Clifford-Deformed Compass Codes

• Clifford-deformed compass codes are quantum error-correcting codes that combine gauge fixing and single-qubit Clifford operations to tailor stabilizer geometry for biased dephasing noise.
• They use patterned Hadamard gates to convert weight-4 X-stabilizers into mixed-basis checks such as XZZX and ZXXZ, thereby improving decoder symmetry and threshold values.
• Numerical studies reveal these codes outperform traditional CSS and XZZX surface codes at moderate bias by achieving higher thresholds and lower logical error rates.

Clifford-deformed compass codes are quantum error-correcting codes obtained by combining two design freedoms in the square-lattice compass-code family: gauge fixing of the 2D quantum compass subsystem code, and local single-qubit Clifford conjugations of the resulting stabilizers. In the formulation studied by Campos and Brown, the central use case is biased noise dominated by dephasing, where elongated compass codes can be tailored further by applying patterned Hadamards so that the deformed stabilizers acquire mixed $X/Z$ support while retaining the same locality and support size. The resulting codes exhibit improved decoder symmetries, thresholds that increase with bias, and lower logical error rates; one of the reported deformations outperforms the XZZX surface code at moderate bias (Campos et al., 2024).

1. Compass-code origin and gauge fixing

Compass codes arise from the 2D quantum compass subsystem code on an $L \times L$ square lattice. In the convention used for the elongated-compass construction, the gauge generators are two-body $XX$ operators on horizontal edges and two-body $ZZ$ operators on vertical edges,

$$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$

Different choices of which gauge operators are fixed convert this subsystem structure into different stabilizer codes, including Bacon–Shor codes, surface codes, and elongated compass codes (Campos et al., 2024).

The elongated family is specified by an elongation parameter $\ell$. The construction fixes $X$-gauges on every $\ell$th diagonal of plaquettes, $i-j \equiv 0 \pmod \ell$, producing weight-4 $X$-stabilizers on those plaquettes. Between these diagonals, products of $L \times L$0-gauges are fixed to form elongated $L \times L$1-stabilizers, and the remaining horizontal $L \times L$2-gauges are then fixed to complete the stabilizer group. The unfixed gauge products become logical degrees of freedom. In the square-lattice description used for circuit-level studies, one can choose

$L \times L$3

with code distance $L \times L$4 (Meinking et al., 26 May 2026).

The significance of this gauge-fixing viewpoint is architectural rather than merely taxonomic. The family inherits a common local gauge origin, but different gauge-fixing choices redistribute syndrome information between $L \times L$5- and $L \times L$6-type checks. In the biased-noise setting emphasized in the literature, elongation is used to obtain more information on the higher-rate error channel, namely dephasing-dominated $L \times L$7 faults (Campos et al., 2024).

2. Clifford deformation as local Pauli-axis rotation

A Clifford deformation applies single-qubit Clifford unitaries $L \times L$8 independently to the data qubits. If a stabilizer is $L \times L$9, then conjugation by $XX$0 yields

$XX$1

which is again a Pauli operator with the same weight and support. In the elongated-compass constructions studied to date, the only nontrivial single-qubit Clifford required is the Hadamard $XX$2, which swaps $XX$3 (Campos et al., 2024).

On each fixed weight-4 $XX$4-plaquette, a patterned choice of Hadamards on opposite corners converts the undeformed $XX$5 stabilizer into a mixed-basis plaquette. One diagonal choice produces an $XX$6-type plaquette, while the other produces a $XX$7-type plaquette. In the notation of the 2024 work, applying $XX$8 on the upper-right and lower-left corners maps

$XX$9

and the complementary diagonal defines the $ZZ$0 deformation (Campos et al., 2024). In the circuit-level formulation, the $ZZ$1 pattern is described equivalently as applying Hadamards on the top-left and bottom-right qubits of each fixed cell so that

$ZZ$2

without changing support; when $ZZ$3, this exactly recovers the XZZX deformation of the rotated surface code (Meinking et al., 26 May 2026).

This local-Clifford mechanism should be distinguished from code deformation in the older topological-computation sense. Earlier work on subsystem color codes used “code deformation” to mean changing which local gauge and face operators are enforced in order to move twists, braid defects, or split and merge patches (Bombin, 2010). By contrast, in Clifford-deformed compass codes the code family is altered by static local conjugation of Pauli axes before decoding is considered. The two notions are related by shared subsystem and compass-model ancestry, but they are operationally distinct.

3. Biased noise, decoding-graph structure, and threshold behavior

The noise model used in the code-capacity analysis is an asymmetric Pauli channel with independent single-qubit errors,

$ZZ$4

Here $ZZ$5 is the dephasing bias. Depolarizing noise corresponds to $ZZ$6, while trapped-ion and cat-qubit experiments are described as reporting $ZZ$7–$ZZ$8 (Campos et al., 2024).

Thresholds are not given in closed form; they are extracted numerically by finite-size scaling of the logical error rate near threshold. The 2024 study writes

$ZZ$9

and determines $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$0 from the crossings and fits (Campos et al., 2024). The central empirical observation is that the deformed elongated compass codes exhibit thresholds that increase with bias and logical error rates that decrease relative to undeformed CSS variants.

The decoder improvement is tied to a structural property of the deformed matching graph. For the $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$1-deformed code, the low-weight edges associated with the more likely dephasing faults form quasi-1D strips or isolated diamonds. High-weight edges associated with less likely $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$2 faults connect these regions only weakly. The paper describes this as a “domain-wall” symmetry that confines dephasing defects to small regions, simplifies matching, and raises thresholds under biased noise (Campos et al., 2024).

The threshold data reflect this reorganization. For $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$3, the reported thresholds at $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$4 are $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$5 for the CSS code, $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$6 for the $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$7 deformation, and $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$8 for the $$G_X(i,j)=X_{i,j}X_{i+1,j}, \qquad G_Z(i,j)=Z_{i,j}Z_{i,j+1}.$$9 deformation. At larger biases the gain becomes more pronounced: at $\ell$0, the values are $\ell$1, $\ell$2, and $\ell$3; at $\ell$4, $\ell$5, $\ell$6, and $\ell$7; and at $\ell$8, $\ell$9, $X$0, and $X$1, again in the order CSS, $X$2, $X$3 (Campos et al., 2024). These figures show that the deformation is not merely a basis relabeling from the decoder’s perspective; it changes the syndrome geometry relevant to biased matching.

4. Relation to the XZZX surface code and logical-error comparisons

The $X$4 member of the deformed elongated family coincides with the rotated-surface-code XZZX construction. This makes the elongated deformations a direct generalization rather than a separate code family. The comparison emphasized in the original study is therefore a controlled comparison between $X$5 Clifford-deformed elongated compass codes and the standard XZZX surface code realized at $X$6 (Campos et al., 2024).

At moderate bias, the $X$7-deformed $X$8 code exceeds the XZZX surface code both in threshold and, in part of parameter space, in logical error rate. The paper reports that at $X$9–$\ell$0, the $\ell$1-deformed $\ell$2 code achieves thresholds from approximately $\ell$3 to $\ell$4, while the XZZX surface code plateaus in the same regime at approximately $\ell$5 to $\ell$6 (Campos et al., 2024). This establishes that elongation and deformation can be complementary rather than redundant.

The logical-error comparison is presented through the ratio

$\ell$7

for physical error rates $\ell$8 and $\ell$9. Above bias $i-j \equiv 0 \pmod \ell$0, the $i-j \equiv 0 \pmod \ell$1 code is reported to achieve the lowest logical error rate, and in some cases $i-j \equiv 0 \pmod \ell$2, corresponding to a ratio below one (Campos et al., 2024). A common misconception is that XZZX exhausts the advantage obtainable from local Clifford tailoring under biased dephasing. The elongated-compass results show that further gauge-fixing freedom can produce additional gains.

5. Circuit-level noise and correlated decoding

Subsequent work studied the same Clifford-deformed elongated compass codes under circuit-level noise rather than code-capacity noise. The circuit model uses a Hybrid Biased-Depolarizing (HBD) channel: bias-preserving $i-j \equiv 0 \pmod \ell$3 gates receive a two-qubit Pauli channel with

$i-j \equiv 0 \pmod \ell$4

while $i-j \equiv 0 \pmod \ell$5 and single-qubit $i-j \equiv 0 \pmod \ell$6 gates are assigned a fully depolarizing channel of strength $i-j \equiv 0 \pmod \ell$7. Idling, state preparation, and measurement use a single-qubit asymmetric channel with

$i-j \equiv 0 \pmod \ell$8

and measurement errors occur with probability $i-j \equiv 0 \pmod \ell$9 (Meinking et al., 26 May 2026).

Syndrome extraction is performed with a single ancilla per stabilizer. For an $X$0-stabilizer of weight $X$1, the circuit is

$X$2

whereas $X$3-stabilizers use $X$4. In the Clifford-deformed case, some of these $X$5 gates become $X$6 gates because of the Hadamard conjugations. Simulations are carried out in Stim, and the resulting detector error models are decoded with MWPM or with PyMatching’s correlated decoder (Meinking et al., 26 May 2026).

The decoder comparison is central. Standard MWPM uses graph weights

$X$7

where $X$8 is the probability attached to an edge in the matching graph. The correlated decoder performs a first matching pass, then updates edge weights using conditional probabilities inherited from shared hyperedges in the detector error model, and performs a second pass. Under circuit-level noise, correlated decoding improves thresholds for all biases studied (Meinking et al., 26 May 2026).

Representative thresholds make the effect explicit. For the CSS elongated code with $X$9, the threshold rises from $L \times L$00 to $L \times L$01 at $L \times L$02, from $L \times L$03 to $L \times L$04 at $L \times L$05, from $L \times L$06 to $L \times L$07 at $L \times L$08, and from $L \times L$09 to $L \times L$10 at $L \times L$11 when moving from standard MWPM to correlated decoding. For the $L \times L$12-deformed code with $L \times L$13, the corresponding improvements are $L \times L$14, $L \times L$15, $L \times L$16, and $L \times L$17 (Meinking et al., 26 May 2026). The same study reports that CSS thresholds continue to rise with bias, whereas the $L \times L$18 thresholds saturate at high $L \times L$19 because the syndrome circuits mix bias-preserving $L \times L$20 operations with depolarizing $L \times L$21 and $L \times L$22 operations. A plausible implication is that the asymptotic code-capacity advantages of Clifford deformation depend sensitively on how faithfully the hardware and extraction circuits preserve the physical bias.

6. Broader theoretical setting and later generalizations

Later work places Clifford-deformed compass-like constructions inside a broader theory of Clifford-deformed zero-rate LDPC codes under pure dephasing. In that framework, if every nontrivial pure-$L \times L$23 logical has weight at least $L \times L$24 and the number of pure-$L \times L$25 logicals grows sub-exponentially in $L \times L$26, then for any $L \times L$27 the decoding failure probability vanishes asymptotically. An alternative sufficient condition uses a basis of logical operators whose supports are disjoint or satisfy controlled overlap conditions. The result is a $L \times L$28 infinite-bias code-capacity threshold (Das et al., 14 May 2026).

This framework is presented as explaining previously known $L \times L$29 biased-noise thresholds for the XY surface code, XZZX surface code, deformed color codes, and some 3D Clifford-deformed codes. It is also extended to tile codes, which are described there as zero-rate LDPC “tile (compass) codes.” Random and translationally invariant Clifford deformations of those codes exhibit regions with estimated threshold $L \times L$30 under pure $L \times L$31 noise, and circuit-level performance is analyzed in terms of a residual bias

$L \times L$32

after a full syndrome-extraction cycle (Das et al., 14 May 2026).

For Clifford-deformed compass codes proper, these later results are best read as theoretical context rather than a direct rederivation of the elongated-compass numerics. They suggest that the high-bias performance of local-Clifford deformations is governed by the geometry of the surviving pure-$L \times L$33 logical sector and by the extent to which syndrome extraction preserves the intended anisotropy. In that sense, the 2024 elongated-compass constructions occupy an intermediate position between the XZZX surface code and more general Clifford-deformed LDPC families: they retain a transparent gauge-fixing origin, admit explicit circuit constructions and decoder analyses, and exemplify how local Pauli-axis rotations can be co-designed with subsystem structure and biased-noise decoding (Campos et al., 2024).