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arxiv: 2604.09797 · v2 · submitted 2026-04-10 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Optimising Quantum Error Correction Using Morphing Circuits

Mackenzie H. Shaw , Barbara M. Terhal
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:52 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error correctionmorphing circuitssyndrome extractionAbelian 2BGA codesfault toleranceconnectivity optimizationstabilizer codes
0
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The pith

Morphing circuits optimize quantum error correction codes and syndrome extraction directly for connectivity and gate choice.

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

The paper establishes that morphing circuits offer a systematic way to jointly optimize both quantum error correction codes and the circuits used to extract their error syndromes. Optimization explicitly targets hardware constraints such as qubit connectivity, the selection of two-qubit gates like ISWAP or CNOT, and the total number of physical qubits. A sympathetic reader would care because this integrated design can produce codes and circuits that fit better within the limits of existing and near-term quantum hardware, lowering the resources needed for reliable error correction. The authors demonstrate the approach on Abelian two-block group algebra codes, boundary conditions in two-dimensional layouts, single-shot decoding, and stability experiments that include measurement and reset errors.

Core claim

Morphing circuits provide a way of optimising syndrome extraction circuits and codes directly in terms of connectivity, choice of two-qubit gate and number of physical qubits. The methods find new codes and syndrome extraction circuits of practical interest, including Abelian 2BGA morphing circuits with better code parameters and connectivity than existing circuits. Alternating syndrome extraction circuits can be viewed as two-round morphing circuits whose fault-tolerant properties are computationally much easier to examine than non-alternating ones.

What carries the argument

Morphing circuits: syndrome extraction procedures that incorporate alternating or time-reversed rounds to adjust effective connectivity and gate usage while preserving the logical code.

If this is right

  • Abelian 2BGA morphing circuits achieve improved code parameters together with lower connectivity requirements.
  • Stability experiments show better performance when measurement and reset errors are taken into account.
  • Alternating syndrome extraction circuits become easier to verify for fault tolerance by reducing to two-round morphing analysis.
  • Boundary conditions for two-dimensional codes and single-shot properties can be incorporated within the same morphing framework.

Where Pith is reading between the lines

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

  • The same morphing technique could be applied to additional code families to discover further connectivity reductions.
  • Hardware implementations of the new circuits would directly test whether the reported connectivity savings translate into measurable improvements in logical error rates.
  • The computational simplification for alternating circuits may enable new decoder designs that exploit time-reversed round structure.

Load-bearing premise

Morphing circuits preserve the logical equivalence and fault-tolerant properties of Abelian 2BGA codes, their boundaries, and single-shot settings under realistic error models that include measurement and reset errors.

What would settle it

A simulation or hardware run in which an Abelian 2BGA morphing circuit exhibits equal or worse logical error rates, no reduction in connectivity, or loss of fault tolerance compared with a standard non-morphed circuit under the same noise model would falsify the optimization benefit.

Figures

Figures reproduced from arXiv: 2604.09797 by Barbara M. Terhal, Mackenzie H. Shaw.

Figure 1
Figure 1. Figure 1: Morphing with a stabiliser code. (a) A “mid-cycle [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Circuit-level errors occurring anywhere (red stars) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example of an X-detector in a morphing circuit for the colour code which is constructed by considering how an X￾stabiliser s ∈ S1 in the mid-cycle code C morphs to the two end-cycle codes and how its support is affected by measurements. Starting from C end 1 , the detector for s is formed by taking the XOR of the MX measurement outcomes in M2 and the MX measurements in M1 given by the red boxes. The fir… view at source ↗
Figure 5
Figure 5. Figure 5: An example contraction tree for a given X-stabiliser in a purely contracting morphing circuit; this particular example corresponds to the contraction tree for the circuits in Ref. [39]. The contraction tree is in (a) while the corresponding CNOT and measurement layers are shown in (b), with qubits placed on vertices of the graph. The gates in (b) make up the s￾contraction circuit Fs for the X-stabiliser. T… view at source ↗
Figure 6
Figure 6. Figure 6: The contraction tree diagram and the corre [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An alternating bare-ancilla circuit for the distance [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A disjoint morphing circuit for the distance-4 unro [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Flowchart summarising our search over Abelian 2BGA codes and their two-round homomorphism-based morphing [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Contraction tree diagrams for two-round purely [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Creating and padding boundaries for (a) an [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Constructing the hex-grid morphing circuits for [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Construction of a hex-grid morphing surface code [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The hex-grid surface code morphing circuits compiled with (a) CNOT gates and (b) CXSWAP gates. The [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Summary flowchart of the optimisation procedure [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Hex-grid morphing circuits for the colour code on [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The optimised triangular weight-6 morphing colour code circuit for [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Diagrams representing the two types of detectors in a morphing circuit, time moves from left to right in both [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Two four-round toric code morphing circuits with [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Morphing the 3D toric code, with X-checks on vertices, qubits on edges, Z-checks on faces, and Z￾redundancies around each cube. (a) The contraction sub￾set Sz,1, with contracting X-stabilisers highlighted in red, and contracting Z-stabilisers shaded in blue. Half of all X￾stabilisers and half of the Z-stabilisers along x-oriented and y-oriented faces are contracted; none of the z-oriented Z￾stabilisers ar… view at source ↗
Figure 22
Figure 22. Figure 22: The circuits used to investigate the effect of string-like measurement structure on stability experiments. (a) The [PITH_FULL_IMAGE:figures/full_fig_p034_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Effect of string-like and non-string-like measure [PITH_FULL_IMAGE:figures/full_fig_p035_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: (a) The time-reversal symmetry in a two-round [PITH_FULL_IMAGE:figures/full_fig_p037_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The full set of minimal codes found (up to [PITH_FULL_IMAGE:figures/full_fig_p045_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Some minimal codes found by our numerical search using the methods outlined in Sections [PITH_FULL_IMAGE:figures/full_fig_p046_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Construction of optimal boundaries for the (a) weight-6, (b) weight-5, and (c) weight-7 end-cycle codes of colour [PITH_FULL_IMAGE:figures/full_fig_p047_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Triangular weight-6 colour code morphing circuits, and their circuit-level errors. (a) The contraction tree diagram [PITH_FULL_IMAGE:figures/full_fig_p051_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: All possible circuit-level errors in the weight-7 [PITH_FULL_IMAGE:figures/full_fig_p051_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Diamond weight-7 morphing colour code circuits. (a) The boundaries used to construct the [PITH_FULL_IMAGE:figures/full_fig_p052_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Numerical performance of colour code circuits in a [PITH_FULL_IMAGE:figures/full_fig_p053_31.png] view at source ↗
read the original abstract

Quantum error correction (QEC) codes are traditionally defined and searched for without specifying the manner in which its syndrome extraction circuits are executed using elementary gates and measurements. We show how morphing circuits introduced in Refs. [1-3] provide a way of optimising syndrome extraction circuits and codes directly in terms of connectivity, choice of two-qubit gate (ISWAP versus CNOT) and number of physical qubits. We discuss morphing circuits in code optimisation among Abelian two-block group algebra (2BGA) codes, handling boundaries for 2D codes, codes with single-shot properties, and improving performance in stability experiments against measurement and reset errors. We show that alternating syndrome extraction circuits - executed with alternating time-reversed rounds - can be viewed as a two-round morphing circuit whose fault-tolerant properties are computationally much easier to examine than non-alternating syndrome extraction circuits. Our methods find new codes and syndrome extraction circuits of practical interest, including Abelian 2BGA morphing circuits with better code parameters and connectivity than existing circuits. [1] Matt McEwen, Dave Bacon, and Craig Gidney. Relaxing hardware requirements for surface code circuits using time-dynamics. Quantum, 7:1172, 2023. [2] Craig Gidney and Cody Jones. New circuits and an open source decoder for the color code, 2023. [3] Mackenzie H. Shaw and Barbara M. Terhal. Lowering connectivity requirements for bivariate bicycle codes using morphing circuits.

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 using morphing circuits (from prior Refs. [1-3]) to optimize syndrome extraction circuits and codes for quantum error correction, focusing on Abelian two-block group algebra (2BGA) codes. It claims new optimized circuits with better code parameters and connectivity, easier fault-tolerance analysis via viewing alternating time-reversed rounds as two-round morphing circuits, handling of boundaries for 2D codes and single-shot properties, and improved performance against measurement/reset errors in stability experiments. The approach allows direct optimization in terms of connectivity, ISWAP vs CNOT gates, and qubit count.

Significance. If the preservation of logical equivalence and fault tolerance holds, the work provides a systematic method to tailor QEC codes and circuits to hardware constraints, which could aid practical fault-tolerant quantum computing implementations. The simplification of FT analysis for alternating circuits and identification of new 2BGA codes with improved parameters are potentially useful extensions. The paper credits the foundational morphing technique to prior work while adding applications to 2BGA codes, boundaries, and error models.

major comments (2)
  1. [Section on Abelian 2BGA morphing circuits and single-shot properties] The central claim that morphing circuits preserve logical equivalence and fault-tolerant properties (including code distance) when applied to Abelian 2BGA codes with boundaries and single-shot settings is load-bearing for all optimization results and new code claims, yet the manuscript supplies no explicit derivation, theorem, or circuit-level verification showing that error propagation remains controlled under realistic noise including measurement and reset errors (see the section discussing Abelian 2BGA morphing circuits and the skeptic note on preservation).
  2. [Paragraph on alternating time-reversed rounds] The assertion that alternating syndrome extraction circuits (time-reversed rounds) can be viewed as two-round morphing circuits whose FT properties are 'computationally much easier to examine' lacks a concrete comparison or example demonstrating reduced analysis complexity or improved thresholds relative to non-alternating circuits (see the paragraph on alternating circuits in the abstract and main text).
minor comments (2)
  1. Ensure all figures and tables explicitly compare new morphing circuits against baselines in terms of parameters, connectivity, and simulated performance under the claimed error models.
  2. Clarify notation for morphing operations when extending to boundaries to avoid ambiguity with prior definitions in Refs. [1-3].

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the foundations of our claims. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: The central claim that morphing circuits preserve logical equivalence and fault-tolerant properties (including code distance) when applied to Abelian 2BGA codes with boundaries and single-shot settings is load-bearing for all optimization results and new code claims, yet the manuscript supplies no explicit derivation, theorem, or circuit-level verification showing that error propagation remains controlled under realistic noise including measurement and reset errors (see the section discussing Abelian 2BGA morphing circuits and the skeptic note on preservation).

    Authors: The core preservation properties of morphing circuits are established in the cited foundational works [1-3]. Our section on Abelian 2BGA morphing circuits extends these by providing explicit circuit constructions for codes with boundaries and single-shot properties, together with stabilizer commutation arguments showing that the logical operators and code distance are unchanged. The stability experiments already incorporate measurement and reset errors and demonstrate improved performance under the morphing construction. We acknowledge that an explicit theorem statement would make the load-bearing claim clearer. We will therefore add a short formal subsection stating the preservation theorem for the 2BGA case with boundaries, including a sketch of the error-propagation argument under depolarizing and measurement noise. revision: yes

  2. Referee: The assertion that alternating syndrome extraction circuits (time-reversed rounds) can be viewed as two-round morphing circuits whose FT properties are 'computationally much easier to examine' lacks a concrete comparison or example demonstrating reduced analysis complexity or improved thresholds relative to non-alternating circuits (see the paragraph on alternating circuits in the abstract and main text).

    Authors: The computational simplification arises because the alternating (time-reversed) schedule reduces to a two-round morphing circuit, allowing direct application of the verification techniques already developed in Refs. [1-3] rather than tracking error propagation across an arbitrary number of rounds. The manuscript illustrates this view in the stability experiments, where the alternating construction yields visibly more stable logical observables under measurement noise. To make the advantage concrete, we will expand the relevant paragraph with a small explicit example using a minimal Abelian 2BGA code: we list the distinct error-propagation cases that must be checked in the non-alternating schedule versus the two-round morphing schedule, showing a reduction from O(n) independent checks to a constant number. This example also indicates how the simplified analysis facilitates threshold estimation against reset and measurement errors. revision: yes

Circularity Check

1 steps flagged

Morphing optimization technique originates in self-cited prior work but new 2BGA applications add independent content

specific steps
  1. self citation load bearing [Abstract]
    "We show how morphing circuits introduced in Refs. [1-3] provide a way of optimising syndrome extraction circuits and codes directly in terms of connectivity, choice of two-qubit gate (ISWAP versus CNOT) and number of physical qubits. We discuss morphing circuits in code optimisation among Abelian two-block group algebra (2BGA) codes, handling boundaries for 2D codes, codes with single-shot properties, and improving performance in stability experiments against measurement and reset errors."

    The optimization framework and its claimed preservation of logical equivalence/fault tolerance are taken from the authors' prior self-cited work [3] on morphing circuits; the present paper's new codes and performance claims for 2BGA codes with boundaries/single-shot properties inherit those properties without re-deriving them independently here.

full rationale

The paper's central method of optimizing syndrome extraction via morphing circuits is introduced and justified in the authors' own prior Ref. [3], with this work extending the technique to Abelian 2BGA codes, boundaries, and single-shot settings. While the applications to new code families and stability experiments provide non-trivial independent analysis, the load-bearing preservation of logical equivalence and fault tolerance under morphing reduces to properties defined in the self-citation. This yields moderate circularity without fully collapsing the derivation to tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the primary unstated premise is that morphing circuits can be applied without compromising code properties. No explicit free parameters, new entities, or additional axioms are detailed.

axioms (1)
  • domain assumption Morphing circuits preserve the logical equivalence and error correction properties of the original codes
    Required for optimization to be valid; invoked implicitly when claiming improved circuits for 2BGA codes and stability experiments.

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