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arxiv: 2406.14445 · v2 · submitted 2024-06-20 · 🪐 quant-ph

High-threshold, low-overhead and single-shot decodable fault-tolerant quantum memory

Thomas R. Scruby , Timo Hillmann , Joschka Roffe This is my paper

Pith reviewed 2026-05-24 00:13 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error correctionLDPC codesfault tolerancesurface codessingle-shot decodingquantum memorylifted product codes
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The pith

Radial codes achieve surface-code error suppression using five times fewer physical qubits, even with single-shot decoding.

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

The paper presents radial codes, a new family of quantum LDPC codes built via the lifted product of selected classical quasi-cyclic codes and parameterized by a pair of integers (r, s). These codes have the explicit parameters [[2r²s, 2(r-1)², ≤2s]], with numerical evidence that the typical distance grows linearly with s. Circuit-level simulations show they suppress errors at rates comparable to surface codes of similar distance while consuming roughly one-fifth as many physical qubits; the same performance holds when decoding is restricted to a single shot. The construction supplies an explicit visual layout, a canonical set of logical operators, and short stabilizer measurement circuits.

Core claim

Radial codes obtained from the lifted product of a specific subset of classical quasi-cyclic codes realize the parameters [[2r²s, 2(r-1)², ≤2s]] with average-case distance linear in s; under circuit-level noise they deliver logical error suppression comparable to surface codes of matching distance while using approximately five times fewer physical qubits, and this advantage remains when the codes are decoded in a single shot.

What carries the argument

The lifted-product construction that defines radial codes from a chosen family of classical quasi-cyclic codes, together with their visual representation and optimal-length stabilizer circuits.

Load-bearing premise

The circuit-level noise simulations and numerical distance studies are performed under representative noise models without post-selection that would change the reported qubit savings.

What would settle it

A circuit-level simulation or hardware experiment in which the logical error rate of a radial code of growing s fails to improve at the same rate as a surface code of equal distance would falsify the claimed performance advantage.

Figures

Figures reproduced from arXiv: 2406.14445 by Joschka Roffe, Thomas R. Scruby, Timo Hillmann.

Figure 1
Figure 1. Figure 1: Radial layouts for two examples of (r, s) = (2, 3) classical codes with parity check matrices given in (5) and (6) respectively. Bits/checks are shown as circles/squares. Rings (and ring indices) are shown in red. Spokes (and spoke indices) are shown in blue. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Structure of the QRC with PCMs (11) and (12). The code is made of four (r, s) = (2, 3) classical radial codes, with two (red and blue) providing Z stabilisers and two (or￾ange and green) providing X stabilisers. An example of a X stabiliser from ring 1 of the orange code (supported on a qubit from each ring of the orange code and a ring 1 qubit in each of the red and blue codes) is also shown. qubits are u… view at source ↗
Figure 4
Figure 4. Figure 4: Representation of the structure of a quantum radial [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical estimates of average distances for QRCs of different [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Distribution of distances for QRC codes of different [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Measurement circuit for two stabilisers from the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Word error rates in a pair of radial codes as a function of (a) number of decoding cycles and (b) physical error rate. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distance distributions for the (4, 11) quantum radial code obtained from QDistRand for an increasing number of samples, i.e., 105 in (a) and 106 in (b). Annotated numbers on each bar indicate the upper bound on the probability that the distance d˜ calculated by QDistRand is greater than the true code distance. distance codes. For r = 4, on the other hand, this bound is much higher, on the order of 0.1 for … view at source ↗
Figure 10
Figure 10. Figure 10: Logical X error rate after 15 decoding rounds with the (3, 1)-overlapping window decoder in (a) for the J90, 8, 10K and in (b) for J352, 18, 20K code. Here we varied the number of maximum iterations in the min-sum decoder, shown in the legend, and fixed the reprocessing routine to OSD-0. A very large number of iterations are necessary before performance begins to plateau. the noise strength proportional t… view at source ↗
read the original abstract

We present a new family of quantum low-density parity-check codes, which we call radial codes, obtained from the lifted product of a specific subset of classical quasi-cyclic codes. The codes are defined using a pair of integers $(r,s)$ and have parameters $[\![2r^2s,2(r-1)^2,\leq2s]\!]$, with numerical studies suggesting average-case distance linear in $s$. In simulations of circuit-level noise, we observe comparable error suppression to surface codes of similar distance while using approximately five times fewer physical qubits. This is true even when radial codes are decoded using a single-shot approach, which can allow for faster logical clock speeds and reduced decoding complexity. We describe an intuitive visual representation, canonical basis of logical operators and optimal-length stabiliser measurement circuits for these codes, and argue that their error correction capabilities, tunable parameters and small size make them promising candidates for implementation on near-term quantum devices.

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 radial codes, a family of quantum LDPC codes obtained as the lifted product of a specific subset of classical quasi-cyclic codes. For parameters (r,s) the codes have explicit parameters [[2r²s, 2(r-1)², ≤2s]], with numerical studies indicating that the average-case distance scales linearly with s. Circuit-level noise simulations are reported to demonstrate error suppression comparable to surface codes of similar distance while using roughly five times fewer physical qubits; the advantage is claimed to persist under single-shot decoding.

Significance. If the reported simulation results hold under standard noise models and without post-hoc selection, the construction would provide a concrete low-overhead, single-shot-decodable alternative to the surface code, with tunable parameters and small block size that could be relevant for near-term hardware.

major comments (2)
  1. [Abstract] Abstract: the central performance claim of 'approximately five times fewer physical qubits' with 'comparable error suppression' rests on circuit-level simulations whose noise model, decoder implementation, sample sizes, fitting procedures, and exact surface-code baselines are not described. This information is load-bearing for the qubit-overhead advantage.
  2. [Abstract] Abstract: the statement that 'numerical studies suggest average-case distance linear in s' does not specify the range of s examined, the ensemble size, the precise distance estimator, or any statistical uncertainty, making it impossible to assess whether the linear scaling is robust or an artifact of the chosen instances.
minor comments (2)
  1. [Abstract] The abstract states the distance bound as ≤2s; it would be helpful to clarify whether this is a proven upper bound or an observed value and to indicate where the proof or evidence appears.
  2. The manuscript mentions an 'intuitive visual representation' and 'canonical basis of logical operators'; these should be cross-referenced to the relevant figures or sections for readers who wish to verify the construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the two major comments below and will revise the manuscript to incorporate additional details that strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central performance claim of 'approximately five times fewer physical qubits' with 'comparable error suppression' rests on circuit-level simulations whose noise model, decoder implementation, sample sizes, fitting procedures, and exact surface-code baselines are not described. This information is load-bearing for the qubit-overhead advantage.

    Authors: We agree that these methodological details are necessary to fully substantiate the performance claims. Although the main text provides an overview of the simulation approach, we will revise the manuscript to include an expanded description of the circuit-level noise model, the specific decoder implementation (including single-shot aspects), the Monte Carlo sample sizes, the procedures used to fit logical error rates, and the precise surface-code parameters and distances used as baselines. These additions will be placed in the numerical results section to make the reported overhead advantage transparent and reproducible. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'numerical studies suggest average-case distance linear in s' does not specify the range of s examined, the ensemble size, the precise distance estimator, or any statistical uncertainty, making it impossible to assess whether the linear scaling is robust or an artifact of the chosen instances.

    Authors: We acknowledge that the abstract statement is brief and lacks these specifics. In the revised manuscript we will expand the relevant section on code parameters and distance to report the range of s values examined, the ensemble size for each s, the exact procedure used to estimate distance (minimum-weight logical operator search), and any associated statistical uncertainty across the ensemble. This will allow readers to evaluate the robustness of the observed scaling. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the radial code family explicitly from the lifted-product construction applied to a specified subset of classical quasi-cyclic codes, yielding the closed-form parameters [[2r²s, 2(r-1)², ≤2s]] directly from the algebraic definition. The distance claim is an explicit upper bound ≤2s together with separate numerical sampling that reports average-case scaling linear in s; neither quantity is obtained by fitting a parameter to the same data later used to assert the result. Circuit-level simulations are presented as independent Monte-Carlo evidence under a stated noise model and are not required to close any derivation. No self-citation is invoked as a load-bearing uniqueness theorem, and no ansatz or renaming step reduces the central claims to their own inputs by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper contributes a new choice of classical quasi-cyclic subset inside an existing lifted-product framework; r and s function as design parameters rather than fitted constants, and no new physical entities are postulated.

free parameters (1)
  • r and s
    Integers that define code length, dimension, and distance; chosen by the authors to generate the family rather than fitted to performance data.
axioms (1)
  • standard math Lifted-product construction applied to classical quasi-cyclic codes yields valid quantum LDPC codes
    Invoked when the radial codes are defined from the chosen classical subset.

pith-pipeline@v0.9.0 · 5697 in / 1250 out tokens · 29798 ms · 2026-05-24T00:13:38.282289+00:00 · methodology

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