontal-parallelogram, and vertical-parallelogram geometries; the boundary geometry changes qubit overhead.Source: Figure 5 · p. 7, A. Pauli boundaries · p. 8

B. Generalised boundaries

Generalised boundaries extend vine-code boundary construction beyond Pauli boundaries by using tunnelling operators to truncate stabilisers. For colour-code-equivalent vine codes, coloured boundaries condense both X and Z anyons of one colour, with explicit vine- and flip-vine-code examples.

Generalised boundaries

  • Tunnelling operators are Pauli operators that flip two stabilisers of a given anyon type, transporting syndrome charge without increasing total syndrome weight; operators of the same type can form logical operators.The boundary construction first identifies anyon types from syndrome-defect equivalence classes and toric-code layer charges.Source: B. Generalised boundaries · p. 8
  • A boundary is formed by choosing a maximal commuting set of tunnelling operators in the vacuum region, measuring them out, and truncating the remaining stabilisers to the code patch.This condenses the corresponding anyons at the boundary, where their logical operators can terminate.Source: B. Generalised boundaries · pp. 8–9

Coloured boundaries

  • For color-code-equivalent vine codes, red, green, or blue boundaries condense both X and Z anyons of one colour, whose tunnelling operators mutually commute.The construction applies to codes encoding two qubits on a Pauli-boundary patch.Source: B. Generalised boundaries · pp. 8–9

Coloured-boundary examples

  • The examples include a two-logical-qubit vine-code patch and a triangular flip-vine patch that reproduces a one-logical-qubit 2D color-code patch.The flip-vine patch remains weakly self-dual and supports transversal single-qubit Clifford gates; its boundary circuit uses routing qubits for truncated stabilisers.Source: B. Generalised boundaries · p. 9

IV. RESULTS

The results section reports an exhaustive search of vine codes through stabiliser weight 9 and evaluates selected codes through qubit-overhead scaling and circuit-level noise simulations. It also presents flip-vine examples with transversal single-qubit Clifford gates.

IV. RESULTS

  • The paper benchmarks selected vine codes against the surface code using qubit-overhead scaling and circuit-level noise simulations, and presents flip-vine examples with transversal single-qubit Clifford gates.The simulations use syndrome-extraction circuits under circuit-level noise; the flip-vine examples are reported in Table III.Source: §IV · pp. 9–10
  • The exhaustive search covers all valid vine codes up to stabiliser weight 9.The paper reports its search results in Section IV A.Source: §IV · p. 9

A. Vine code search

The search enumerates vine-code sequences through stabiliser weight 9, filters for validity and symmetry, and reports 147 sequences. Its canonical representatives are not selected to optimise hook errors.

A. Vine code search

  • The search filters for commutativity, independently measured stabilisers, topological order, and full patch connectivity, then removes sequences related by D4 rotation or reflection.It reports one representative for each remaining sequence in Table IV.Source: A. Vine code search · p. 10
  • 147 step sequences were found up to stabiliser weight 9: 96 have weight 9 and 51 have weight 8 or less.The paper focuses on weight-8-or-less sequences in the overhead analysis.Source: A. Vine code search · p. 10
  • Before equivalent data-qubit supports are consolidated, the search has 906 vine codes; representatives are not optimised for hook errors, which can change circuit distance.Different entangling sequences with the same data-qubit support may have different hook errors and circuit distances.Source: A. Vine code search · p. 10

B. Qubit overhead scaling

Selected vine-code patches use fewer qubits than equivalent surface-code patches at verified distances, with savings that the authors expect to grow at higher distances. The comparison depends on boundary geometry, and some distance-10 results are extrapolated rather than verified.

B. Qubit overhead scaling

  • Small vine-code instances at circuit distance 6 or 7 already use fewer qubits than equivalent surface-code patches, with efficiencies expected to increase at higher distances.The overhead comparisons include routing qubits when reporting total qubit counts.Source: B. Qubit overhead scaling · p. 10
  • At distance 7, a rotated nSeWwSs patch requires 88% of the total qubits of equivalent surface-code patches.The reported total includes data, measure, and routing qubits.Source: B. Qubit overhead scaling · p. 10, Table I · p. 11
  • The fitted asymptotic total-qubit overhead ratio relative to the surface code is C/2k; nSeWwSs on a rotated patch has C/2k = 0.797.The fits model total qubits and data qubits with quadratic and linear terms.Source: B. Qubit overhead scaling · p. 10
  • At distance 10, the rotated nSeWwSs geometry requires 82% of the total qubits of equivalent surface-code patches, with circuit distance conjectured rather than verified.The distance-10 comparison extrapolates from smaller patches without distance-reducing hook errors.Source: B. Qubit overhead scaling · p. 10, Table II · p. 11
  • Eleven examples have fitted C/2k < 1, suggesting lower qubit counts than the surface code as code distance increases.These are among codes already using fewer data and measure qubits than the surface-code comparison at circuit distance 6.Source: B. Qubit overhead scaling · p. 10
  • Boundary-cut geometry changes the data and total qubit counts needed for the same vine code and circuit distance.Different cuts can also bias the lengths of the boundary vectors used to achieve equal distances in both bases.Source: B. Qubit overhead scaling · p. 10, Table I · p. 11

C. Circuit-Level Noise Simulations

Circuit-level simulations compare vine-code memories with equivalent surface-code memories under SI1000 noise. At a noise rate of 10^-3, Z-memory performance is substantially better for one vine code, while X-memory performance is roughly equivalent to the distance-7 surface code.

C. Circuit-Level Noise Simulations

  • The simulations use SI1000 noise, BPOSD decoding for vine codes, and PyMatching for surface codes, with memory experiments in both X and Z bases.The plotted patches have circuit distances 3, 5, and 7; the same patches are used for X- and Z-memory experiments.Source: C. Circuit-Level Noise Simulations · p. 11, Figure 7 · p. 12
  • At p = 10^-3, the nSeWwSs vine code’s Z-memory performance is almost an order of magnitude better than the same-distance surface code, while its X-memory performance is roughly equivalent to a distance-7 surface code.The authors attribute the asymmetry to logical-operator structure and different entropic factors for the two memory bases.Source: C. Circuit-Level Noise Simulations · p. 12, Figure 7 · p. 12
  • The comparisons scale the per-round logical error rate of one surface-code patch to estimate the error rate for k logical qubits encoded in k surface-code patches.The scaling formula is Pk = 1−(1−PL)^k.Source: C. Circuit-Level Noise Simulations · pp. 12–13

D. Flip-vine codes

The paper searches periodic-boundary flip-vine codes on tori, checking that wrapped qubits retain their types and that the resulting stabilisers commute. It reports code parameters and data-qubit efficiency relative to toric codes.

D. Flip-vine codes

  • The search examines valid step sequences up to length 8 and constructs example flip-vine codes on tori with specified lattice vectors.The search covers skew and rectangular tori across the stated ranges of v, w, and p.Source: D. Flip-vine codes · p. 13
  • Each torus wrapping is checked to preserve data-qubit and measure-qubit types and to produce commuting stabilisers.The checks also determine the number of logical qubits and the code distance, with X and Z distances equivalent.Source: D. Flip-vine codes · p. 13
  • The reported data-qubit efficiency compares flip-vine and toric codes encoding the same number of logical qubits at the same distance.The comparison is the ratio of their required data-qubit counts.Source: D. Flip-vine codes · p. 13

V. CONCLUSION

The conclusion presents vine codes as lower-overhead planar-grid qLDPC codes with circuit-level performance equal to or better than surface-code patches in the reported comparisons. It also identifies open questions about boundaries, fault-tolerant operations, connectivity, and hardware implementation.

V. CONCLUSION

  • With routing qubits counted, some vine codes encoding k ≥ 6 logical qubits use fewer physical qubits than same-distance surface-code patches and perform equally or better under circuit-level noise.The authors report utility-scale savings up to ≲20.3% based on their fits, while noting the asymptotic bound for two-dimensional translationally invariant codes.Source: §V · p. 13
  • The authors have not investigated generalised boundaries or differently shaped patches in depth, including Pauli-Y and further coloured-boundary constructions.They also call for general prescriptions to construct syndrome-extraction circuits for generalised boundaries.Source: §V · p. 13
  • The conclusion identifies adaptation of lattice surgery, magic-state cultivation, and distillation as open work; arbitrary Clifford gates or multi-Pauli measurements within one patch may be challenging without large overheads or long-range connectivity.Single-logical-qubit patches with more exotic boundaries are offered as an alternative direction.Source: §V · p. 13
  • The work assumes a square-grid layout; connectivity-scaling tradeoffs and codes using five- or six-valent tilings remain possible research directions.The conclusion also raises hyperbolic embeddings and planar-grid Floquet-code constructions as theoretical avenues.Source: §V · pp. 13–14
  • The fairness of charging all routing qubits to vine codes is unclear because surface-code lattice surgery also requires boundary buffer qubits.The authors suggest some counted routing qubits might be absorbed into a corridor already needed for lattice surgery.Source: §V · p. 14
  • Current-device implementation requires calibrating CZ and iSWAP gates and achieving high fidelities for both simultaneously, though some sequences can use iSWAP alone at the cost of potentially more routing qubits.The paper notes demonstrations of devices operating both gate types together.Source: §V · p. 14

Appendix A: Valid sequences

  • Valid sequences must satisfy commutativity, independent measurement, topological order, and patch connectivity conditions.The search excludes codes failing topological order and omits patches with too few final data qubits because of disconnected regions.Source: Appendix A · p. 14
  • Some sequences previously called valid fail topological order because their codewords remain fixed weight as system size grows; some examples may instead have boundary-induced fixed-weight logicals.Such examples are treated cautiously, and the exhaustive search also excludes codes producing multiple disconnected regions.Source: Appendix A · pp. 14–15

Appendix B: Exhaustive vine code sequence search

The appendices describe how vine-code sequences are enumerated and filtered, how boundaries and stabiliser compatibility are handled, and how routing and SWAP overheads are reduced. They also set out the qubit-scaling comparison with equivalent surface-code patches.

Appendix B: Exhaustive vine code sequence search

  • The exhaustive search considers vine sequences up to length 9, removes invalid sequences, and applies D4 and equivalent-support filters to select representatives.The support filter can identify sequences with equivalent data support but different entangling orders; representatives are not optimised for hook errors.Source: Appendix B · pp. 15–16
  • Applying validity conditions 1–4 and reductions 5–7 yields 146 valid vine codes; without step 7, while retaining reductions 1–6, the count would be 906.The 146 are canonical representatives, with one representative for each set of codes with equivalent stabiliser support.Source: Appendix B · p. 15
  • The qubit-scaling analysis fits data and total qubit counts across circuit distances against an ansatz and compares them with equivalent surface-code patches.When the fitted quadratic coefficient C for total qubits is below 2k, the authors expect the vine code's efficiency over the surface code to improve at large distances.Source: Appendix B · p. 16

Appendix C: Vine code ∆sets

  • Measure qubits separated by a vector in ∆odd must have the same Pauli type, because such separations correspond to stabiliser supports that overlap oddly.The appendix defines ∆odd from displacement vectors occurring an odd number of times.Source: Appendix C · p. 16

Appendix D: Corners

  • At corners, X stabilisers with support in the Z-boundary region are removed while Z stabilisers there are truncated, with the converse rule at X boundaries.The corner domain wall is set by the angular bisector of the two outward boundary rays.Source: Appendix D · p. 16

Appendix E: Minimising routing qubits and SWAP gates

  • Routing and SWAP reduction shortens qubit paths when equivalent circuits preserve all non-SWAP gate locations, then removes redundant measurement/reset swaps and unused routing qubits.The procedure has three phases: find shorter paths, remove SWAP-then-measure or reset-then-SWAP events, and delete routing qubits no longer used by gates.Source: Appendix E · pp. 17–18

Appendix F: Generalised boundary details

The appendix constructs coloured boundaries for vine-code examples and describes their syndrome circuits, then specifies the simulation setup and the scope of the listed code sequences. It uses color-code-equivalent examples to illustrate the boundary constructions.

Appendix F: Generalised boundary details

  • The sEEEn code is color-code-equivalent and supports six boundary types, constructed by measuring tunnelling operators for selected anyons.The appendix describes measuring b-anyon operators at the sides and r-anyon operators at the top and bottom, then truncating compatible stabilisers to the patch.Source: Appendix F · p. 19
  • For the sENWSw flip-vine code, the triangular patch has coloured boundaries whose boundary stabilisers are simple truncations of bulk stabilisers.The example is also equivalent to the 2D color code.Source: Appendix F · p. 20

Appendix G: Numerical Simulation Details

  • The simulations use Stim memory circuits run for max(5, dcircuit) syndrome rounds, with native-equivalent gates and an SI1000 noise model.The setup includes routing-qubit resets and measurements each round, flags their measurements, and decodes using BPOSD.Source: Appendix G · pp. 20–21

Weight 4

  • Table IV lists topologically ordered vine-code sequences up to length 9, filtered to representatives under the stated search criteria.The search assumes alternating rows of X- and Z-stabilisers.Source: Table IV · p. 26
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