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arxiv: 2604.05126 · v1 · submitted 2026-04-06 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

In-Situ Simultaneous Magic State Injection on Arbitrary CSS qLDPC Codes

Kun Liu , Shifan Xu , Tomas Jochym-O'Connor , Zhiyang He , Shraddha Singh , Yongshan Ding
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:52 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error correctionqLDPC codesmagic state injectionfault-tolerant computingCSS quantum codessyndrome extractionin-situ preparation
0
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The pith

Logical magic states can be prepared directly inside any CSS qLDPC code using only syndrome extraction resources.

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

This paper proposes the first in-situ scheme for injecting logical magic states into quantum LDPC codes. Instead of preparing raw magic states externally and transferring them, the method creates them inside the target code block by adapting the existing syndrome extraction circuit. The protocol generalizes to all CSS qLDPC codes and is demonstrated on two example codes with circuit simulations. Under realistic noise models, the resulting logical error rates per injected state are low enough to be useful for fault-tolerant computation, while requiring no extra qubits. A reader would care because this lowers the hardware cost of supplying the non-Clifford resources needed for universal quantum computing on these high-rate codes.

Core claim

The authors claim that an in-situ magic state injection protocol exists for arbitrary CSS qLDPC codes, where logical magic states are prepared within the memory block using only ancilla qubits required for syndrome extraction, as verified by examples on the bivariate bicycle code and hypergraph product code with reported injection error rates of 1.62e-3 and lower under depolarizing and asymmetric noise.

What carries the argument

Repurposing the syndrome extraction circuit for simultaneous preparation of multiple logical magic states inside the qLDPC block.

Load-bearing premise

The scheme works in a regime where the contribution of correlated injection errors is negligible compared to other error sources.

What would settle it

A noise simulation or hardware test that shows the correlated error rate during simultaneous magic state injection to be substantially higher than the 1% of injection error reported in the paper.

Figures

Figures reproduced from arXiv: 2604.05126 by Kun Liu, Shifan Xu, Shraddha Singh, Tomas Jochym-O'Connor, Yongshan Ding, Zhiyang He.

Figure 1
Figure 1. Figure 1: FIG. 1. High-level summary of this work. (a) In-logical overlap: Typical injection scheme for a single logical qubit. For surface [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Circuit-level simulation results of [[144 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Histogram of the change in total injection error rate [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Circuit-level simulation results of [[144 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Circuit-level simulation on the HGP code [[225 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

Quantum low-density parity-check (qLDPC) codes can encode many logical qubits within a single code block at low physical qubit overhead, yet magic state injection into such codes remains largely underexplored. Existing state injection proposals for qLDPC codes predominantly follow an external prepare-and-transfer paradigm, in which raw magic states are prepared outside the target code block and subsequently injected via inter-code operations. We propose the first \emph{in-situ} magic state injection: a scheme in which logical magic states are directly prepared within a qLDPC memory block, only using resources required for syndrome extraction. We show that our scheme is generalizable to any CSS qLDPC code, with examples of circuit-level simulations on the $[[144,12,12]]$ Bivariate Bicycle (BB) code and the $[[225,9,4]]$ Hypergraph Product code. We focus on a regime where correlated injection errors are negligible. In the BB code, this corresponds to a configuration that simultaneously injects four logical $|Y\rangle$ states. Under a uniform depolarizing noise model with physical error rate $10^{-3}$, this achieves an injection error rate of $1.62 \times 10^{-3}$ per logical qubit, while the correlated-error contribution is only $2 \times 10^{-5}$ per logical qubit (about $1\%$ of the injection error rate). Under a hardware-motivated asymmetric noise model where single-qubit gate errors are $10\%$ of two-qubit gate errors, the injection error rate per logical qubit falls to $ 6.7 \times 10^{-4} $, below the error rate ($ 10^{-3} $) of the two-qubit gates used to encode the magic states. Its simplicity allows our scheme to be applied to arbitrary CSS qLDPC codes using only the ancilla qubits native to syndrome extraction, and yield a reduction in space overhead relative to both prepare-and-transfer approaches and surface-code-based magic state injection schemes.

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 proposes the first in-situ magic state injection scheme for arbitrary CSS qLDPC codes, preparing logical magic states directly inside the memory block using only native syndrome-extraction ancillas. It asserts that the scheme generalizes to any CSS qLDPC code and supports this with circuit-level simulations on the [[144,12,12]] bivariate bicycle code (simultaneously injecting four logical |Y⟩ states) and the [[225,9,4]] hypergraph product code, reporting per-logical-qubit injection error rates of 1.62×10^{-3} (depolarizing noise at p=10^{-3}) and 6.7×10^{-4} (asymmetric noise), with correlated-error contributions at ~1% of the total.

Significance. If the generality claim holds, the approach offers a meaningful reduction in space overhead relative to prepare-and-transfer injection or surface-code methods by reusing syndrome ancillas. The concrete error-rate numbers under both uniform depolarizing and hardware-motivated asymmetric noise models, together with the explicit regime of negligible correlated errors, supply useful benchmarks for qLDPC-based fault-tolerant architectures.

major comments (2)
  1. [General construction and claim of generality] The central generality claim (that the scheme applies to any CSS qLDPC code without extra resources or code-specific modifications) rests on a high-level description plus simulations restricted to the BB and HGP families. An explicit, code-agnostic construction—e.g., a procedure expressed solely in terms of the parity-check matrix and ancilla scheduling that works for arbitrary CSS codes—is required to confirm that no implicit reliance on BB/HGP stabilizer weights, connectivity, or four-qubit |Y⟩ simultaneity exists.
  2. [Simulation results and noise models] Circuit-level simulation section: the exact circuit construction for in-situ |Y⟩ preparation, the precise noise-model definitions (including how single-qubit vs. two-qubit error rates are applied during injection), and the data-exclusion rules used to obtain the quoted error rates (1.62×10^{-3} and 6.7×10^{-4}) must be provided in sufficient detail for independent reproduction; without them the numerical results cannot be verified as independent checks rather than artifacts of the chosen configuration.
minor comments (2)
  1. [Abstract and introduction] The abstract states that correlated injection errors are negligible in the chosen BB configuration; the main text should quantify the precise threshold at which this approximation breaks and whether the same regime can be identified for arbitrary CSS codes.
  2. [Noise model definitions] Notation for the asymmetric noise model (ratio of single- to two-qubit gate errors) should be defined once and used consistently when reporting the 6.7×10^{-4} figure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address each major comment below. Where the comments identify areas requiring greater explicitness, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [General construction and claim of generality] The central generality claim (that the scheme applies to any CSS qLDPC code without extra resources or code-specific modifications) rests on a high-level description plus simulations restricted to the BB and HGP families. An explicit, code-agnostic construction—e.g., a procedure expressed solely in terms of the parity-check matrix and ancilla scheduling that works for arbitrary CSS codes—is required to confirm that no implicit reliance on BB/HGP stabilizer weights, connectivity, or four-qubit |Y⟩ simultaneity exists.

    Authors: We thank the referee for this observation. The scheme is constructed to depend only on the CSS property (to identify the logical Pauli operators and the controlled-phase or controlled-Hadamard operations that prepare the magic state on the data qubits) together with the ancilla qubits and measurement schedule already required for syndrome extraction; no additional qubits or code-family-specific connectivity assumptions are used. To make this fully explicit and address the request for an algorithmic description, the revised manuscript adds a new subsection that states the injection procedure directly in terms of the parity-check matrices H_X and H_Z and the standard ancilla scheduling. The procedure is written as a sequence of steps that any CSS qLDPC code can follow without reference to stabilizer weights, graph structure, or the particular choice of injecting four states simultaneously (the latter is used only in the BB simulation to illustrate multi-logical-qubit injection). We believe this addition removes any ambiguity while leaving the underlying scheme unchanged. revision: yes

  2. Referee: [Simulation results and noise models] Circuit-level simulation section: the exact circuit construction for in-situ |Y⟩ preparation, the precise noise-model definitions (including how single-qubit vs. two-qubit error rates are applied during injection), and the data-exclusion rules used to obtain the quoted error rates (1.62×10^{-3} and 6.7×10^{-4}) must be provided in sufficient detail for independent reproduction; without them the numerical results cannot be verified as independent checks rather than artifacts of the chosen configuration.

    Authors: We agree that the original simulation section lacked sufficient detail for independent reproduction. The revised manuscript expands Section 4 and adds an appendix containing the full circuit diagrams (including all single- and two-qubit gates applied to data and ancilla qubits during the injection round), the exact noise-model specifications (depolarizing model applies equal X/Y/Z probabilities to each gate; asymmetric model sets single-qubit error probability to 0.1 times the two-qubit error probability, with errors applied after every gate), and the post-selection rules (runs are retained only if no syndrome errors are detected on the ancilla measurements performed during injection; logical error rates are then obtained by averaging the decoded outcome over 10^7 Monte Carlo shots). These additions allow the quoted per-logical-qubit error rates to be verified directly. revision: yes

Circularity Check

0 steps flagged

No circularity: scheme and simulations are self-contained

full rationale

The paper presents a construction for in-situ magic state injection using only syndrome-extraction ancillas on CSS qLDPC codes, supported by a general high-level argument plus explicit circuit-level simulations on two concrete codes under stated depolarizing and asymmetric noise models. No equations define a quantity in terms of itself, no fitted parameters are relabeled as predictions, and no load-bearing steps reduce to self-citations or prior ansatzes by the authors. The simulations constitute independent numerical verification rather than tautological outputs of the input noise model. The generalizability statement rests on the explicit resource claim (native ancillas only) rather than on any definitional equivalence or imported uniqueness theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract introduces no new physical particles or forces. The scheme relies on standard properties of CSS codes and the ability to perform syndrome extraction; no free parameters are fitted or mentioned.

axioms (1)
  • domain assumption CSS qLDPC codes admit transversal or low-weight operations sufficient for in-situ state preparation
    The generality claim rests on this standard property of CSS codes.

pith-pipeline@v0.9.0 · 5685 in / 1382 out tokens · 53704 ms · 2026-05-10T18:52:33.552474+00:00 · methodology

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Reference graph

Works this paper leans on

71 extracted references · 33 canonical work pages · 4 internal anchors

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    Prepare thekcarrier qubits{Q i }in the +1 eigen- states ofM Qi for alli∈[k]

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    (Such Clifford could be free

    Apply the Clifford circuitCon the carrier qubits. (Such Clifford could be free. See discussion in Ap- pendix D)

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    Perform a furtherr 2 rounds of SE and decode using all retained syndrome information. This protocol is chosen because the dominant contribu- tion to the injection error arises from faults before and during the first SE round. Additional SE rounds can further suppress higher-order contributions, while poten- tially introducing more memory faults. As a poin...

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