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arxiv: 2606.20263 · v1 · pith:7SQFNBE3new · submitted 2026-06-18 · 🪐 quant-ph · cond-mat.str-el

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

Georgia M. Nixon , Campbell K. McLauchlan , Charles C. L. van Rest This is my paper

Pith reviewed 2026-06-26 16:49 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords vine codesquantum LDPC codessurface codeplanar gridqubit overheadcircuit distanceopen boundary conditionssuperconducting qubits
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The pith

Vine codes are planar qLDPC codes that cut data and measure qubits by up to 28% versus surface codes at circuit distance 7 while using only nearest-neighbor gates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces vine codes as quantum low-density parity-check codes that operate on a planar square grid with open boundaries. They generalize earlier directional codes, which required a torus, by adding routing qubits to support nearest-neighbor iSWAP and CZ gates. The central claim is that specific vine codes achieve verified circuit distances with substantially lower qubit counts than the surface code and maintain or improve logical error rates under circuit-level noise at 10^{-3}. The work supplies an exhaustive list of all such codes up to stabilizer weight 9 and introduces flip-vine variants that admit single-qubit transversal Clifford gates. If the numeric results hold, the construction would make low-overhead qLDPC codes hardware-compatible without long-range connections.

Core claim

Vine codes are qLDPC codes on a planar square grid with open boundaries constructed using routing qubits and nearest-neighbor iSWAP and CZ gates. The authors identify candidate codes such as [[121,4,6]], [[221,6,7]], and [[234,9,6]] whose circuit distances yield up to a 28% reduction in data and measure qubits relative to the surface code at distance 7. Even after including routing qubits the total qubit count remains lower, for example by roughly 18% at distance 10. Circuit-level simulations show superior performance to the surface code at a noise rate of 10^{-3}. Flip-vine codes are defined to support single-qubit transversal Clifford gates, and generalized open boundaries are constructed

What carries the argument

Vine codes, qLDPC codes on a planar square grid with open boundaries realized through routing qubits and nearest-neighbor two-qubit gates.

If this is right

  • Data and measure qubits are reduced by up to 28% relative to the surface code at circuit distance 7.
  • Total qubit count including routing qubits drops by approximately 18% at circuit distance 10.
  • At a noise rate of 10^{-3}, vine codes achieve lower logical error rates than the surface code in circuit-level simulations while using fewer qubits.
  • Flip-vine codes admit single-qubit transversal Clifford gates useful for fault-tolerant logic.
  • An exhaustive enumeration of all unique vine codes up to stabilizer weight 9 is supplied, together with constructions for generalized open boundaries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Overhead reductions are stated to increase at higher distances, which would compound resource savings for larger-scale error-corrected computation.
  • The planar nearest-neighbor requirement aligns directly with existing superconducting qubit layouts that lack long-range couplers.
  • Transversal Clifford support in flip-vine codes may lower the cost of magic-state cultivation within the same grid architecture.
  • The supplied catalog up to weight 9 supplies a concrete starting point for further code searches or hardware mapping studies.

Load-bearing premise

The numeric searches have found codes whose circuit distances and logical error rates under the stated noise model are accurately represented by the reported figures.

What would settle it

Circuit-level simulation at distance 7 under the same depolarizing noise model at 10^{-3} that shows a vine code requires more total qubits or produces a higher logical error rate than the surface code.

Figures

Figures reproduced from arXiv: 2606.20263 by Campbell K. McLauchlan, Charles C. L. van Rest, Georgia M. Nixon.

Figure 1
Figure 1. Figure 1: FIG. 1. Vine codes are defined with qubits on a square lattice where each measure qubit has four nearest neighbour data ⃗ ⃗ ⃗⃗ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The qubit overhead scaling of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The qubit overhead scaling for three different open boundary patches of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. An example of an [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Examples of coloured boundaries applied to (flip-)vine codes that are equivalent (up to a finite-depth circuit) to the 2D [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparing the performance of a vine code quantum memory circuit encoding [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: In our exhaustive search, we build vine codes us [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Qubit number versus circuit distance plots for all codes in Table I. The conventions are equivalent to those in Fig. 2 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The Pauli boundaries at the corners of the → [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. An example of the SWAP reduction techniques ap → [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The three minimal-weight tunnelling operator motifs for the [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Syndrome extraction circuits for the [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Syndrome measurement circuit for the [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
read the original 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 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Vine Codes, a family of qLDPC codes implementable on a planar square grid using only nearest-neighbor iSWAP and CZ gates. The codes generalize Directional Codes to open boundaries via routing qubits. Numeric searches yield candidates such as [[121,4,6]], [[221,6,7]], and [[234,9,6]]. The authors claim these achieve up to ~28% reduction in data+measure qubits relative to the surface code at circuit distance 7 (and ~18% total-qubit reduction at distance 10), verify the circuit distances, and show via circuit-level simulations that vine codes can outperform the surface code at p=10^{-3}. They also provide an exhaustive list of vine codes up to stabilizer weight 9 and introduce Flip-Vine Codes possessing transversal Clifford gates.

Significance. If the reported qubit reductions and logical-error-rate advantages are reproducible, the work would offer a concrete route to lower-overhead qLDPC codes on superconducting hardware without long-range connections. The explicit planar constructions, the exhaustive enumeration up to weight 9, and the introduction of codes with single-qubit transversal gates are clear strengths that could be built upon.

major comments (2)
  1. [Abstract / numeric-search description] Abstract and the numeric-search paragraph: the central overhead claims (~28% data+measure reduction at circuit distance 7 for [[221,6,7]] and ~18% total-qubit reduction at distance 10) rest on numeric searches whose selection criteria, distance-verification procedure (decoder-based or exhaustive, including routing and measurement errors), and noise-model parameters are not described. Without these details the reported advantage over the surface code cannot be independently assessed.
  2. [Circuit-level noise simulations paragraph] Circuit-level noise simulations paragraph: the statement that vine codes 'can perform better than the surface code' at p=10^{-3} is load-bearing for the practical-advantage claim, yet the precise noise model (gate error rates for iSWAP/CZ, measurement error rates, routing-qubit error handling, and decoder) is not specified, preventing verification that the simulated logical error rates are not artifacts of the search or decoder choice.
minor comments (2)
  1. [Results on qubit overhead] The distinction between 'data and measure qubits' and 'total qubits including routing qubits' should be made explicit with a table or equation that lists both quantities for each reported code and the corresponding surface-code baseline.
  2. [Introduction] The reference to Gehér et al. (2025) Directional Codes should include a brief contrast of boundary conditions and routing-qubit usage to clarify the generalization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for recognizing the potential significance of Vine codes for planar qLDPC implementations. We agree that the two major comments identify genuine gaps in the description of our methods. Below we respond point-by-point and commit to a major revision that supplies the requested details.

read point-by-point responses

  1. Referee: [Abstract / numeric-search description] Abstract and the numeric-search paragraph: the central overhead claims (~28% data+measure reduction at circuit distance 7 for [[221,6,7]] and ~18% total-qubit reduction at distance 10) rest on numeric searches whose selection criteria, distance-verification procedure (decoder-based or exhaustive, including routing and measurement errors), and noise-model parameters are not described. Without these details the reported advantage over the surface code cannot be independently assessed.

    Authors: We agree that the numeric-search methodology and verification protocol were described too briefly. In the revised manuscript we will add a dedicated subsection that specifies: (i) the exact selection criteria and objective function used in the exhaustive and heuristic searches, (ii) the distance-verification procedure (decoder-based Monte-Carlo sampling that explicitly includes routing-qubit errors and measurement errors, cross-checked against exhaustive enumeration for small instances), and (iii) the precise noise-model parameters employed when computing the reported overhead reductions. These additions will enable independent reproduction of the ~28 % and ~18 % figures. revision: yes

  2. Referee: [Circuit-level noise simulations paragraph] Circuit-level noise simulations paragraph: the statement that vine codes 'can perform better than the surface code' at p=10^{-3} is load-bearing for the practical-advantage claim, yet the precise noise model (gate error rates for iSWAP/CZ, measurement error rates, routing-qubit error handling, and decoder) is not specified, preventing verification that the simulated logical error rates are not artifacts of the search or decoder choice.

    Authors: We concur that the circuit-level simulation details must be stated explicitly. The revised manuscript will include a new paragraph (or expanded methods section) that lists: the per-gate error rates assigned to iSWAP and CZ, the measurement-error probability, the model for routing-qubit errors, the decoder algorithm and its hyperparameters, and the precise depolarizing or circuit-level noise model used at p = 10^{-3}. We will also state how many shots were run and how logical-error rates were extracted, thereby removing any ambiguity about possible artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions, external citation, and independent benchmarks

full rationale

Vine codes are defined by generalizing directional codes (cited to Gehér et al. 2025, external authors) to open boundaries with routing qubits. Candidates such as [[121,4,6]] are located by numeric search, circuit distances are verified directly, and qubit-count/performance claims are obtained by explicit comparison to the independently defined surface code plus circuit-level simulations at p=10^{-3}. No equation, parameter fit, or self-citation chain reduces any reported advantage to a quantity defined inside the paper; the surface-code baseline and noise model are external. This is the normal non-circular case for a construction-plus-benchmark paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the correctness of the numeric search procedure and the realism of the circuit-noise model; both are domain assumptions rather than derived quantities.

axioms (2)
  • domain assumption The numeric search procedure locates all or the best vine codes up to the stated stabilizer weight and correctly computes their circuit distances.
    Performance numbers are produced by this search; if the search is incomplete or the distance metric is misapplied the overhead claims fail.
  • domain assumption The chosen circuit-level noise model at 10^{-3} is representative of near-term superconducting hardware.
    The simulation comparison to the surface code depends on this model matching reality.
invented entities (1)
  • routing qubits no independent evidence
    purpose: Enable open-boundary conditions on the planar grid while preserving nearest-neighbor connectivity.
    New auxiliary qubits introduced to close the boundary problem; no independent experimental evidence is supplied in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Bunny Codes: Broadening Superconducting Quantum Error Correction Capability through Advanced Control Engineering

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    Bunny codes are qLDPC codes found via exhaustive search that achieve ~3x higher code rate than toric codes (periodic) and ~2x over rotated surface codes (open) when using CNOT+CXSWAP on nearest-neighbor connectivity, ...

Reference graph

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