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Optimising Quantum Error Correction Using Morphing Circuits

Mackenzie H. Shaw, Barbara M. Terhal

arXiv:2604.09797v2quant-ph

TL;DR

The paper addresses the gap between defining QEC codes and specifying how their syndrome-extraction circuits are physically executed. It uses morphing circuits to optimize codes and extraction circuits across connectivity, gate choice, qubit count, boundaries, and noise, finding improved circuits and a simpler analysis for alternating rounds.

  • Problem

    QEC codes are traditionally defined and searched without specifying how their syndrome-extraction circuits are executed using elementary gates and measurements.

  • Method

    The paper develops morphing-circuit methods for optimizing QEC codes and syndrome-extraction circuits, including detector analysis, boundary design, and biased-noise stability experiments.

  • Results

    The methods find Abelian 2BGA circuits with strictly improved code parameters over bare-ancilla circuits, improved colour-code morphing circuits, and alternating-round circuits whose distance analysis is computationally cheaper.

  • Takeaways & Limitations

    Morphing circuits provide a practical framework for designing QEC codes and syndrome-extraction circuits jointly with hardware connectivity and fault-tolerance requirements.

  • Takeaways & Limitations

    The reported circuit-level distances may be lower than the plotted distances, and detailed numerical studies to identify specific codes of interest remain future work.

Abstract

from arXiv · show

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.

1 Introduction

This work develops morphing circuits as an alternative to bare-ancilla syndrome extraction, optimizing QEC circuits and codes for hardware connectivity and other implementation constraints. It introduces systematic searches and analyses that find improved circuits, simplify fault-tolerance evaluation, and address boundaries, leakage, single-shot properties, and biased measurement/reset noise.

  • Morphing circuits: Morphing circuits use a QEC code as input but produce syndrome extraction circuits that move between potentially different QEC codes.They have also been called middle-out, dynamic, or ancilla-free circuits.
  • Morphing circuits: Compared with bare-ancilla circuits, morphing can lower connectivity requirements, reduce leakage, and adapt to qubit and coupler dropouts.These advantages have been demonstrated across surface, colour, and Abelian 2BGA codes.
  • Code optimisation: Morphing-circuit searches for Abelian 2BGA codes found code parameters that strictly improve upon bare-ancilla circuits.The authors describe this as the first systematic demonstration of such an improvement for morphing circuits.
  • Code optimisation: Manual boundary optimisation improved colour-code morphing-circuit parameters and produced multiple circuits potentially useful in QEC experiments.The optimisation targets the boundary geometry of the morphing circuits.
  • Fault-tolerance analysis: Alternating time-reversed syndrome-extraction rounds form a two-round morphing circuit whose circuit-level distance is determined by errors within one round.This property makes calculating circuit-level distance computationally much cheaper than for non-alternating circuits.
  • Fault tolerance and noise: The paper develops general detector-setting methods, including for single-shot codes, and uses them to bound lattice-surgery rounds and improve performance under biased measurement and reset noise.For biased-noise stability experiments, the authors seek circuits in which measurement errors trigger more than two defects.

2 Morphing Circuits: Features and Variations

Morphing circuits organize QEC around a mid-cycle code and contracting rounds that produce end-cycle codes, enabling circuit design, detector construction, and fault-tolerance analysis. The framework also connects alternating syndrome extraction to two-round morphing circuits and supports code, connectivity, and decoding analyses.

  • Morphing design principle: Two-round morphing circuits use a mid-cycle code C and contracting circuits to produce end-cycle codes with associated syndrome extraction circuits.The mid-cycle-code perspective supports design, while the end-cycle-code perspective emphasizes the resulting circuits and codes.
  • Code parameters: End-cycle distances are often the limiting factor in a morphing circuit’s circuit-level distance because end-cycle codes contain fewer qubits than the mid-cycle code.This observation motivates designing morphing circuits with larger circuit-level distance.
  • Fault-tolerance analysis: A two-round morphing circuit’s circuit-level distance can be determined from a one-round memory experiment, substantially reducing the computational cost of distance calculation.Errors spanning multiple rounds can be folded into errors spanning one measurement round, and propagated errors can then be tested against the mid-cycle code.
  • Detectors and decoding: Morphing detectors may involve measurements from multiple rounds, and measurement errors can trigger more than two detectors, requiring decoder adaptations.For two-round circuits, detector regions can still have duration two, matching bare-ancilla circuits.
  • Detectors and time-overhead: Two-round morphing circuits have the same lattice-surgery time overhead as normal bare-ancilla circuits because their detectors last the same duration.More than two contraction rounds can require multiple measurement rounds to measure each end-cycle code’s stabilisers.
  • Alternating circuits and constructions: Alternating normal syndrome extraction circuits can be viewed as two-round morphing circuits, while contraction-tree diagrams specify purely contracting morphing circuits.Redundancies in the mid-cycle code typically produce redundancies in each end-cycle code, supporting stability experiments and single-shot error correction.

3 Constructing Morphing Circuits

The paper develops general and practical methods for constructing morphing circuits, including disjoint constructions, simultaneous contraction searches, and numerical searches over Abelian 2BGA codes. These methods produce circuits with improved code parameters and lower connectivity than bare-ancilla circuits, while important distance and contraction-round limitations remain.

  • 3.1 Existence of Disjoint Morphing Circuits: Disjoint morphing circuits exist for arbitrary quantum LDPC code families with disjoint stabiliser supports and a constant number of contraction rounds.Their contraction circuits can use minimal-depth contraction trees, with each end-cycle code distance scaling as Ω(d) relative to the mid-cycle distance.
  • 3.1 Existence of Disjoint Morphing Circuits: For surface and toric codes, the disjoint construction uses J = 4 rounds, while colour and studied weight-6 bivariate bicycle codes use J = 6 rounds.Constant-depth J = 2 circuits are known for these families, but the conditions guaranteeing such circuits for growing code families remain unknown.
  • 3.2 Practical Construction: Simultaneous contractibility of stabilisers provides a practical way to place contraction trees into fewer rounds and fully specify a morphing circuit.The conditions require parallelisable CNOT layers and properness, meaning neither contracting circuit disrupts the other stabiliser's support.
  • 3.3 Case Study: Numerical Search for Morphing Abelian 2BGA Codes: The numerical search brute-forces two-round purely contracting homomorphism-based circuits for Abelian 2BGA mid-cycle codes and evaluates the resulting end-cycle parameters.The two end-cycle codes are identical up to a qubit permutation, and the end-cycle code remains an Abelian 2BGA code.
  • 3.3 Case Study: Numerical Search for Morphing Abelian 2BGA Codes: Morphing circuits strictly improve the corresponding bare-ancilla code parameters for the searched weights, while requiring connectivity w −1 instead of w.Minimal end-cycle and mid-cycle codes need not coincide, so optimising the two cycle types can produce different outcomes.
  • 3.3 Case Study: Numerical Search for Morphing Abelian 2BGA Codes: The reported code distances are end-cycle distances and may exceed the circuit-level distances, which the authors leave for future numerical study.The authors additionally check that the mid-cycle distance is at least the end-cycle distance as a tighter bound for morphing circuits.

4 Adapting Morphing Circuits For Leakage Reduction

The paper adapts morphing circuits to reduce leakage by changing measurement assignments, adding teleportation gadgets when needed, and replacing CNOTs with CXSWAP gates. These adaptations preserve the end-cycle code up to qubit permutations while introducing specific detector, boundary, or qubit-count trade-offs.

  • 4.1 Choosing between Data and Ancilla Qubits: Morphing circuits can be adapted so every qubit is measured in at least one contraction round, supporting leakage removal through measurement and reset operations.If direct reassignment is insufficient, a single-bit teleportation gadget moves the qubit state to an added ancilla before measurement.
  • 4.1 Choosing between Data and Ancilla Qubits: Prop. 6 allows selected final-layer CNOT directions to be reversed while preserving the end-cycle code up to a qubit permutation.The transformation can alter conditional Pauli corrections and therefore the detector structure, even though the code itself is preserved.
  • 4.1 Choosing between Data and Ancilla Qubits: The all-qubits-measured adaptation preserves the end-cycle code but may change detector structure through conditional Z operations.This detector change is an explicit limitation of the measurement-assignment transformation.
  • 4.1 Choosing between Data and Ancilla Qubits: For the surface code, only one additional qubit is needed for the all-qubits-measured construction, unlike the walking circuit's O(d) additional qubits.The walking circuit nevertheless provides a spatially shifted patch, which is needed for chaining macroscopic movement steps.
  • 4.2 CXSWAP Gates: Any CNOT morphing circuit can be rewritten using CXSWAP gates while preserving its end-cycle code up to qubit permutations.The construction recursively inserts SWAP operations so each compiled layer consists of simultaneously executable CXSWAP gates.
  • 4.2 CXSWAP Gates: The hex-grid surface-code CXSWAP construction uses the same number of qubits as its CNOT circuit but changes the boundary connections required.CXSWAP is equivalent to ISWAP up to single-qubit Clifford gates and may be less prone to leakage than CZ on superconducting platforms.

5 Improving the End-Cycle Code Distance of Morphing Circuits

The section improves morphing-circuit end-cycle distance by optimising boundaries and, in some codes, increasing contraction rounds, while balancing qubit efficiency, connectivity, and time overhead.

  • Boundary optimisation: Optimising end-cycle boundaries can reduce the physical-qubit count needed for a target distance, even when the mid-cycle code has suboptimal boundaries.The optimisation transfers boundary conditions from the end-cycle code back to the mid-cycle code.
  • Colour-code case study: Hex-grid colour-code morphing uses connectivity three rather than six and can achieve more qubit-efficient circuits by optimising the end-cycle geometry.The optimised circuits retain hex-grid connectivity and can reach circuit-level distance equal to the end-cycle distance.
  • Colour-code case study: The weight-7 diamond circuit uses n_tot = 3(d_end)^2 + O(d_end) physical qubits to encode two logical qubits, making it more qubit-efficient than the fault-tolerant triangular weight-6 circuit.Its multi-logical-qubit encoding limits implementation of the full two-qubit Clifford group using mid-cycle transversal gates.
  • Additional contraction rounds: Increasing contraction rounds can raise end-cycle distance by reducing the number of rows contracted per round, but stabilisers are then measured less frequently.The Bacon-Shor example realises this trade-off through additional contraction rounds.
  • Additional contraction rounds: More contraction rounds increase lattice-surgery time overhead, and examples where they improve end-cycle distance beyond Bacon-Shor and diamond surface-code circuits are not generally straightforward to find.The paper identifies the time overhead as the principal cost of this strategy.

6 Circuit-Level Distance for Stability Experiments using Morphing Circuits

This section derives circuit-level stability-distance bounds for morphing circuits and shows how detector structure, contraction-round ordering, and single-shot properties determine fault-tolerant performance.

  • General bounds: For a depth-T stability experiment, the circuit-level distance is bounded using detector duration and the number of contractions of each mid-cycle stabiliser.These propositions provide general bounds for morphing-circuit stability experiments.
  • Two-round circuits: Two-round morphing circuits have coincident upper and lower bounds, making their stability distance exactly computable and formally time-like equivalent to complete bare-ancilla circuits.This applies to the paper’s hex-grid surface-code, colour-code, and Abelian 2BGA examples.
  • More than two rounds: For J > 2, stability-distance bounds need not coincide; two J = 4 toric-code circuits can have d_stab(T) = floor(T/3) or floor(T/2) − 1.The difference arises from the ordering of contracting operations.
  • Dropout-compatible circuits: Grouping gauge checks within the same contracting subsets can improve the lower bound to d_stab(T) ≥ T/2 − O(1), reducing lattice-surgery rounds for target distance d to at most 2d + O(1).Without this grouping, the stated bound is d_stab(T) ≥ T/4 − O(1).
  • Single-shot error correction: Single-shot codes remain single-shot under morphing: their morphing-circuit stability distance matches the phenomenological distance up to a constant factor.Thus, when the phenomenological distance scales with d for T = O(1), the morphing circuit retains single-shot behaviour.

7 Morphing Circuits under Biased Measurement versus Reset Noise

The section investigates whether non-string-like measurement structures improve stability experiments when measurement and reset noise are biased. Such structures can increase protection against measurement errors, but added circuit depth can offset the benefit in the toric code.

  • Colour code: In the colour code, non-string-like circuits improve logical performance without increasing syndrome-extraction depth.The colour-code comparison uses two circuits with identical end-cycle codes but different measurement structures.
  • Noise model: The study varies measurement noise using a modified SI1000 model, with η = 0.4 denoting equal measurement and reset error rates.The model is designed to assess resilience to measurement/reset errors even when CNOT-error resilience does not improve.
  • Mechanism: Non-string-like structures spread measurement-error effects across more detectors, allowing measurement-error-only stability distance to exceed circuit-level stability distance.This distinction arises because measurement and reset errors can produce different syndromes in morphing circuits.
  • Colour code: Transversal (HS†)^⊗n rounds improve both colour-code circuits, while the non-string-like circuit has a lower logical error rate than the string-like circuit outside the 99% confidence intervals.The transversal gate makes X, Z, and Y stabilisers appear in consecutive rounds rather than alternating only between X and Z.
  • Toric code: In the toric code, additional CNOT gates create non-string-like structures by commuting with contracting but not expanding stabilisers.The added gates increase contraction-circuit depth, so the advantage appears only for sufficiently large η.

8 Discussion

The discussion identifies scope boundaries and open design problems for morphing circuits. These include extending results beyond stabiliser codes, handling alternative reset assumptions, and automating circuit and boundary searches.

  • Scope: The formal results are proved for stabiliser codes, while broader consequences for subsystem and Floquet codes remain unexplored.The authors note that generalisations may require corresponding changes to the definitions and analysis.
  • Experimental assumptions: The analysis assumes unconditional resets, whereas avoiding resets after projective measurements could be advantageous on some experimental platforms.Under that alternative, a classical measurement error can halve the circuit-level stability-distance upper bound.
  • Scope: Non-string-like measurement structures may also benefit superdense circuits under biased noise models, but this possibility is left for future work.The discussion notes that some superdense-circuit measurement errors trigger more than two detectors.
  • Open problems: Adapting ancilla-overhead-efficient Pauli-based computation methods to morphing remains open because mid-cycle qubits and stabilisers lose their group-element labels.That loss makes numerical searches for fault-tolerant reduced-connectivity two-round circuits more difficult.
  • Automation: Automated searches over morphing circuits and boundary geometries remain challenging because the valid contraction-tree and boundary design spaces are large.The paper describes existing connectivity-aware search ideas but leaves efficient search-space design and automated boundary construction unresolved.

B Propositions and Their Proofs

The appendix proves structural and distance results for morphing circuits, including time-reversal reductions, detector construction, and bounds for stability and code-family constructions. These results formalise why alternating circuits can be analysed through shorter experiments and how detector structure controls distance.

  • Alternating circuits: For corresponding alternating and non-alternating normal circuits, the alternating circuit cannot have a lower circuit-level distance.The proof folds arbitrary multi-round logical errors into one-round errors, yielding the relevant distance comparison.
  • Time reversal: Time reversal maps an undetectable logical error spanning multiple rounds to an undetectable error confined to one measurement round without increasing its weight.Repeated reflection preserves the logical effect, while cancellations can reduce the resulting weight.
  • Detector construction: A circuit-level error is analysed by propagating it to the mid-cycle code, whose syndrome identifies regular detectors and whose redundancy parity can trigger meta-check detectors.Constant-depth contraction circuits and LDPC structure keep the propagated error and syndrome constant-weight.
  • Distance bounds: For constant-round, constant-depth morphing circuits on quantum LDPC codes, the morphing stability distance satisfies dmorph_stab(T) = Ω(1) with respect to T and the code distance d.The statement assumes T is an integer multiple of the number of contraction rounds J.
  • Distance bounds: The circuit-level distance of a stability experiment has a lower bound determined by the temporal duration of regular detectors and an upper bound determined by stabiliser-contraction counts.The lower-bound proof uses non-overlapping observing regions, while the upper-bound proposition tracks how often each mid-cycle stabiliser is contracted.
  • Code constructions: Disjoint morphing circuits exist for any quantum stabiliser code in an LDPC family, with the original code as the mid-cycle code and code-family-scale contraction structure.The proposition also specifies contraction rounds and end-cycle-code structure for the construction.

C.2 The Lattice Presentation of Abelian 2BGA Codes

The lattice presentation redefines Abelian 2BGA codes using lattice data, making their structure, symmetries, and systematic enumeration more transparent.

  • Lattice presentation: An Abelian 2BGA code is specified by a lattice presentation (mA, mB, Λ, ncopies), where Λ is a full-rank lattice in Z^m.The associated group is G = Z^m/Λ, with m = mA + mB.
  • Lattice presentation: The lattice-to-group construction uses cosets [ei], with A and B formed from selected canonical-basis cosets.Cosets may coincide when ei − ej ∈ Λ.
  • Code size: For ncopies > 1, the code consists of multiple copies of the ncopies = 1 code, and n = 2|G| = 2 det(Λ).The determinant refers to a generating-vector matrix for Λ.
  • Completeness: Every Abelian 2BGA group presentation can be converted into a lattice presentation using the subgroup generated by A and B and its relations.The quotient by that subgroup determines ncopies.
  • Enumeration: Hermite Normal Form enables systematic lattice enumeration while reducing presentation symmetries that scale with group size.The remaining symmetries include ZX-duality and permutations of A and B.
  • Geometric interpretation: A code with stabiliser weight w is nearest-neighbour in w − 2 dimensions, and scaling Λ defines code families.Periodic boundary conditions are determined by Λ, with ncopies qubits per location when ncopies > 1.

C.3 Homomorphism-Based Morphing Circuits

Homomorphism-based morphing circuits restrict two-round Abelian 2BGA searches using translation and ZX-duality symmetries, making circuit construction and verification tractable.

  • Search framework: The search restricts the intractably large space of two-round morphing circuits to homomorphism-based circuits for fixed stabiliser weight and qubit count.This restriction does not guarantee a purely contracting CNOT circuit in general.
  • Definition: A homomorphism-based circuit requires contracting rounds to be closed under qubit translations and translated ZX-duality.ZX-duality uses H⊗nSWAPZX and maps s(X, g) to s(Z, g−1).
  • Definition: The contracting X- and Z-stabilisers in each round are preimages under a homomorphism φ of selected elements of an Abelian group H.This homomorphism condition simplifies the contracting subsets.
  • Examples: For the unrotated toric code, two-round purely contracting circuits use H = Z2 and J = 2, with SWAPZX implementing the required rotation.The example has G = Z4 × Z4 at the stated size.
  • Scope: Many Abelian 2BGA codes lack two-round purely contracting morphing circuits; odd-order G makes one defining condition impossible.The BB-code circuits provide a contrasting example where such circuits exist.
  • End-cycle structure: Two-round homomorphism-based circuits have end-cycle codes equal up to qubit permutation, and those codes are themselves Abelian 2BGA codes.The equality follows from translational symmetry.

C.4 Numerical Search Results for Weight-5 and 6 Abelian 2BGA Codes

The numerical search examines two-round homomorphism-based purely contracting morphing circuits for weight-5 and 6 Abelian 2BGA codes, finding circuits that match or exceed bare-ancilla parameters with lower connectivity.

  • Search results: The search covers Abelian 2BGA codes of fixed stabiliser weight and qubit count using the methods developed for two-round homomorphism-based circuits.The complete numerical results are shown in Figs. 25 and 26.
  • Figures: Figure 25 presents the full set of minimal codes found up to k = 12.The search methods are those outlined in Section 3.3 and Appendix C.
  • Figures: Figure 26 adds minimal codes with d = 3 or 4, including all unplotted minimal codes with k > 12.These codes were found by the numerical search.

C.5 Infinite Abelian 2BGA Codes

Infinite Abelian 2BGA codes extend the lattice framework to non-full-rank lattices, supporting morphing circuits for infinite surface and colour codes and compatible boundary optimisation.

  • Infinite codes: An infinite Abelian 2BGA code has an infinite group G, which corresponds to a lattice Λ that is not full rank.The infinite surface code uses G = Z × Z, A = {1, x}, and B = {1, y}.
  • Examples: The infinite hexagonal-lattice colour code is represented by G = Z × Z, A = {1, x, y}, and B = A−1.These constructions describe codes on infinite planes before boundaries are added.
  • Morphing circuits: Homomorphism-based morphing circuits remain valid for infinite codes when H is finite, ensuring finitely many contraction rounds.The finite-group requirement is added to Item (c).
  • Boundary optimisation: For two-round homomorphism-based circuits, the end-cycle codes are equal and are themselves infinite Abelian 2BGA codes.This makes the optimal boundary conditions for both end-cycle codes the same.

D Optimising Distance versus Number of Qubits for the End-Cycle Codes of Morphing Colour Code Circuits

The appendix optimises end-cycle boundaries by analysing logical error-string geometry and matching it to optimal planar code boundaries. Weight-6 follows colour-code geometry, while weight-5 and weight-7 follow square-grid geometry and favour diamond boundaries.

  • Boundary optimisation: Boundary optimisation begins by analysing each end-cycle code’s logical error-string structure, then matching it to known optimal planar code boundaries.The method compares error-string geometry with rotated surface-code and triangular colour-code boundaries.
  • Weight-6 end-cycle code: Weight-6 end-cycle error strings resemble colour-code strings, so triangular colour-code boundaries provide the relevant optimal geometry.The analysis identifies same-colour stabiliser pairs as building blocks of the error strings.
  • Weight-5 end-cycle code: Weight-5 error strings move along a square grid despite having three colours, motivating diamond boundaries analogous to the rotated surface code.The proposed diamond boundaries can be formed by alternating X- and Z-type boundaries.
  • Weight-7 end-cycle code: Weight-7 error strings also move along a square grid when four-detector strings are allowed, leading to diamond boundary conditions.The resulting error-string structure matches that of the weight-5 end-cycle code.

D.1 Optimal Morphing Circuits for Weight-4 Abelian 2BGA Codes

The appendix complements numerical searches over weight-5 and weight-6 Abelian 2BGA morphing circuits with an optimisation procedure for weight-4 circuits. The resulting hex-grid circuits achieve optimal end-cycle parameters for the stated circuit class.

  • Weight-4 Abelian 2BGA codes: For purely contracting, homomorphism-based two-round morphing circuits with weight-4 mid-cycle codes, the corresponding end-cycle codes are also weight-4 Abelian 2BGA codes.This property was checked numerically for all circuits in the stated class.
  • Optimal circuits: The hex-grid toric-code morphing circuits use mid-cycle codes whose end-cycle codes coincide with rotated toric codes.The table distinguishes even- and odd-distance constructions.
  • Optimal circuits: These two-round, purely contracting, homomorphism-based circuits achieve optimal end-cycle parameters for weight-4 Abelian 2BGA codes.The claim applies to the circuit class described in the table caption.

E Circuit-Level Distance and Numerical Simulations of Morphing Circuits for Colour Codes

This section analyses circuit-level distance and numerically simulates optimised morphing circuits for planar hexagonal-lattice colour codes. The evaluated properties are summarised across the circuits in Table 6 using memory experiments.

  • Circuit-level distance: The section evaluates circuit-level distance for triangular weight-6 and diamond weight-7 morphing colour codes using one measurement round.The analysis invokes the single-round distance result stated as Proposition 1.
  • Numerical simulations: Numerical results are reported for memory experiments conducted on the optimised colour-code morphing circuits.Table 6 provides an overview of the properties of each analysed circuit.

E.1 Triangular Weight-6 Morphing Colour Code

The triangular weight-6 circuit’s end-cycle boundary optimisation does not by itself guarantee optimal circuit-level distance because circuit-level error mechanisms can shorten logical errors. Extra gate ordering and boundary re-optimisation address these mechanisms, but performance comparisons remain partly unresolved and use more qubits than a superdense alternative.

  • Circuit-level limitations: End-cycle boundary optimisation can yield lower circuit-level distance when circuit-level errors create shorter effective logical error strings.One mechanism combines three circuit-level errors into an end-cycle weight-four string, reducing some weights by a factor of 3/4.
  • Circuit modifications: The Extra Layer circuit reorders CNOT gates, adding one contraction-round layer without adding CNOT gates, while the Extra Qubits circuit also re-optimises boundaries using additional qubits.The Extra Qubits circuit was checked for distances 3, 5, 7, and 9; Extra Layer reaches dcirc = ⌈3dend/4⌉ at higher distances.
  • Numerical comparison: The Extra Qubits circuit outperforms the Original and Extra Layer circuits for dend = 5 and 7, although the source does not isolate the contribution of circuit-level distance from other effects.Further investigation at lower physical error rates is identified as necessary.
  • Open direction: The authors were investigating a newer triangular weight-6 construction that uses fewer qubits than Extra Qubits, but its status is not established here.The construction was described as a new idea under active investigation during manuscript preparation.
  • Comparison with existing circuits: Both morphing circuits significantly outperform the superdense colour-code circuit at the same distance, but use significantly more physical qubits.The comparison is presented in Fig. 31(b) and Table 1.

E.2 Diamond Weight-7 Morphing Colour Code

The diamond weight-7 morphing colour code is analysed through propagated circuit-level errors and boundary optimisation. The resulting circuits are numerically comparable to the triangular weight-6 circuit at the tested distances.

  • Circuit analysis: The authors conjecture that optimised diamond weight-7 boundary conditions achieve dcirc = dend.Circuit-level errors are propagated to the mid-cycle code to inspect possible error mechanisms.
  • Circuit analysis: Circuit-level errors are represented at the mid-cycle code after forward- or backward-propagation through the morphing circuit.The analysis classifies propagated errors using a contraction-tree representation and mid-cycle code.
  • Boundary construction: Pauli boundaries were chosen because they introduce fewer irregularities in the diamond boundary geometry.Constructing explicit planar boundaries that preserve circuit-level distance was challenging.
  • Boundary construction: A brute-force numerical search selected diamond boundaries by varying the y-intercepts and gradients of the four boundary edges.The resulting arrangements and circuit sizes are reported in Fig. 30 and Table 7.
  • Numerical performance: Diamond weight-7 morphing circuits had numerical performance comparable to the triangular weight-6 circuit at the three tested distances.The comparison was performed in a numerical benchmark of the diamond weight-7 circuits.
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