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arxiv: 2408.14828 · v4 · pith:B53SIQHTnew · submitted 2024-08-27 · 🪐 quant-ph

Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code

Christopher Gerhard , Todd A. Brun This is my paper

Pith reviewed 2026-05-23 21:40 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error detectionweak fault tolerance[[n,n-2,2]] codestabilizer measurementsuniversal quantum computationNISQ devicesgate constructions
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The pith

Constructions in the [[n,n-2,2]] code detect any single faulty gate error using only end-of-computation measurements to achieve weak fault tolerance.

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

The paper develops gate constructions inside a quantum error-detecting code that encodes n-2 logical qubits into n physical ones. These constructions ensure that any error from a single faulty gate can be detected by measuring the code stabilizers and extra ancillas after the full computation finishes. This weak form of fault tolerance improves the success rate of small quantum calculations compared to running them without any encoding, provided the physical error rate is low enough. The approach requires far fewer extra qubits and operations than schemes that achieve complete fault tolerance.

Core claim

In the [[n,n-2,2]] quantum error-detecting code, specific constructions for a universal set of gates allow detection of any error arising from a single faulty gate. Detection occurs by measuring the stabilizer generators of the code together with additional ancillas at the conclusion of the computation. This property holds up to analog imprecision on the physical rotation gate and yields weak fault tolerance.

What carries the argument

The [[n,n-2,2]] error-detecting code together with gate constructions that keep single errors detectable by final stabilizer and ancilla measurements.

Load-bearing premise

End-of-computation measurements alone suffice to catch every possible error produced by any single faulty gate during the entire circuit.

What would settle it

An explicit single-gate error in one of the constructions that produces an output state indistinguishable from the correct one after the final measurements.

Figures

Figures reproduced from arXiv: 2408.14828 by Christopher Gerhard, Todd A. Brun.

Figure 1
Figure 1. Figure 1: FIG. 1: Circuits for the logical CNOT gate in the [[4 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Circuits for the logical Phase gate in the [[4 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Circuits for the logical Hadamard gate in the [[4 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Weakly fault-tolerant circuits for the ZZ rotation gate. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Weakly fault-tolerant circuits for the XX rotation gate. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Logical [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: A rotation gate that rotates a quantum state [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Measurement of the eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Weakly fault-tolerant circuit for initialization into the [[ [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Weakly fault-tolerant Bell basis measurement [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Weakly fault-tolerant circuit for readout of the [[ [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Log-log plot of the undetectable error probability for the physical (blue), encoded (orange), and weakly [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Log-log plot of the post-selection rate for the physical (blue), encoded (orange), and weakly fault-tolerant [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Log-log plot of the undetectable-error probability for the physical (blue), encoded (orange), and weakly [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Log-log plot of the post-selection rate for the physical (blue), encoded (orange), and weakly fault-tolerant [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
read the original abstract

Many current quantum error-correcting codes that achieve full fault tolerance suffer from having low ratios of logical to physical qubits and significant overhead. This makes them difficult to implement on current noisy intermediate-scale quantum (NISQ) computers and results in the inability to perform quantum algorithms at useful scales with near-term quantum processors. As a result, calculations are generally done without encoding. We propose a middle ground between these two approaches: constructions in the $[[n,n-2,2]]$ quantum error-detecting code that can detect any error from a single faulty gate by measuring the stabilizer generators of the code and additional ancillas at the end of the computation. This achieves weak fault tolerance. As we show, this yields a significant improvement over no error correction for small computations with low enough physical error probabilities and requires much less overhead than codes that achieve full fault tolerance. We give constructions for a set of gates that achieve universal quantum computation in this error-detecting code, while satisfying weak fault tolerance up to analog imprecision on the physical rotation gate.

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 paper proposes explicit gate constructions for universal quantum computation inside the [[n,n-2,2]] quantum error-detecting code. These constructions are claimed to achieve 'weak fault tolerance': any error arising from a single faulty physical gate is detected by measuring the code stabilizers together with additional ancillas only at the very end of the circuit. The authors argue that the resulting overhead is far lower than that of full fault-tolerant codes while still yielding a net improvement over bare, uncoded computation for small algorithms when the physical error rate is sufficiently low.

Significance. If the weak-fault-tolerance property can be rigorously established, the approach would supply a practical middle ground for near-term devices: modest error detection with qubit overhead linear in the number of logical qubits rather than the quadratic or higher overhead required for full fault tolerance. The claim is therefore potentially relevant to NISQ-era algorithm execution, provided the detection guarantee holds for the supplied gate decompositions.

major comments (2)
  1. [constructions for universal gates (abstract and § on gate implementations)] The central claim—that every single physical fault (including during non-Clifford rotations) produces a detectable non-trivial syndrome when stabilizers and ancillas are measured only at the final time—rests on the specific gate constructions. Because the code distance is 2, a fault inside a multi-qubit gate can in principle spread to a weight-2 error that lies in the codespace or commutes with the final checks. The manuscript must therefore supply an explicit case-by-case verification (or inductive argument) showing that no such undetectable propagated error occurs for the chosen decompositions; without this verification the weak-fault-tolerance property is unproven.
  2. [performance claims] The performance comparison with uncoded computation is stated to be 'significant' for small computations at low physical error rates. No numerical simulation, threshold calculation, or explicit error-probability model is provided to support this quantitative claim, making it impossible to assess whether the improvement is realized under realistic noise models that include the analog imprecision on the physical rotation gate.
minor comments (2)
  1. Notation for the additional ancillas and the precise timing of their measurements should be defined consistently throughout the text.
  2. The abstract states that the scheme works 'up to analog imprecision on the physical rotation gate'; the precise tolerance on this imprecision should be stated explicitly in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments below and will revise the manuscript to strengthen the presentation of the weak fault-tolerance property and the supporting performance analysis.

read point-by-point responses

  1. Referee: [constructions for universal gates (abstract and § on gate implementations)] The central claim—that every single physical fault (including during non-Clifford rotations) produces a detectable non-trivial syndrome when stabilizers and ancillas are measured only at the final time—rests on the specific gate constructions. Because the code distance is 2, a fault inside a multi-qubit gate can in principle spread to a weight-2 error that lies in the codespace or commutes with the final checks. The manuscript must therefore supply an explicit case-by-case verification (or inductive argument) showing that no such undetectable propagated error occurs for the chosen decompositions; without this verification the weak-fault-tolerance property is unproven.

    Authors: We agree that an explicit verification is required to rigorously establish the claim. The gate decompositions were selected so that any single physical fault produces a non-trivial syndrome detectable by the final stabilizer and ancilla measurements, but the manuscript presents the constructions without a dedicated case-by-case or inductive argument. In the revised manuscript we will add a new subsection that performs this verification for each gate (including the non-Clifford rotations, accounting for analog imprecision), confirming that no weight-2 error lies in the codespace or commutes with the checks. revision: yes

  2. Referee: [performance claims] The performance comparison with uncoded computation is stated to be 'significant' for small computations at low physical error rates. No numerical simulation, threshold calculation, or explicit error-probability model is provided to support this quantitative claim, making it impossible to assess whether the improvement is realized under realistic noise models that include the analog imprecision on the physical rotation gate.

    Authors: The claim rests on an analytical model in which the undetected-error probability scales as O(p²) versus O(p) for the bare circuit. We will expand the relevant section to include the explicit error-probability expressions for small circuits under this model, together with a brief discussion of how analog imprecision on rotations is bounded by the detection property. Full Monte-Carlo simulations under detailed hardware noise models are beyond the scope of the present work; we will therefore qualify the performance statement as analytical and note the limitation. revision: partial

Circularity Check

0 steps flagged

No circularity detected; explicit constructions rest on standard QEC assumptions without self-referential reduction.

full rationale

The paper proposes explicit gate constructions in the [[n,n-2,2]] code to achieve weak fault tolerance via end-of-computation stabilizer and ancilla measurements. No evidence of self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appears in the abstract or context. The derivation chain relies on direct construction of gates satisfying the detection property under stated assumptions (including analog imprecision for rotations), which are independent of the target result and do not reduce to the inputs by definition. This is a standard constructive proposal in quantum error correction and scores as self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that end-of-computation stabilizer measurements suffice to detect single faulty gate errors in this code family; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Measuring the stabilizer generators and additional ancillas at the end of the computation detects any error from a single faulty gate in the [[n,n-2,2]] code.
    This premise underpins the weak fault tolerance claim.

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Forward citations

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