Clifford-Deformed Elongated Compass Codes

• Clifford-deformed elongated compass codes are 2D stabilizer codes crafted via anisotropic gauge fixing and selective local Clifford deformations to optimize error detection under Z-biased noise.
• The construction leverages weight transformations and tailored gauge choices to preserve crucial short X-type stabilizers while reorganizing higher-weight checks into decoder-friendly patterns.
• Circuit-level studies and correlated decoding reveal enhanced thresholds and lower logical error rates, demonstrating practical benefits across varying noise biases.

Clifford-deformed elongated compass codes are a family of two-dimensional stabilizer codes obtained by combining anisotropic gauge fixing of the quantum compass model with local single-qubit Clifford transformations. In the construction studied most directly, one starts from elongated compass codes tailored to $Z$-biased noise and then applies Hadamards on selected qubits so that the higher-weight $X$ stabilizers acquire XZZX- or ZXXZ-type patterns while the weight-2 $X$ stabilizers that are crucial for detecting many $Z$ errors are preserved. The resulting XZZX$_\square$- and ZXXZ$_\square$-deformed elongated compass codes are designed for dephasing-biased noise, exhibit thresholds that increase with bias, and display lower logical error rates; circuit-level studies further show that correlated minimum-weight perfect matching enhances thresholds for all noise biases relative to standard MWPM (Campos et al., 2024, Meinking et al., 26 May 2026).

1. Gauge-theoretic origin and elongated compass structure

Clifford-deformed elongated compass codes inherit their base geometry from the two-dimensional quantum compass model on an $L\times L$ square lattice with qubits on vertices. The underlying Hamiltonian is

$$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$

and the associated gauge group is generated by vertical $X$-type and horizontal $Z$-type weight-2 operators. In the broader compass-code framework, this subsystem structure interpolates between Bacon–Shor and surface-code-like limits through gauge fixing; the surface code and the Bacon–Shor code represent two extremes of possible codes depending on how many gauge qubits are fixed (Campos et al., 2024, Li et al., 2018).

The elongated compass-code subfamily is parameterized by an integer elongation $X$0. Its defining feature is an asymmetric gauge-fixing pattern that increases the number of short $X$1-type checks while aggregating $X$2-type gauges into longer checks. In the earlier compass-code literature this asymmetry was introduced precisely to trade locality for asymmetry and gauge degrees of freedom for stabilizer syndrome information under biased Pauli noise (Li et al., 2018). In the specific elongated construction used for Clifford deformation, $X$3 reproduces the standard rotated surface code, while larger $X$4 produces a progressively more anisotropic code in which information about $X$5 errors is fine-grained and information about $X$6 errors is coarse-grained (Campos et al., 2024).

This gauge-theoretic origin matters because the later Clifford deformation does not discard the elongated compass-code rationale. The deformation is applied after the bias-tailored gauge fixing has already selected a stabilizer pattern that is intended to obtain more information on high-rate errors (Campos et al., 2024).

2. Explicit construction and local Clifford deformation

The elongated compass code $X$7 is built on plaquettes labeled by coordinates $X$8. The gauge-fixing rule is organized around the congruence

$X$9

On those plaquettes, vertical $X$0-type gauges are fixed into weight-4 stabilizers

$X$1

Between successive such plaquettes, horizontal $X$2 gauges are combined into elongated $X$3 stabilizers of weight $X$4, schematically of the form

$X$5

All remaining vertical weight-2 $X$6 gauges around those $X$7 rectangles are then fixed to ensure commutativity, yielding an ordinary CSS stabilizer code (Campos et al., 2024).

The Clifford-deformed variants are obtained by applying local Hadamards to selected corners of each weight-4 $X$8 plaquette. Two deformations are singled out. In the XZZX$X$9 deformation, Hadamards are applied to the top-right and bottom-left qubits of each weight-4 $Z$0 stabilizer. Thus

$Z$1

In the ZXXZ$Z$2 deformation, Hadamards are applied instead to the top-left and bottom-right qubits, giving

$Z$3

These transformations preserve the support of each stabilizer and preserve total stabilizer weight, but typically break CSS structure by producing mixed-type checks (Campos et al., 2024).

In the general framework of Clifford-deformed surface codes, such a deformation is written as

$Z$4

with transformed stabilizer group

$Z$5

A local Clifford circuit of this kind leaves the code parameters $Z$6 unchanged, while changing the Pauli type and orientation of stabilizers and logical operators (Dua et al., 2022). In the elongated compass setting, that invariance is exploited selectively: the deformation preserves the weight-2 $Z$7 stabilizers that are crucial for detecting many $Z$8 errors, but deforms the higher-weight $Z$9 stabilizers into XZZX- or ZXXZ-type patterns introducing strong geometric symmetries in the syndrome graph (Campos et al., 2024).

3. Noise model, effective inhomogeneity, and decoder geometry

The physical noise model is an independent single-qubit Pauli channel

$_\square$0

with total rate $_\square$1, bias

$_\square$2

and the simplifying assumption $_\square$3. The depolarizing point is $_\square$4, and the regime of interest is $_\square$5-biased noise with $_\square$6 (Campos et al., 2024).

A useful feature of local Clifford deformation is that the deformed code can be decoded on the undeformed CSS code with an effective inhomogeneous Pauli channel. If the local Clifford on qubit $_\square$7 is $_\square$8, then the effective probabilities on the original code are

$_\square$9

Thus Clifford deformation is equivalent to changing the noise to be spatially inhomogeneous but still Pauli and independent (Campos et al., 2024).

Decoding is performed with minimum-weight perfect matching. In the deformed non-CSS code, syndromes are mapped back to the original CSS code under the Clifford action, and MWPM weights are assigned using the inhomogeneous error probabilities above. The deformations are then interpreted through the geometry of the matching graph. For the XZZX surface code, the low-weight graph is a set of parallel lines and the high-weight graph is a set of parallel lines oriented orthogonally. For XZZX$_\square$0 on elongated codes, the low-weight graph partitions the lattice into regions bounded by diagonal XZZX plaquettes, and within each region one sees chains of diamonds that behave similarly to repetition codes for $_\square$1 errors. For ZXXZ$_\square$2, the low-weight graph forms disjoint strings, essentially decoupled repetition codes, while the high-weight graph is also partitioned rather than fully dense (Campos et al., 2024).

The underlying design principle is explicit: the Clifford deformations enhance decoder performance by introducing symmetries, while the stabilizers of compass codes can be selected to obtain more information on high-rate errors (Campos et al., 2024).

4. Thresholds, bias dependence, and comparison with XZZX

At no bias, CSS, XZZX$_\square$3, and ZXXZ$_\square$4 are equivalent from the decoder’s perspective. Representative thresholds at $_\square$5 are $_\square$6 for $_\square$7, $_\square$8 for $_\square$9, $L\times L$0 for $L\times L$1, and $L\times L$2 for $L\times L$3 (Campos et al., 2024).

For CSS elongated compass codes without Clifford deformation, each elongation has an optimal bias $L\times L$4 at which the threshold is maximal. The values reported are

$L\times L$5

with corresponding CSS thresholds $L\times L$6 for $L\times L$7, $L\times L$8 for $L\times L$9, $$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$0 for $$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$1, and $$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$2 for $$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$3 (Campos et al., 2024).

The Clifford-deformed families behave differently. Thresholds for both XZZX$$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$4 and ZXXZ$$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$5 increase monotonically with bias for all $$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$6. Representative values are summarized below.

Family	Selected $$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$7	Threshold
$$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$8 XZZX surface code	$$\hat{H} = -J_X \sum_{i=0}^{L-2} \sum_{j=0}^{L-1} \hat{X}_{i,j}\hat{X}_{i+1,j} -J_Z \sum_{i=0}^{L-1} \sum_{j=0}^{L-2} \hat{Z}_{i,j}\hat{Z}_{i,j+1},$$9	$X$0
$X$1 XZZX$X$2	$X$3	$X$4
$X$5 ZXXZ$X$6	$X$7	$X$8

These numbers exhibit the main comparative phenomenon. For $X$9, XZZX$Z$0 and ZXXZ$Z$1 coincide with the standard XZZX surface code. For $Z$2, they define distinct elongated families. ZXXZ$Z$3 grows much more rapidly with bias than XZZX$Z$4, and for $Z$5 its thresholds near $Z$6–$Z$7 are about $Z$8 (Campos et al., 2024).

Logical-error-rate data follow the same pattern. At fixed $Z$9 below threshold, logical error rates decrease exponentially with distance. For $X$00, distance $X$01, the normalized logical error rate $X$02 shows that at low bias the CSS compass code performs best, but for $X$03 both XZZX$X$04 and ZXXZ$X$05 outperform CSS, and ZXXZ$X$06 has the lowest logical error rates. The 2024 study states explicitly that one of the Clifford deformations explored yields QEC codes with better thresholds and logical error rates than those of the XZZX surface code at moderate biases (Campos et al., 2024).

A common misconception is to treat XZZX-like deformation as uniformly optimal once bias is present. The reported data do not support that conclusion. In the elongated compass setting, deformation type and elongation both matter: XZZX$X$07 and ZXXZ$X$08 have distinct syndrome-graph geometries and distinct threshold trajectories (Campos et al., 2024).

5. Circuit-level behavior and correlated decoding

A subsequent circuit-level study considers Clifford-deformed elongated compass codes under a hybrid biased-depolarizing model. In this model, CZ gates are followed by a biased two-qubit Pauli channel in which the pure-dephasing errors $X$09 occur with probability

$X$10

while the remaining two-qubit errors occur with probability $X$11. By contrast, CNOT and $X$12 are treated as bias-breaking gates and are followed by depolarizing noise; idling qubits undergo the asymmetric single-qubit Pauli channel, and measurements fail with probability $X$13 (Meinking et al., 26 May 2026).

The decoding comparison is between standard MWPM and correlated MWPM. At code capacity, the CSS correlated decoder exploits the conditional probabilities

$X$14

and updates second-pass matching weights to

$X$15

At circuit level, PyMatching’s correlated mode performs an analogous two-pass procedure on a detector error model, using conditional probabilities reconstructed from hyperedge decompositions such as

$X$16

These formulas encode not only $X$17 correlations but also the structured space-time correlations introduced by the syndrome-extraction circuits (Meinking et al., 26 May 2026).

The principal circuit-level finding is unambiguous: correlated decoding enhances thresholds for all noise biases relative to standard MWPM under circuit-level noise. The same work further concludes that correlated decoding leads to a higher relative gain in thresholds compared to standard MWPM when applied to codes with asymmetric stabilizers under biased noise (Meinking et al., 26 May 2026). For CSS elongated compass codes, the maximum relative gain reported is about $X$18 for the $X$19 code at $X$20. For the ZXXZ$X$21 family, the relative gain increases systematically with $X$22 for all $X$23, which directly ties decoder advantage to stabilizer asymmetry (Meinking et al., 26 May 2026).

This circuit-level analysis also tempers code-capacity conclusions. Standard MWPM thresholds for CSS elongated compass codes continue to increase with $X$24, but for ZXXZ$X$25 with $X$26 the advantage seen at code capacity is largely suppressed by the non bias-preserving $X$27 and CNOT gates required in the syndrome circuits. Correlated decoding partially compensates for this effect, but does not remove the hardware-level dependence on gate set and extraction schedule (Meinking et al., 26 May 2026).

6. Broader theoretical context and open directions

Clifford-deformed elongated compass codes sit at the intersection of two larger research programs. The first is the study of local Clifford deformations as a general design knob for bias-tailored stabilizer codes. In the surface-code setting, applying site-dependent single-qubit Cliffords produces Clifford-deformed surface codes whose $X$28 parameters are unchanged but whose stabilizer Pauli content, effective distance under biased noise, and threshold behavior can differ dramatically. Random and translation-invariant Clifford-deformed surface-code families exhibit phase-diagram structure under $X$29-biased noise, including regions with $X$30 threshold at infinite bias (Dua et al., 2022).

The second program is the gauge-fixing view of compass codes. Two-dimensional compass codes were introduced as a broad class of local codes obtained from Bacon–Shor by gauge fixing, with explicit elongated constructions that improve thresholds against asymmetric noise. In that framework, the elongation parameter $X$31 controls an effective asymmetrization of Kitaev’s toric code in the bulk with extended $X$32-body plaquette operators, and the code family remains local while trading stabilizer weight against directional syndrome information (Li et al., 2018).

More recent zero-rate LDPC theory provides a broader explanation for why Clifford-deformed codes can approach $X$33 threshold under pure dephasing. If the number of biased logical operators grows slowly enough or if there exists a basis of logical operators whose overlap satisfies suitable scaling conditions, then the code-capacity threshold of the Clifford-deformed variant under i.i.d. pure dephasing noise approaches $X$34. This framework explicitly explains previously known examples such as XY surface code, XZZX surface code, color code, and some 3D Clifford-deformed codes, and it suggests that elongated compass-code constructions belong to a larger class of bias-tailored deformations of zero-rate LDPC codes (Das et al., 14 May 2026).

A related but distinct line studies explicit local Clifford layers on geometric CSS codes. Recent work on “Quantum Logic Codes” gives a depth-1 transversal logical $X$35 on rotated surface codes and a depth-1 intra-block logical $X$36 on the 2D toric code, framing these as Clifford deformations of planar and toric geometries. This suggests a broader program in which geometric codes, including compass-like layouts, are engineered to support more logical Clifford structure through local Clifford patterns and symmetry constraints (Holmes, 11 Jun 2026).

Several open issues remain. Circuit-level performance depends strongly on whether the hardware natively preserves dephasing bias during syndrome extraction. The 2026 circuit-level study identifies non bias-preserving $X$37 and CNOT gates as a central limitation for Clifford-deformed elongated compass codes (Meinking et al., 26 May 2026). A plausible implication is that fully bias-preserving extraction schedules, or hardware platforms with native CZ-like interactions, may be decisive for realizing the full advantage indicated by code-capacity thresholds. Another active direction is decoder design: the current evidence indicates that code design and correlation-aware decoding must be co-optimized, rather than treated as separate layers of the architecture (Meinking et al., 26 May 2026).

In that sense, Clifford-deformed elongated compass codes are best understood not as a single code, but as a bias-tailored design paradigm: start from a compass-code gauge fixing that favors the dominant error channel, apply local Clifford deformations that reorganize the syndrome graph into decoder-friendly structures, and then match the resulting asymmetry with a decoder that can exploit the induced correlations (Campos et al., 2024, Meinking et al., 26 May 2026).