Optimising Quantum Error Correction Using Morphing Circuits

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.

• The paper develops a unified morphing-circuit framework that measures stabilisers without dedicated ancillas by contracting them onto data qubits, resetting those qubits, and expanding the code back.
• The paper proves that two-round morphing circuits have circuit-level distance determined by a single measurement round and that alternating circuits are never worse than non-alternating designs.
• The paper’s searches and boundary optimisations find morphing codes that outperform comparable bare-ancilla circuits under connectivity constraints, while highlighting unresolved circuit-level verification and automated design challenges.

Overview

This paper by Shaw and Terhal develops a systematic framework for designing and optimising syndrome extraction circuits via the "morphing" design principle, in which a stabiliser code $C$ is measured without dedicated ancilla qubits by contracting subsets of stabilisers onto single qubits, measuring and resetting them, and expanding back. The work consolidates, generalises, and extends prior morphing constructions for the surface code (2604.09797), colour code, and bivariate bicycle codes into a unified theory with several provable structural results. The central contributions are: (i) a numerical search over morphing circuits for Abelian two-block group algebra (2BGA) codes that yields end-cycle codes strictly outperforming bare-ancilla circuits at equal connectivity budget; (ii) hand-optimised boundary geometries for hex-grid colour code morphing circuits; (iii) proofs that two-round morphing circuits have circuit-level distance computable from a single measurement round, and that alternating normal syndrome extraction circuits are exactly two-round morphing circuits whose distance is never worse than their non-alternating counterparts; and (iv) an analysis of stability experiments showing that two-round morphing circuits match bare-ancilla time-like fault tolerance, while $J>2$ circuits incur quantifiable overheads.

The morphing design principle

A morphing circuit partitions the stabiliser generators $S$ of a CSS code $C$ into $J$ contracting subsets $S_j$, each containing independent commuting stabilisers, together with constant-depth Clifford contraction circuits $F_j$ mapping each $s \in S_j$ to a single-qubit Pauli. Each measurement round consists of $F_j$, single-qubit measurements $M_j$, resets $J>2$0, and $J>2$1. The codes visited between rounds — the end-cycle codes $J>2$2 — have $J>2$3 qubits, unchanged logical dimension $J>2$4, and distance lower-bounded by $J>2$5. In practice the end-cycle distances frequently bound the circuit-level distance from above, motivating much of the optimisation effort in the paper.

A key structural result is that for any two-round morphing circuit,

$J>2$6

i.e., the circuit-level distance of a $J>2$7-round memory experiment equals that of a one-round experiment. The proof exploits a time-reversal symmetry of two-round morphing circuits to "fold" any multi-round undetectable logical error into a single-round one without increasing its weight. This reduces the computational cost of distance certification from $J>2$8 rounds to a single round — a substantial practical saving for distance-finding algorithms.

Alternating normal circuits as morphing circuits

The paper proves that any alternating normal syndrome extraction circuit — one executed with time-reversed rounds every other cycle — is a two-round morphing circuit. Consequently, such circuits satisfy

$J>2$9

Alternating circuits therefore never have smaller circuit-level distance than non-alternating ones, despite identical depth, gate count, and connectivity. The rotated surface code with interleaved check measurements provides a concrete instance where alternation strictly improves distance, since hook errors then span multiple rounds in the non-alternating case. The authors note the caveat that larger distance does not automatically imply better logical performance at all error rates, since the count of low-weight logical operators also matters.

Numerical search over Abelian 2BGA codes

The paper introduces a lattice presentation of Abelian 2BGA codes — specifying a code by a lattice signature $S$0, a full-rank lattice $S$1, and a copy number — which absorbs translational and automorphism symmetries and enables efficient enumeration. Restricting to homomorphism-based two-round purely contracting circuits, the authors brute-force search mid-cycle codes of weight 5 ($S$2) and weight 6 ($S$3).

The headline result is that all minimal morphing end-cycle code parameters found strictly dominate the corresponding bare-ancilla parameters at equal total qubit count, while requiring connectivity degree $S$4 rather than $S$5. Representative examples include a $S$6 end-cycle code at degree-4 connectivity and a $S$7 code at degree-5 connectivity. An important caveat stated plainly by the authors: circuit-level distances were not verified for these searched codes and may fall below the reported end-cycle distances; identifying specific codes of practical interest is deferred to future work.

Boundary optimisation for topological codes

For surface and colour codes, the paper describes how to impose boundaries on infinite-code morphing circuits via anyon condensation, using a heuristic requiring each finite-code contraction tree to be a rooted subtree of the infinite-code tree, with "padding" where necessary. A second technique optimises boundaries of the end-cycle code rather than the mid-cycle code, transferring optimal geometry back through the contraction structure.

Applied to the hexagonal-lattice colour code, this yields three novel circuit families:

Circuit	$S$8 scaling	$S$9	$C$0	Connectivity
Gidney–Jones morphing	$C$1	1	$C$2	Hex grid
Triangular weight-6 (original)	$C$3	1	$C$4	Hex grid
Triangular weight-6 (extra qubits)	$C$5	1	$C$6	Hex grid
Diamond weight-7	$C$7	2	conjectured $C$8	Hex grid

The original triangular weight-6 circuit suffers a circuit-level error of weight $C$9; adding a gate layer and re-optimising boundaries restores full distance at the cost of additional qubits. The diamond weight-7 circuit encodes two logical qubits, precluding full transversal Clifford implementation in the mid-cycle code — a limitation the authors acknowledge explicitly. Numerical memory experiments confirm viability, and both morphing circuits outperform the superdense colour code circuit at equal distance, though they use more physical qubits.

Time-like fault tolerance and stability experiments

Using regular detectors (duration $J$0) and meta-check detectors derived from mid-cycle redundancies (duration $J$1), the paper bounds the circuit-level distance $J$2 of depth-$J$3 stability experiments:

$J$4

where $J$5 is the maximum regular-detector duration and $J$6 counts contractions of stabiliser $J$7. For two-round circuits these bounds coincide, giving $J$8 — formally establishing that two-round morphing circuits match bare-ancilla time-like fault tolerance, so lattice surgery requires no extra depth. For $J$9 the bounds scale differently, $S_j$0, and need not be tight: two $S_j$1 toric code circuits differing only in the ordering of contracting subsets achieve $S_j$2 versus $S_j$3. As a corollary, the original LUCI dropout scheme needs only $S_j$4 rounds for lattice surgery rather than the $S_j$5 previously suggested.

For single-shot codes, exemplified by a six-round morphing circuit for the 3D toric code, the paper proves that single-shot properties are preserved: $S_j$6. Notably, this particular 3D toric code morphing circuit increases connectivity to degree eight relative to bare ancillas — an honest counterexample to the claim that morphing always reduces connectivity. Numerical verification of the exact stability distance was inconclusive due to computational cost, and the preservation result rests on the phenomenological-to-circuit reduction rather than direct simulation.

Biased measurement–reset noise

Under a modified SI1000 noise model with independently scaled measurement error rate ($S_j$7 parameter), the paper shows that "non-string-like" measurement structures — where each measurement error triggers more than two detectors — improve stability-experiment performance when measurement errors dominate. In the colour code this is achieved at no depth cost (a strict win); in the toric code it requires deeper contraction circuits, with crossover at roughly $S_j$8 (standard SI1000) for a non-planar variant and $S_j$9 for a hex-grid variant. These circuits may be advantageous specifically during lattice surgery on devices with measurement–reset bias. One trade-off is noted: fixing the final CNOT direction precludes simultaneous data–ancilla swapping in the weight-6 colour code circuit.

Limitations and open questions

Several limitations are conceded directly. Formal propositions are proved only for stabiliser codes, though subsystem and Floquet generalisations are sketched. The existence of two-round constant-depth morphing circuits for general LDPC families remains open. Circuit-level distances for the searched Abelian 2BGA codes are unverified. The diamond weight-7 circuit's distance is conjectural, and why some target distances require disproportionately many qubits is unexplained. The analysis assumes unconditional resets; omitting resets would halve the stability-distance upper bound due to classical measurement errors, and whether that trade-off pays off is left open. Finally, automated search over contraction tree diagrams — identified as the "holy grail" of morphing design — remains unrealised because the search space is prohibitively large.

Conclusion

This paper establishes morphing circuits as a principled and computationally tractable framework for co-optimising QEC codes, connectivity, gate choice, and fault tolerance. Its strongest quantitative claims — strict parameter dominance of morphing over bare-ancilla Abelian 2BGA circuits at reduced connectivity, single-round distance certification for two-round morphing circuits, and exact time-like equivalence with bare-ancilla circuits — are backed by proof or systematic search, while the remaining gaps (circuit-level verification at scale, automated design, subsystem/Floquet theory) are clearly delineated as open problems.