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Optimising Quantum Error Correction Using Morphing Circuits

Mackenzie H. Shaw, Barbara M. Terhal

arXiv:2604.09797v2quant-ph

TL;DR

QEC codes are often searched without specifying how their syndrome-extraction circuits are implemented, leaving code parameters and circuit design to be considered together. This paper uses morphing circuits to optimize such designs and reports Abelian 2BGA circuits with code parameters that strictly improve on bare-ancilla circuits.

  • Problem

    For morphing colour codes, boundary geometry can leave a mid-cycle-optimized design requiring more physical qubits at a given circuit-level distance, motivating end-cycle boundary optimization.

    Source: §5.1 · p. 20, §5.2 · p. 21

  • Method

    The paper searches morphing circuits for Abelian 2BGA codes, optimizes morphing-circuit boundaries, and analyzes alternating time-reversed syndrome rounds as two-round morphing circuits.

    Source: §1 · p. 2

  • Results

    The search found Abelian 2BGA morphing circuits with code parameters strictly better than bare-ancilla circuits, and boundary optimization improved colour-code morphing-circuit parameters.

    Source: §1 · p. 2

  • Takeaways & Limitations

    For alternating time-reversed rounds, circuit-level distance is set by errors within one syndrome round, drastically reducing the computational cost of calculating it.

    Source: §1 · p. 2

    Circuit-level distances for both bare-ancilla and morphing circuits may be lower than shown, and detailed numerical studies to identify specific codes remain future work.

    Source: §3.3 · p. 16

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

The paper studies morphing as an alternative to bare-ancilla syndrome extraction: circuits take a QEC code as input and move between codes while measuring stabilisers. It develops methods and case studies for circuit construction, fault tolerance, and performance.

1 Introduction

  • The search finds Abelian 2BGA morphing circuits with 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 in the literature, as far as they are aware.Source: §1 · p. 2
  • The paper optimises colour-code morphing-circuit boundary geometry, improving the parameters of circuits from Ref. [20] and providing multiple circuits for QEC experiments.Source: §1 · p. 2
  • Alternating time-reversed syndrome-extraction rounds form a two-round morphing circuit whose distance is set by errors in one round and is never lower than its non-alternating counterpart.This single-round characterization makes calculating circuit-level distance substantially less computationally costly.Source: §1 · p. 2
  • The authors give general detector-setting methods, including for single-shot codes, and use them to bound lattice-surgery rounds and improve performance under biased measurement and reset errors.Source: §1 · p. 2

1.1 Relationship to Existing Literature

  • The first surface-code morphing circuit [36] demonstrated reduced connectivity requirements, compilation using CXSWAP rather than CNOT gates, and swapping data and ancilla qubits each QEC round.End-cycle BB codes had distance and qubit-count parameters matching another code in the BB family. Morphing addressed qubit and coupler dropouts in surface codes [28], and this scheme was generalized to arbitrary CSS codes.Source: §1.1 · p. 3

1.2 Some Definitions

  • The paper calls a circuit normal when it measures a complete or overcomplete stabiliser-generator set once using ancillas; bare-ancilla circuits use one ancilla per measured stabiliser.The analysis restricts itself to CSS codes with purely X and purely Z checks, and treats resets and measurements separately because their experimental error rates may differ.Source: §1.2 · p. 3

2 Morphing Circuits: Features and Variations

This section defines morphing circuits for stabilizer codes and examines how their structure affects circuit-distance analysis, end-cycle codes, connectivity, detectors, and decoding. It explains how morphing accommodates redundancies and alternating extraction circuits, extends to subsystem and Floquet codes, and motivates additional contraction rounds for robust syndrome measurement.

2.1 The Morphing Design Principle

  • Morphing measures stabilizers without ancillas by partitioning generators into independent contracting subsets, mapping each subset to single-qubit Paulis, measuring and resetting those qubits, then undoing the contraction.Two rounds are the smallest useful choice: one subset requires decoding the code, while one stabilizer per subset measures stabilizers infrequently.Source: §2.1 · p. 4

2.2 Morphing Circuits

  • End-cycle-code distances often, but not always, limit a morphing circuit’s circuit-level distance, motivating efforts to improve those codes.The end-cycle codes have fewer qubits than the mid-cycle code; examples include rotated surface-code end cycles and lower-distance hexagonal colour-code end cycles.Source: §2.2 · p. 5

2.2.1 The End-Cycle-Code Perspective

  • The end-cycle perspective treats each round as syndrome extraction while transitioning between end-cycle codes; adjacent contraction layers may compile together so the state never occupies the mid-cycle code.Source: §2.2.1 · p. 5

2.2.2 Computational Cost of Determining The Circuit-level Distance

  • A two-round morphing circuit’s circuit-level distance can be determined from a one-measurement-round memory experiment, reducing the computational cost of the calculation.Time-reversal symmetry folds any undetectable logical error spanning multiple rounds into one round without increasing its weight.Source: §2.2.2 · pp. 5–6

2.2.3 Connectivity

  • Morphing can reduce connectivity, but not inherently: prior 2BGA circuits connect each qubit to w −1 others versus degree w for bare-ancilla circuits, while a 3D toric-code circuit increases connectivity.Reduced-connectivity circuits were obtained for most, but not all, BB codes [8].Source: §2.2.3 · p. 6

2.3 Detectors and the Time-Overhead of Morphing Circuits

  • Two-round morphing detectors span two measurement rounds, matching bare-ancilla duration, but may combine varying numbers of outcomes and let measurement errors trigger more than two detectors.With more than two contraction rounds, typically more than one measurement round is needed to measure each end-cycle code’s stabilizers, affecting lattice-surgery time.Source: §2.3 · p. 7

2.3.1 Decoding

  • For constant-depth J-round morphing circuits measuring every generator exactly once, each circuit-noise hyperedge decomposes into a constant number of phenomenological-noise hyperedges.This permits applying phenomenological-noise decoders, but decomposition can lose correlations; bare-ancilla decoders do not immediately transfer because errors may trigger more than two detectors.Source: §2.3.1 · p. 8

2.3.2 Redundancies and Stability Experiments

  • Mid-cycle redundancies typically induce a redundancy in each end-cycle code and provide extra detectors that support stability experiments and single-shot correction.Separate detectors can be formed from the contracting stabilisers in each subset using measurements from one round; they enable single-shot correction when the mid-cycle code is single-shot.Source: §2.3.2 · pp. 8–9

2.4 Purely Contracting Morphing Circuits as Contraction Tree Diagrams

  • Purely contracting circuits require every CNOT layer to contract at least one stabiliser without expanding any contracting stabiliser, and their contraction-tree diagrams uniquely specify the circuit.Each tree represents a stabiliser’s contraction, with vertices for its qubits, edges for contracting CNOTs, and a root for the measured qubit.Source: §2.4 · p. 9

2.5 Alternating Normal Syndrome Extraction Circuits are Two-Round Morphing Circuits

  • Alternating normal circuits have circuit-level distance at least that of corresponding non-alternating circuits, while reducing distance analysis to a single round.They have the same depth, CNOT count, and connectivity requirements, and can have larger circuit-level distance; this does not guarantee better logical performance at every physical error rate.Source: §2.5 · p. 11

2.6 Beyond Two-Round Morphing for Stabiliser Codes

  • The framework moves beyond two-round morphing for stabiliser codes to address cases such as qubit dropouts, where subsystem or Floquet codes and additional rounds may be useful.The paper motivates these extensions as ways to generalise morphing to subsystem and Floquet codes and to more than two contraction rounds.Source: §2.6 · p. 12

2.6.1 Morphing for Subsystem and Floquet Codes

  • Morphing extends to subsystem and Floquet codes by compiling sequences of commuting, independent multi-qubit Pauli measurement sets and defining mid- and end-cycle codes as subsystem codes.The generalised codes use detector-region time-slices as stabilisers and operators that flip neither detectors nor logical operators as gauge operators.Source: §2.6.1 · p. 12

2.6.2 The Need for More than Two Contraction Rounds

  • More than two contraction rounds are straightforward to construct but increase lattice-surgery measurement rounds and preclude the efficient circuit-distance method for two-round circuits.They can be needed for subsystem codes, whose contracting subsets must contain mutually commuting operators, and can improve end-cycle distance, as in Bacon–Shor.Source: §2.6.2 · p. 12
  • When stabilisers outnumber mid-cycle qubits, using half the qubits as ancillas cannot measure half the stabilisers in one contraction round, necessitating J > 2.This applies to codes whose locally redundant parity checks must all be measured for robust syndrome measurement.Source: §2.6.2 · p. 13

3 Constructing Morphing Circuits

The section presents constructive and practical methods for designing morphing circuits, applies a numerical search to Abelian 2BGA codes, and extends the construction to finite topological codes with boundaries.

3.1 Existence of Disjoint Morphing Circuits

  • For any stabilizer code in an LDPC family, Proposition 4 establishes a disjoint morphing circuit with the code as its mid-cycle code and a constant number of contraction rounds.The disjoint construction can be suboptimal in round count: it gives J = 4 for surface and toric codes and J = 6 for colour codes and the studied weight-6 bivariate bicycle codes; whether growing families admit constant-depth J = 2 circuits remains open.Source: §3.1 · p. 13

3.2 Constructing Purely Contracting Morphing Circuits Using Contraction Tree Diagrams

  • A practical design method groups stabilizers into rounds when their chosen contraction trees are simultaneously executable: gate layers are parallelisable and neither circuit affects the other's stabilizer.Once the sets of simultaneously contractible stabilizers are identified, they specify the full contraction tree diagram and morphing circuit.Source: §3.2 · pp. 13–14

3.3 Case Study: Numerical Search for Morphing Abelian 2BGA Codes

  • At equal stabilizer weight, all minimal morphing end-cycle code parameters found strictly improve on bare-ancilla parameters, with connectivity w −1 rather than w.The reported code distance is the end-cycle distance and may exceed circuit-level distance; the authors also check that mid-cycle distance is at least the end-cycle distance.Source: Figure 11 · p. 15, §3.3 · p. 16
  • The Abelian 2BGA search found minimal end-cycle codes with strictly better parameters than same-weight bare-ancilla codes, using connectivity w −1 rather than w.The search compared end-cycle parameters; circuit-level distance may be lower, and morphing circuits may have greater depth without typically requiring more CNOT gates per measurement round.Source: Figure 11 · p. 15, §3.3 · p. 16

3.4 Adding Boundaries to Topological Codes

  • For finite topological codes, create boundaries by measuring beyond-boundary qubits in a boundary-specific basis or Bell pairs, then adapt contraction trees by truncating or padding them to rooted subtrees.Padding reinserts selected stabilizers and qubits when removed support prevents a finite-code contraction tree from being a rooted subtree of its infinite-code counterpart.Source: §3.4 · pp. 16–17

4 Adapting Morphing Circuits For Leakage Reduction

The section presents two leakage-reduction adaptations that preserve the end-cycle code up to qubit permutations: choosing which qubits are measured and compiling CNOT circuits with CXSWAP gates. Both adaptations also apply to alternating normal circuits.

4.1 Choosing between Data and Ancilla Qubits

  • Reversing any CNOT directions in the final contraction layer preserves the end-cycle code up to qubit permutations, enabling choices of which qubits are measured.The conditional Z gate can change detector structure, so the transformation does not preserve the circuit’s detectors.Source: §4.1 · p. 18
  • For the hex-grid surface code, the measurement adaptation needs only one additional qubit, versus O(d) for the walking circuit [36].The construction first tries changing final-layer CNOT directions; if that does not measure every qubit, it uses a single-bit teleportation gadget. Measuring every qubit may remove ancilla leakage without a dedicated data-qubit leakage-reduction operation.Source: §4.1 · p. 18, Figure 14 · p. 19

4.2 CXSWAP Gates

  • Any CNOT morphing circuit, for any number of rounds, can be rewritten using CXSWAP gates while preserving the end-cycle code up to qubit permutations.The CXSWAP gate is equivalent to ISWAP up to single-qubit Clifford gates and may be less prone to leakage than CZ on superconducting platforms [36].Source: §4.2 · p. 19
  • The hex-grid surface-code CXSWAP circuit uses the same number of qubits as its CNOT circuit, but alters boundary connections while keeping maximum qubit degree three.The altered boundary connectivity may prevent the circuit from being a chunk of a larger surface-code lattice because it requires different couplers.Source: §4.2 · pp. 19–20, Figure 15 · p. 20

5 Improving the End-Cycle Code Distance of Morphing Circuits

The section presents two ways to improve morphing circuits’ end-cycle distance: optimise topological-code boundary geometry and use more contraction rounds. Both can improve circuit efficiency or distance, but they entail tradeoffs such as added qubits or less frequent stabiliser measurements.

5.1 Optimising the Boundary Geometry of Topological Morphing Circuits

  • Optimising end-cycle boundaries can reduce the physical-qubit cost for a target distance: choose boundaries to maximise logical-string length for a fixed qubit count, then transfer them to the mid-cycle code.The mid-cycle code may have suboptimal boundaries and additional qubits; its distance must still meet the desired end-cycle distance.Source: §5.1 · pp. 20–21

5.2 Case Study: the Hex-grid Colour Code

  • A boundary-adjusted triangular weight-6 colour-code circuit retains hex-grid connectivity and mid-cycle transversal Clifford gates, and correcting its circuit-level error mechanism makes its circuit-level distance equal its end-cycle distance.The uncorrected circuit has a circuit-level error of weight 3dend/4 + O(1); the corrected version requires additional qubits and is less qubit-efficient than the superdense colour-code circuit.Source: §5.2 · pp. 21–22
  • The diamond weight-7 circuit encodes two logical qubits with ntot = 3(dend)2+O(dend), but cannot implement the full two-qubit Clifford group transversally and has conjectured circuit-level distance equal to its end-cycle distance.It is more qubit-efficient than the fault-tolerant triangular weight-6 circuit, but logical operations would require lattice surgery.Source: §5.2 · p. 22

5.3 Adding More Contraction Rounds: Bacon-Shor Code Example

  • More contraction rounds reduce ancilla use per round and can raise end-cycle distance, but stabilisers are measured less often, increasing time overhead for lattice surgery.The text also identifies four-round surface-code diamond circuits as another example that use fewer physical qubits for a given end-cycle distance than the standard two-round circuit.Source: §5.3 · pp. 23, 25
  • In the Bacon-Shor example, contracting fewer gauge-operator rows per round increases end-cycle distance, so achieving that increase requires more contraction rounds.For mid-cycle distance d, the end-cycle distance is dend = d − Nrows, where Nrows is the number of rows contracted in a round; the same principle applies to Z-gauge operators.Source: §5.3 · pp. 23, 25

6 Circuit-Level Distance for Stability Experiments using Morphing Circuits

This section examines fault-tolerance properties of morphing circuits for stability experiments, formalizing circuit-level distance and establishing conditions for preserving single-shot error correction. It develops detector constructions and bounds, and illustrates distance, detector-duration, and connectivity considerations in example circuits.

6.1 Additional Conventions

  • The section’s experimental conventions assume access to unconditional resets on ancilla qubits.The experiments begin in the end-cycle code; the text notes that reset-free differences are discussed in Section 8.Source: §6.1 · p. 25

6.2 Detectors in Morphing Circuits: Regular and Meta-Check

  • Regular detectors track a stabilizer between consecutive contractions for duration J, while meta-check detectors combine a redundancy’s stabilizers for duration J−1.The detector construction uses mid-cycle detecting-region time-slices that correspond to code stabilizers.Source: §6.2 · pp. 27–28

6.3 Circuit-level Distance of Stability Experiments

  • For depth-T experiments, regular-detector duration δ gives a lower bound floor(T/(δ−1)), while redundancy-participating stabilizers give an upper bound via errors of weight 2N(s).The upper-bound construction places measurement errors on each contraction and reset errors on each following reset for a participating stabilizer.Source: §6.3 · pp. 28–29
  • For two-round morphing circuits measuring each stabilizer once, stability-experiment distance is exactly T, matching complete bare-ancilla circuits in time-like distance.This formalizes the equivalence for circuits including the hex-grid surface code, colour-code circuits, and Abelian 2BGA circuits.Source: §6 · p. 25, §6.3 · p. 29
  • In two J = 4 toric-code circuits, changing only contraction-subset order yields distances ⌊T/3⌋ and ⌊T/2⌋−1, making careful ordering important.The first ordering admits an undetectable error string; the reordered circuit removes that mechanism, with its distance found numerically.Source: Figure 20 · p. 29, §6.3 · pp. 29–30

6.3.1 A Tighter Bound for CSS Morphing Circuits

  • For CSS morphing circuits, a type-specific bound using δX can be tighter than Proposition 7; when X-stabilizers contract in only two rounds, the bounds match up to a constant.The same result holds for X-type circuit-level distance; omitting Z-basis measurements does not change the Z-type distance.Source: §6.3.1 · p. 30

6.3.2 Consequences for Combating Qubit and Coupler Dropouts

  • For LUCI, arranging stabilizers as products of gauge operators from the same contracting subsets improves the lower bound from dstab(T) ≥ T/4 − O(1) to dstab(T) ≥ T/2 − O(1).This implies at most 2d + O(1) measurement rounds to reach distance d, rather than 4d + O(1).Source: §6.3.2 · p. 30

6.4 Single-Shot Error Correction and the 3D Toric Code

  • The section concludes by examining morphing circuits’ interplay with single-shot error correction using the 3D toric code as an example.Source: §6.4 · p. 30
  • In the 3D toric code, cube-wise products of six face Z-stabilizers detect Z-measurement errors in one round, but the code is not single-shot for X-stabilizer measurement errors.The construction places qubits on edges, X-stabilizers at vertices, and Z-stabilizers on faces.Source: §6.4 · p. 30
  • The 3D toric-code morphing circuit uses six contraction rounds; every X-check is measured every two rounds, while every Z-check is measured twice per six rounds.The six circuits are ordered Sx,1, Sy,2, Sz,1, Sx,2, Sy,1, Sz,2.Source: §6.4 · p. 31
  • Because the 3D toric code has more stabilizer generators than qubits, using half the qubits as ancillas requires J > 2; J = 2 instead uses more than half and is expected to reduce end-cycle distance.The contraction subsets in the six-round example contract half of the X-stabilizers but only a third of the Z-stabilizers in one circuit.Source: §6.4 · p. 31
  • Regular X-detectors have duration two, while regular Z-detectors alternate between durations two and four as Z-stabilizers are contracted in different rounds.The duration variation follows from the selected contraction-subset order.Source: §6.4 · p. 31
  • The example requires a degree-eight connectivity graph, compared with degree six for data qubits and X-ancillas and degree four for Z-ancillas in a bare-ancilla circuit, limiting its likely use in superconducting devices.The authors note it may still be of interest in mobile-qubit platforms.Source: §6.4 · pp. 31–32
  • Although its end-cycle code has half as many physical qubits, each mid-cycle redundancy yields an end-cycle redundancy, producing two Z-stabilizer redundancies per cube.An additional weight-six Z-stabilizer traverses each cube.Source: §6.4 · pp. 31–32

6.5 Morphing Circuits Preserve Single-Shot Error Correction

  • Numerical optimization of the 3D toric-code circuit-level stability distance was inconclusive, and BP-OSD upper bounds exceeded an explicit circuit-level logical error constructed by hand.The attempted upper bounds followed Ref. [8].Source: §6.5 · p. 32
  • For quantum LDPC mid-cycle codes with constant J and constant-depth contractions, morphing-circuit stability distance matches phenomenological stability distance up to a constant factor, so single-shotness scales with code distance d when T = O(1).Source: §6.5 · p. 32

7 Morphing Circuits under Biased Measurement versus Reset Noise

Morphing circuits with non-string-like measurement structure can improve stability experiments when measurement noise is biased relative to reset noise. In the colour code this can come without added syndrome-extraction depth, whereas toric-code variants require deeper circuits and help only at sufficiently large bias.

7 Morphing Circuits under Biased Measurement versus Reset Noise

  • Non-string-like measurement structures make measurement errors trigger more than two defects, preventing purely measurement-error string-like logical failures and increasing measurement-error-only stability distance.The study targets two-round morphing circuits without single-shot error correction, where measurement and reset errors can produce different syndromes.Source: §7 · pp. 33–34
  • In the d = 4 toric colour-code experiment, the two circuits have almost equal logical error rates at η = 0.4; elsewhere their rates overlap within 99% confidence intervals, with the non-string-like variant slightly better.η = 0.4 is where measurement and reset error rates are equal; the experiment uses a minimum-weight decoder and the error rate rises with η.Source: §7 · p. 33, Table 2 · p. 33
  • With a transversal (HS†)^⊗n gate, the colour-code non-string-like circuit beats its string-like counterpart outside 99% confidence intervals; toric circuits II and III cross over at η = 1 and roughly η = 2, respectively.The toric Circuit II crossover occurs under the standard SI1000 model but requires non-planar connectivity; Circuit III retains hex-grid connectivity.Source: §7 · p. 34
  • The transversal gate makes X, Z, and Y stabilisers measured in consecutive rounds, allowing detectors to compare a Y outcome with prior X and Z outcomes and improving protection against measurement errors.Both circuits with the transversal gate outperform both circuits without it.Source: §7 · pp. 33–34
  • In the toric code, added CNOTs create non-string-like measurement structure: Circuit II uses a depth-3 contraction circuit with non-nearest-neighbour connectivity, while Circuit III has depth 4 and hex-grid connectivity.The added gates commute with contracting stabilisers but not expanding ones, spreading measurement-error effects; their extra depth adds noise relative to standard Circuit I.Source: §7 · p. 34

8 Discussion · A Time-Reversal of Normal Circuits

The discussion identifies open directions for extending morphing-circuit design, while the following section shows that reversing a normal syndrome-extraction circuit preserves its normality. These passages highlight broader code classes, circuit trade-offs, automated design, and time reversal.

8 Discussion

  • The paper’s formal results are proved for stabiliser codes, but may extend to subsystem and Floquet codes; broader mid- and end-cycle definitions could enable more complex morphing circuits.The authors developed definitions sufficient for their Bacon-Shor analysis but did not explore them more broadly.Source: §8 · pp. 34–35
  • Avoiding resets after projective measurements may suit some platforms, but classical measurement errors halve the circuit-level stability-distance upper bound in Proposition 8.A classical measurement error has the same effect as combined measurement and reset errors; the discussion leaves the trade-off open.Source: §8 · p. 35
  • Non-string-like measurement structure improved morphing-circuit stability experiments under biased reset and measurement errors, and may also benefit superdense circuits under biased noise.The passage notes that some superdense-circuit measurement errors trigger more than two detectors.Source: §8 · p. 35
  • Adapting ancilla-overhead-efficient methods for logical Pauli measurements to morphing circuits remains open because mid-cycle qubits and stabilisers lack group-element labels, complicating numerical searches.The passage contrasts these methods with the morphing-design approach for measuring Pauli operations in BB codes.Source: §8 · p. 35
  • Automated searches should explore morphing circuits across device connectivities, but the valid contraction-tree space is extremely large and needs careful restriction.Ref. searches using a device connectivity graph as input; the desired search would also determine what connectivity is necessary.Source: §8 · p. 35
  • Automated boundary design could improve on the paper’s hand-optimised colour-code boundaries; existing syndrome-extraction automation [40] [41] does not use morphing design, and a triangular weight-6 circuit improvement is under investigation.The boundary optimisation targeted end-cycle and circuit-level distance.Source: §8 · pp. 35–36

A Time-Reversal of Normal Circuits

  • Reversing a normal circuit preserves normality because its circuit stabilisers are invariant under swapping inputs and outputs.The stabilisers comprise the code stabilisers on the input and output, and logical operators placed on both.Source: A Time-Reversal of Normal Circuits · p. 36

B Propositions and Their Proofs · C Morphing Abelian 2BGA Codes · C.1 Abelian 2BGA Codes

This appendix establishes propositions and structural guarantees for morphing-circuit fault tolerance, including results on memory- and stability-experiment distances and decoding-edge complexity. It also introduces Abelian 2BGA codes and search-simplifying symmetries, and presents the numerical-search results.

B Propositions and Their Proofs

  • Time-reversal symmetry folds any undetectable logical fault in a T-round memory experiment into one round without increasing weight, establishing equal circuit-level distance.The argument uses the two-round circuit's time-reversal symmetry; the stated limitation is that this symmetry does not apply to stability experiments.Source: B Propositions and Their Proofs · pp. 36–37, Figure 24 · p. 37
  • For a quantum LDPC code with constant-depth J-round morphing and each stabilizer measured once, every circuit-level decoding hyperedge is a product of constantly many phenomenological hyperedges under a detector bijection.The proof maps a circuit-level error to a constant-weight mid-cycle Pauli error, then constructs a constant-size set of phenomenological errors with matching detectors.Source: B Propositions and Their Proofs · pp. 38–39
  • With constant J and constant-depth contraction circuits, the stability circuit-level distance is bounded below by a constant independent of T and code distance d.This proposition concerns a quantum LDPC mid-cycle code and its corresponding morphing circuit.Source: B Propositions and Their Proofs · p. 39
  • When regular detectors span at most δ measurement rounds, the T-round stability experiment has circuit-level distance at least floor(T/(δ−1)), because logical faults must hit nonoverlapping observable regions.The proof uses repeated representatives of the same logical observable whose observing regions do not overlap.Source: B Propositions and Their Proofs · p. 40
  • The stability-experiment proposition upper-bounds circuit-level distance using contraction counts N(s) for mid-cycle stabilizers contained in redundancies.Its proof constructs an undetectable time-like error from alternating measurement and reset errors for a stabilizer that participates in a redundancy.Source: B Propositions and Their Proofs · p. 40
  • Every LDPC stabilizer-code family admits a disjoint morphing circuit with the code as its mid-cycle code, a constant number of contraction rounds, and end-cycle distance Ω(d).Here d is the mid-cycle code's distance.Source: B Propositions and Their Proofs · p. 40

C.1 Abelian 2BGA Codes

  • An Abelian 2BGA code is specified by an Abelian group G and subsets A, B; it has 2|G| physical qubits and |G| X- and |G| Z-stabilizers, which may be dependent.Qubits are labeled L or R and by group elements; the construction ensures that all stabilizers commute.Source: §C.1 · p. 41
  • Translation, ZX-duality, and automorphism symmetries identify equivalent Abelian 2BGA codes, reducing the search space and motivating an alternative presentation.The stated equivalences act on (G, A, B) by translating A and B, replacing them with B^-1 and A^-1, or applying a group automorphism.Source: §C.1 · p. 41

C.2 The Lattice Presentation of Abelian 2BGA Codes

The section defines Abelian 2BGA codes through a lattice presentation and shows how this representation captures code construction, supports systematic enumeration, and reveals their geometric locality.

C.2 The Lattice Presentation of Abelian 2BGA Codes

  • An Abelian 2BGA code is specified by (mA, mB, Λ, ncopies), where mA and mB are positive integers, Λ is a full-rank lattice in Z^m for m = mA + mB, and ncopies counts code copies.The lattice presentation is introduced as an alternative definition of the code.Source: §C.2 · p. 41
  • The lattice quotient G = Z^m/Λ is finite Abelian, and the code’s qubits are labelled by its elements, with n = 2|G| = 2 det(Λ).The sets A and B consist of cosets of canonical basis vectors; multiple copies repeat the ncopies = 1 code.Source: §C.2 · p. 43
  • Every Abelian 2BGA code given by a group presentation can be converted to a lattice presentation by forming the subgroup generated by A and B and representing its relations as a lattice.The construction sets ncopies to the size of the quotient of the original group by that generated subgroup.Source: §C.2 · p. 43
  • Hermite Normal Form gives a unique specification for each lattice and enables systematic enumeration; the representation reduces relevant symmetries to ZX-duality and permutations of A and B.For fixed stabiliser weight, these symmetries scale with the number of elements in A and B rather than with group size, making numerical enumeration more attractive than using group presentations.Source: §C.2 · p. 43
  • A code with stabiliser weight w is nearest-neighbour in w − 2 dimensions, with Λ supplying periodic boundary conditions; for weight 4, this makes the connection to toric codes explicit [42].When ncopies > 1, the embedding places ncopies qubits at each location.Source: §C.2 · p. 43
  • Multiplying Λ by a scaling factor provides a way to define families of Abelian 2BGA codes.Source: §C.2 · p. 43

C.3 Homomorphism-Based Morphing Circuits · C.3.1 Two-round Morphing

Homomorphism-based morphing circuits make the search over otherwise intractably many two-round circuits tractable by imposing symmetry and homomorphism conditions. In two rounds, these conditions also ensure an Abelian 2BGA end-cycle code equivalent up to a qubit permutation.

C.3 Homomorphism-Based Morphing Circuits

  • Restricting the search to homomorphism-based circuits makes finding two-round morphing circuits tractable, but their definition does not guarantee a purely contracting CNOT circuit exists.For J = 2, prior work proved that a particular homomorphism ensures a purely contracting homomorphism-based circuit.Source: §C.3 · p. 43
  • Condition (a) requires translations of each contracting circuit to remain in the circuit family, letting one contraction tree specify all X-stabiliser trees.Translational symmetry lets the other X-stabiliser trees be inferred from the tree for s(X, 1).Source: §C.3 · pp. 43–44
  • Condition (b) requires the SWAPZX-conjugated contracting circuit to remain in the family, enforcing ZX-duality between X- and Z-stabiliser contractions.SWAPZX exchanges left and right qubits with inverse group labels, mapping s(X, g) to s(Z, g^-1).Source: §C.3 · p. 44
  • Condition (c) requires contracting X- and Z-stabilisers in each round to be preimages under a homomorphism ϕ, simplifying the subsets to check.A homomorphism-based circuit can be specified by one contraction tree for s(X, 1), with other trees inferred from translation and ZX-duality.Source: §C.3 · p. 44
  • The unrotated toric code has two-round purely contracting homomorphism-based examples, but being homomorphism-based does not itself reduce connectivity.For G = Z4 × Z4 and H = Z2, one example has degree-3 connectivity and the other degree-4 connectivity; both satisfy the translation and ZX-duality conditions.Source: §C.3 · p. 44
  • Some Abelian 2BGA codes lack two-round purely contracting morphing circuits; for example, odd-order G makes condition (c) impossible.BB-code circuits introduced in Ref. are another example of two-round purely contracting homomorphism-based morphing circuits.Source: §C.3 · p. 44

C.3.1 Two-round Morphing

  • Two-round homomorphism-based morphing circuits yield equal end-cycle codes up to a qubit permutation, and the resulting code is itself Abelian 2BGA.The end-cycle group is isomorphic to ker(ϕ), with the surviving data-qubit labels forming one of the two cosets of ϕ.Source: §C.3.1 · p. 44

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

The numerical search found two-round homomorphism-based purely contracting morphing circuits whose code parameters consistently match or beat bare-ancilla circuits despite lower connectivity degree. The reported search does not assess circuit-level distance.

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

  • The search covered two-round homomorphism-based purely contracting morphing circuits for Abelian 2BGA codes at fixed weight and qubit count.It followed Fig. 9’s flowchart and the methods outlined above.Source: §C.4 · p. 44
  • Circuit-level distance was not checked for either bare-ancilla or morphing circuits, and the authors leave investigation of particular codes or circuits to future work.Minimal morphing circuits with connectivity degree above w −1 did not have significantly better code parameters.Source: §C.4 · p. 44
  • The searched morphing circuits consistently match or beat bare-ancilla code parameters while requiring lower-degree connectivity.The complete results cover the minimal codes shown in Figs. 25 and 26; only morphing circuits with connectivity degree w −1 are plotted.Source: §C.4 · p. 44, Figure 25 · p. 45, Figure 26 · p. 46

C.5 Infinite Abelian 2BGA Codes

The section defines infinite Abelian 2BGA codes to build morphing circuits for surface and colour codes on infinite planes before adding boundaries.

C.5 Infinite Abelian 2BGA Codes

  • An infinite Abelian 2BGA code is any Abelian 2BGA code with infinite G; in the lattice presentation, this is equivalent to Λ not being full rank.For Generalised Bicycle codes, G becomes Z.Source: §C.5 · p. 46
  • The infinite surface code has G = Z × Z, A = {1, x}, B = {1, y}; the infinite hexagonal-lattice colour code has G = Z × Z, A = {1, x, y}, B = A−1.Source: §C.5 · p. 46
  • Homomorphism-based morphing circuits remain defined for infinite Abelian 2BGA codes, with finite Abelian H required to ensure finitely many contraction rounds.Source: §C.5 · p. 46
  • For any two-round homomorphism-based morphing circuit on an infinite Abelian 2BGA code, 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.Source: §C.5 · p. 46

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

The appendix optimises colour-code morphing-circuit boundaries by matching end-cycle error-string geometry to known optimal surface- or colour-code boundaries, then transferring those boundaries to the mid-cycle code. This yields distinct optimal boundary geometries for the weight-6, weight-5, and weight-7 end-cycle codes.

Boundary-optimisation method

  • The method matches end-cycle string geometry to the rotated surface code’s diamond boundaries or the colour code’s equilateral-triangular boundaries, then transfers the optimized boundaries to the mid-cycle code.Surface-code strings traverse a square grid, while colour-code strings traverse a triangular grid and can split between colours.Source: D Optimising Distance versus Number of… · pp. 46, 48

Weight-6 end-cycle code

  • Weight-6 end-cycle strings follow a triangular grid, so equilateral-triangular end-cycle boundaries transfer to right-angled-triangle boundaries on the mid-cycle code.The end-cycle strings’ triangular grid is not aligned with the mid-cycle colour-code grid.Source: D Optimising Distance versus Number of… · p. 48

Weight-5 end-cycle code

  • Weight-5 end-cycle strings follow a square grid, favoring diamond boundaries; alternating X/Z or red/blue boundaries encode two logical qubits, while triangular boundaries yielded no competitive [[n, k, d]] codes.The two-logical-qubit option prevents transversal implementation of the full Clifford group; triangular boundaries were originally used in Ref. [20].Source: D Optimising Distance versus Number of… · p. 48

Weight-7 end-cycle code

  • Weight-7 end-cycle strings follow the weight-5 square-grid structure when strings triggering four detectors are allowed, so diamond boundaries are optimal.The four detectors are two on each end of the string.Source: D Optimising Distance versus Number of… · p. 48

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

Optimal two-round, purely contracting, homomorphism-based morphing circuits for weight-4 Abelian 2BGA codes can be constructed by transferring optimized rotated-toric periodic boundary conditions back to the mid-cycle code. The hex-grid examples attain optimal end-cycle code parameters for even and odd distances.

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

  • Numerical checks found that all considered purely contracting, homomorphism-based two-round circuits with weight-4 mid-cycle codes also have weight-4 end-cycle codes, unlike the frequent weight increase for weight-5 and weight-6 codes.Source: §D.1 · p. 48
  • Because weight-4 Abelian 2BGA codes are equivalent to toric-code copies and toric periodic boundaries can be optimized, the optimized end-cycle boundaries can be transferred back to choose the mid-cycle code.Source: §D.1 · p. 48
  • The hex-grid morphing circuits use listed mid-cycle codes to produce end-cycle codes matching optimized rotated toric codes, achieving optimal end-cycle parameters for even and odd distances.The circuits are two-round, purely contracting, and homomorphism-based.Source: Table 5 · p. 49, §D.1 · p. 49

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

The section analyzes circuit-level distance for triangular weight-6 and diamond weight-7 morphing colour codes, then reports numerical memory-experiment results under a specified noise and decoding setup.

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

  • Circuit-level distance is analyzed for triangular weight-6 and diamond weight-7 morphing colour codes considering one measurement round.The section also presents an overview of the analyzed circuits’ properties in Table 6.Source: E Circuit-Level Distance and Numerical… · p. 49
  • Numerical memory experiments run for d measurement rounds under SI1000 depolarising noise and use a minimum-weight decoder with Gurobi.The noise model is referenced in Table 2.Source: E Circuit-Level Distance and Numerical… · p. 49

E.1 Triangular Weight-6 Morphing Colour Code

The triangular weight-6 morphing circuit’s end-cycle-optimized boundaries leave circuit-level error paths that reduce distance. Gate reordering and boundary re-optimization address these paths in the Extra Layer and Extra Qubits variants.

E.1 Triangular Weight-6 Morphing Colour Code

  • End-cycle-optimized boundaries allow two circuit-level failures: a top-boundary CNOT error lowers distance by one, while three circuit errors reduce some weight-four logical strings to 3/4 weight.The boundary optimization equalizes end-cycle logical-error-string weights, but circuit-level errors can reduce some of those weights.Source: §E.1 · p. 49
  • Extra Qubits matches expected circuit-level distance for end-cycle distances 3, 5, 7, and 9; Extra Layer reaches only ceil(3d_end/4) at higher distances.Extra Layer reorders CNOTs, adding a gate layer but no CNOT gates; Extra Qubits also re-optimizes boundaries using additional qubits.Source: §E.1 · p. 49
  • At distances 5 and 7, Extra Qubits outperformed Original and Extra Layer; both morphing circuits also outperformed superdense at equal distance but used more physical qubits.The authors note that the Extra Qubits advantage may reflect either improved circuit-level distance or more qubits reducing low-weight logical operators; further investigation at lower physical error rates is needed.Source: §E.1 · p. 49
  • The authors were investigating a further triangular weight-6 circuit-level-distance improvement that uses fewer qubits than Extra Qubits.The idea arose during final manuscript preparation, and further information was available from the authors.Source: §E.1 · p. 49

E.2 Diamond Weight-7 Morphing Colour Code

Diamond weight-7 morphing circuits performed comparably to triangular weight-6 circuits at the three tested distances, while boundary-free error checks motivate a bulk fault-tolerance conjecture. Planar boundary optimization produced circuits with mixed qubit counts relative to other codes.

E.2 Diamond Weight-7 Morphing Colour Code

  • For the boundary-free bulk, propagated Z-error checks found no circuit-level error string lighter than an end-cycle-code error, motivating a bulk fault-tolerance conjecture.The errors are propagated to the mid-cycle code, where they are equivalent to a single-qubit error, an end-cycle error, or a CNOT error in the first contraction-circuit gate layer.Source: §E.2 · p. 51, Figure 29 · p. 51
  • The authors chose Pauli boundaries to reduce irregularities, then optimized edge gradients and brute-force searched for the smallest diamonds meeting target circuit-level distances.The boundary locations are specified by y-intercepts; Table 7 also lists total data and ancilla qubits.Source: §E.2 · pp. 51–52, Figure 30 · p. 52, Table 7 · p. 52
  • For some target circuit-level distances, diamond weight-7 circuits used fewer qubits than surface and superdense colour codes, while for others they used more; the cause remains open.The authors suggest boundary-related circuit-level error mechanisms as a possibility, leaving this investigation to future work.Source: §E.2 · p. 52
  • Numerical tests found diamond weight-7 morphing circuits comparable in performance to triangular weight-6 circuits at all three tested distances.Source: §E.2 · p. 52
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