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arxiv: 2605.21898 · v2 · pith:FRMJMAEOnew · submitted 2026-05-21 · 🪐 quant-ph

Concatenating Algebraic Codes over High-Rate Quantum LDPC Codes

Adam Wills , Michael E. Beverland , Lev S. Bishop , Jay M. Gambetta , Patrick Rall , Vikesh Siddhu , Andrew W. Cross This is my paper

Pith reviewed 2026-06-30 17:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error correctioncode concatenationLDPC codesReed-Solomon codesGalois quditsfault tolerancesyndrome extraction
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The pith

Modeling gross code blocks as Galois qudits enables Reed-Solomon concatenation that reaches the teraquop regime at 10^{-3} noise with reduced overhead.

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

The paper shows that high-rate quantum LDPC inner codes can be concatenated with algebraic outer codes by treating each code block as one logical Galois qudit. This representation manages correlated errors within blocks and permits the use of list decoders from quantum Reed-Solomon codes. The resulting memory system reaches the teraquop regime at uniform 10^{-3} physical noise, which prior constructions could not access, while requiring lower space overhead than the 288-qubit two-gross code. A Galois qudit Shor scheme with time-like Reed-Solomon protection handles fault-tolerant syndrome extraction.

Core claim

By treating each gross code block as a single logical Galois qudit, quantum Reed-Solomon outer codes can be concatenated over it, allowing the system to reach the teraquop regime at 10^{-3} noise with lower space overhead than the two-gross code.

What carries the argument

The logical Galois qudit representation of each inner LDPC code block, which enables concatenation with algebraic outer codes possessing list decoders.

Load-bearing premise

The modeling choice that treats each inner-code block as a single logical Galois qudit correctly captures the correlated errors so that the outer Reed-Solomon list decoder can suppress them.

What would settle it

A calculation or simulation of the logical error rate for the concatenated gross code at uniform 10^{-3} physical noise that checks whether it falls below the threshold for teraquop operation with the stated space overhead.

Figures

Figures reproduced from arXiv: 2605.21898 by Adam Wills, Andrew W. Cross, Jay M. Gambetta, Lev S. Bishop, Michael E. Beverland, Patrick Rall, Vikesh Siddhu.

Figure 1
Figure 1. Figure 1: The space overheads of various systems, allowing a maximum size of half a million physical qubits. Concatenation allows the gross code to work in the teraquop regime relevant for the execution of many large-scale quantum algorithms. There, it offers an improved space overhead versus the two-gross code, with several engineering benefits. We show the surface code, as well as the “yoked” (concatenated) surfac… view at source ↗
Figure 2
Figure 2. Figure 2: An overview of the moving parts of the construction and analysis. Blue text is hyperlinked. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: This figure illustrates the key technical idea of packaging the logical information in one gross code into a higher-dimensional qudit. The first logical qubit, the pivot, is sacrificed to enable the logical operations required to operate the memory. The 11 non-pivot logical qubits become one larger 2 11-dimensional qudit. Outer Code Fault-Tolerant Syndrome Extraction To operate these outer codes using logi… view at source ↗
Figure 4
Figure 4. Figure 4: The left-hand figure illustrates our global memory system. 2 rows of the system are sacri￾ficed for the fault-tolerant generation of the cat states used for syndrome extraction of neighbouring outer codeblocks. These ancillary blocks move through the system over the course of time to extract the syndrome of other codeblocks. On the right, we illustrate a Z error affecting an outer codeblock, labelled by so… view at source ↗
Figure 5
Figure 5. Figure 5: Depiction of the fault-tolerant preparation and verification of the qudit cat state for n = 8 and R = 1. Every wire denotes a qudit, which may be imagined as the s non-pivot qubits of a cat state row gross code. The circuit proceeds via an initial non-fault-tolerant preparation, and then R rounds of fault-tolerant checking. Each set of n − 1 measurements can be executed in two “layers”. In this diagram, al… view at source ↗
Figure 6
Figure 6. Figure 6: The fault-tolerant Z αZ β qudit measurement sub-routine acts on two cat state row qudits/gross codes (the first and fourth wires on the right-hand side), using the adjacent two an￾cilla row qudits/gross codes (the second and third wires) as ancillas. All operations depicted are qudit operations, in particular, boxes denote the measurement of the qudit operator by which they are labelled. Again, boxes witho… view at source ↗
Figure 7
Figure 7. Figure 7: The compilation of the measurement Z vZ v , where v ∈ F s 2 , on the s non-pivot qubits in adjacent gross code modules. The pivot qubits in each module are used as ancillas. The measurement of Z vZ v on the two groups of s qubits may be inferred via the XOR of the three Z-type measurements on the right-hand side. Note that the X-type measurements are measurement projections, that is, measurements followed … view at source ↗
Figure 8
Figure 8. Figure 8: The space overheads of various systems at 10−3 physical noise. All systems are chosen with the optimal configuration while keeping their size below 500, 000 physical qubits. The space overhead of the concatenated gross code memory crosses over that of the two-gross code at a logical error rate of 4.10 × 10−15 per logical qubit-round. outer code distances than d = 9 could become useful if one allowed larger… view at source ↗
Figure 9
Figure 9. Figure 9: Further information on the performance of the concatenated gross code system at 10−3 physical noise. The left-hand panel shows the places where outer codes of various distances are used in the optimal system, and the right-hand panel shows the places where the optimal system does (red), and does not (grey), make use of some non-trivial post-selection strategy [PITH_FULL_IMAGE:figures/full_fig_p064_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The performance of our scheme at a uniform physical noise strength 10−4 . We show the space overhead of the optimal system at each target logical error rate per logical qubit-round. In the left-hand panel, we show where different outer code distances are used in the optimal system, and in the right-hand panel, we show that some post-selection is used in all optimal systems. We also plot the performance of… view at source ↗
Figure 11
Figure 11. Figure 11: The compilation of a Z viZ wi measurement on two adjacent gross codes into a gate set higher than the bicycle instructions. All boxes denote qubit measurements; those without meter sym￾bols denote “measurement projections” [Yod+25], meaning measurements followed by the appropriate frame updates to ensure we recover the +1-eigenstate (in the absence of faults). The outcome of the whole measurement is taken… view at source ↗
Figure 12
Figure 12. Figure 12: Histogram of sampled measurement sequences compiled to the bicycle instructions using our compilation algorithm with num-decomposition-attempts = 100. This histogram shows the time to perform the full qudit Z αZ β measurement, in units of measurement lengths, noting that all in and inter-module measurement instructions take the same time: τMeas = 120 timesteps (see [PITH_FULL_IMAGE:figures/full_fig_p073_… view at source ↗
Figure 13
Figure 13. Figure 13: The same plot as [PITH_FULL_IMAGE:figures/full_fig_p077_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The same plots as [PITH_FULL_IMAGE:figures/full_fig_p078_14.png] view at source ↗
read the original abstract

Different quantum error correction schemes trade off overhead, error suppression, and hardware connectivity. Code concatenation can relax these tradeoffs by using an outer code whose non-local connectivity is supplied by logical operations of an inner code rather than directly by hardware. Prior works showed that this can reduce memory overhead for local low-rate inner codes such as the surface code. Here, we study concatenation over non-local, high-rate inner codes. Such inner codes experience correlated errors among the many logical qubits in a single codeblock. We handle this by treating each block as a single logical Galois qudit, enabling concatenation with algebraic outer codes with excellent parameters and, crucially, list decoders. In particular, we consider a memory system formed by concatenating quantum Reed-Solomon outer codes over the gross code. For fault-tolerant syndrome extraction, we develop a Galois qudit Shor scheme using "time-like" Reed-Solomon protection against measurement errors. Interestingly, a lightweight fault tolerance scheme, that would fail for qubits, works well for large-alphabet qudits, suggesting a very different theory of fault tolerance for such qudits. The whole protocol is optimised via improved bicycle instruction logical error rates, novel compilation strategies, and recent decoder post-selection rules. At uniform $10^{-3}$ physical noise, the concatenated gross code reaches the teraquop regime, which it previously could not access, with a lower space overhead than the $288$-qubit two-gross code, while offering several advantages from the engineering standpoint. Beyond our main case study, we believe the core ideas of Galois qudits, quantum Reed-Solomon outer codes, and list decoding, will prove generically powerful and highly transferable ideas across high-rate quantum architectures.

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 / 1 minor

Summary. The paper proposes concatenating quantum Reed-Solomon outer codes over high-rate quantum LDPC inner codes such as the gross code by representing each inner-code block as a single logical Galois qudit. This modeling enables algebraic outer codes and list decoding to suppress correlated logical errors. The authors develop a Galois-qudit Shor syndrome extraction scheme protected by time-like Reed-Solomon codes against measurement errors, optimize via improved bicycle-instruction error rates and decoder post-selection, and report that the resulting memory system reaches the teraquop regime at uniform 10^{-3} physical noise with lower space overhead than the 288-qubit two-gross code.

Significance. If the Galois-qudit modeling accurately captures the correlated error statistics after inner-code decoding, the approach would demonstrate a concrete route to teraquop performance with reduced overhead for high-rate LDPC architectures while introducing transferable techniques (Galois qudits, quantum Reed-Solomon list decoding, and qudit-specific fault tolerance). The work supplies an explicit construction and optimization pipeline rather than an abstract existence argument.

major comments (2)
  1. [concatenation protocol and Galois-qudit representation] The central performance claim (teraquop regime at 10^{-3} physical noise with lower overhead than the 288-qubit two-gross code) rests on the premise, stated in the concatenation and syndrome-extraction sections, that logical errors after inner decoding are faithfully represented by a single Galois-qudit error symbol so that the outer Reed-Solomon list decoder achieves its designed distance and list-decoding radius. No explicit verification or bound is supplied showing that the actual support of post-decoding logical errors matches this single-symbol model rather than producing multi-symbol or time-like patterns outside the assumed channel.
  2. [simulation results and optimization] The abstract and results paragraphs report optimized simulations reaching teraquop performance, yet supply neither error bars on the logical error rates, an explicit statement of the full noise model (including measurement-error rates and correlations), nor any indication that the simulation code or raw data are available. Without these, the quantitative overhead advantage cannot be independently assessed.
minor comments (1)
  1. [fault-tolerant syndrome extraction] Notation for the Galois-qudit Shor scheme and time-like Reed-Solomon protection is introduced without a compact summary table relating physical, inner-logical, and outer-symbol error rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We respond point-by-point to the two major comments below.

read point-by-point responses

  1. Referee: [concatenation protocol and Galois-qudit representation] The central performance claim (teraquop regime at 10^{-3} physical noise with lower overhead than the 288-qubit two-gross code) rests on the premise, stated in the concatenation and syndrome-extraction sections, that logical errors after inner decoding are faithfully represented by a single Galois-qudit error symbol so that the outer Reed-Solomon list decoder achieves its designed distance and list-decoding radius. No explicit verification or bound is supplied showing that the actual support of post-decoding logical errors matches this single-symbol model rather than producing multi-symbol or time-like patterns outside the assumed channel.

    Authors: The Galois-qudit model is introduced precisely to capture the dominant residual error after inner decoding, with each block acting as one logical qudit. We agree that an explicit discussion of the approximation's validity would strengthen the presentation. In the revised manuscript we will add a short subsection in the concatenation section that (i) recalls the inner decoder's distance and failure modes, (ii) bounds the probability of multi-qudit logical errors under the assumed noise model, and (iii) notes that time-like patterns are separately handled by the outer time-like Reed-Solomon protection. revision: yes

  2. Referee: [simulation results and optimization] The abstract and results paragraphs report optimized simulations reaching teraquop performance, yet supply neither error bars on the logical error rates, an explicit statement of the full noise model (including measurement-error rates and correlations), nor any indication that the simulation code or raw data are available. Without these, the quantitative overhead advantage cannot be independently assessed.

    Authors: We will add statistical error bars (derived from the Monte-Carlo sample sizes already used) to all logical-error-rate plots and will expand the methods section with an explicit enumeration of every noise rate and correlation coefficient employed. The simulation parameters are fully specified in the text; we will add a statement that raw data files can be supplied upon request. We do not plan to release the full simulation codebase at this time, as it builds on proprietary extensions of existing decoders. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results rest on explicit modeling assumptions and simulation

full rationale

The paper presents its performance claims (teraquop regime at 10^{-3} noise with reduced overhead) as outcomes of numerical simulation under a chosen modeling assumption that each gross-code block acts as one logical Galois qudit. This modeling choice is stated explicitly to enable Reed-Solomon concatenation and list decoding rather than being derived from or reducing to the simulation outputs themselves. No equations or steps in the provided text reduce a prediction to a fitted parameter by construction, invoke self-citations as the sole justification for uniqueness, or smuggle ansatzes via prior work. The derivation chain for fault-tolerant syndrome extraction and outer-code suppression is therefore independent of the target performance numbers.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of the Galois-qudit abstraction for handling correlated errors, the existence of efficient list decoders for the outer codes, and the accuracy of the simulation noise model at 10^{-3}.

free parameters (1)
  • uniform physical noise rate
    The simulation is performed at a fixed 10^{-3} error rate per physical operation; this value is chosen as the operating point rather than derived.
axioms (2)
  • standard math Quantum Reed-Solomon codes admit efficient list decoders that can correct the effective errors after inner-code decoding
    Invoked when the authors state that list decoding enables the outer code to handle the residual errors from the inner gross code.
  • domain assumption The gross code can be treated as a single logical Galois qudit without losing essential error correlations
    This modeling step is introduced to enable concatenation and is load-bearing for the entire protocol.
invented entities (1)
  • Galois qudit representation of an LDPC code block no independent evidence
    purpose: To abstract correlated errors inside a high-rate inner code block so that algebraic outer codes can be applied
    New modeling construct introduced in the paper; no independent experimental evidence is supplied in the abstract.

pith-pipeline@v0.9.1-grok · 5873 in / 1682 out tokens · 39233 ms · 2026-06-30T17:42:03.354247+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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    quant-ph 2026-06 unverdicted novelty 7.0

    Presents planar open-boundary quantum LDPC codes with nearest-neighbor iSWAP-based syndrome extraction that outperform rotated surface codes in code-efficiency and logical error rate on finite instances like [[323,14,15]].

  2. Full Extractors for Logical Processing in Hypergraph Product Codes

    quant-ph 2026-06 unverdicted novelty 6.0

    Full extractors for HGP codes are built to enable logical processing via PBC without compilation overhead, with sizes 50-80% of base codes and low error rates in simulations.

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages · cited by 2 Pith papers

  1. [1]

    measurement projections

    2025 (cit. on pp. 19, 71, 72). 68 [Rai02] Eric M Rains. ‘Nonbinary quantum codes’. In:IEEE Transactions on Information Theory 45.6 (2002), pp. 1827–1832 (cit. on pp. 6, 13). [Reu+18] Albert Reuther et al. ‘Interactive supercomputing on 40,000 cores for machine learning and data analysis’. In:2018 IEEE High Performance extreme Computing Conference (HPEC). ...

    work page arXiv 2025

  2. [2]

    gamma parameters

    Using the tools in [Qis25], it is possible by brute force to find the shortest sequence of native rotationsR1, . . . , Rk on the11non-pivot qubits such that(R 1 . . . Rk)ZZ w1(R1 . . . Rk)†,(R 1 . . . Rk)ZZ w2(R1 . . . Rk)† and(R 1 . . . Rk)ZZ w3(R1 . . . Rk)† are all native measurements. We also optimise our rotation sequence length using the absorptions...

    2048