Clifford-Deformed Codes

• Clifford-deformed codes are quantum stabilizer constructions that apply local Clifford conjugation to adjust stabilizers and logical operators without changing the parameters [[n,k,d]].
• They have been applied to surface, compass, tile, and bicycle codes, enabling tailored error thresholds and improved resilience under dephasing-biased noise.
• These constructions also facilitate fault-tolerant operations such as transversal gate implementation, logical gate deformation via braiding, and advanced representation-theoretic methods.

Clifford-deformed codes are code constructions in which Clifford structure alters stabilizers, logical operators, or decoding while preserving a controlled error-correcting framework. In the most common stabilizer-code usage, a tensor product of single-qubit Clifford unitaries $U=\bigotimes_{i=1}^n U_i$ maps a Pauli stabilizer group $\mathsf S\subset \mathsf P_n$ to $\mathsf S' = U \mathsf S U^\dagger$, so the deformed code remains a Pauli stabilizer code with the same parameters $[[n,k,d]]$ but with locally rotated Pauli axes (Das et al., 14 May 2026). This mechanism has been used to tailor surface, compass, tile, and bivariate-bicycle-derived qLDPC codes to dephasing-biased noise (Dua et al., 2022). In parallel, related literature uses closely allied Clifford language for code conversion and doubling that strengthen transversal gate sets (Jain et al., 2024), for code deformation that realizes logical Clifford gates by moving and braiding defects (Bombin, 2010), and for representation-theoretic Clifford codes formulated through projective error models and operator algebra quantum error correction (Eidesen et al., 1 Jun 2026).

1. Terminological scope

The phrase “Clifford-deformed code” does not denote a single universally fixed construction. The sources instead organize several distinct, though related, uses of Clifford structure.

Setting	Mechanism	Representative source
Pauli stabilizer deformation	Apply $U=\bigotimes_i U_i$ and conjugate $\mathsf S$ to $U\mathsf S U^\dagger$	(Das et al., 14 May 2026)
Gate-oriented code conversion	Use doubling, folding, or related transformations to enlarge transversal gate support	(Jain et al., 2024)
Geometric code deformation	Change measured checks over time so that twist motion or braiding implements logical Clifford gates	(Bombin, 2010)
Representation-theoretic Clifford codes	Define codes from induced or projective representations, sometimes beyond stabilizer form	(Eidesen et al., 1 Jun 2026)
Circuit-to-code constructions	Derive outcome and spacetime codes from Clifford circuits; the source does not use “Clifford-deformed code” as its own term	(Delfosse et al., 2023)

A recurrent misconception is to identify all of these with surface-code-style code deformation. One source explicitly states that its “Clifford-deformed” theme is not “code deformation” in the surface-code/lattice-surgery sense, but rather code conversion and doubling between classical and quantum code families (Jain et al., 2024). This suggests that the term is best read as a family resemblance across constructions in which Clifford conjugation, Clifford symmetry, or Clifford representation theory is the operative mechanism.

2. Local Clifford conjugation as a code transformation

For Pauli stabilizer codes, the basic construction is elementary: starting from $\mathsf S\subset \mathsf P_n$, apply $U=\bigotimes_{i=1}^n U_i$ with each $U_i$ Clifford, and set $\mathsf S\subset \mathsf P_n$0. Because each $\mathsf S\subset \mathsf P_n$1 maps Pauli operators to Pauli operators, $\mathsf S\subset \mathsf P_n$2 is again a Pauli stabilizer group with unchanged $\mathsf S\subset \mathsf P_n$3, while the local Pauli frame is permuted qubit by qubit (Das et al., 14 May 2026). Under biased dephasing, the point of the transformation is not to alter the topology “in a gross sense,” but to align stabilizers and logicals more favorably against dominant $\mathsf S\subset \mathsf P_n$4-type noise.

For surface codes, the relevant local Clifford choices under the bias model $\mathsf S\subset \mathsf P_n$5 are $\mathsf S\subset \mathsf P_n$6, $\mathsf S\subset \mathsf P_n$7, and $\mathsf S\subset \mathsf P_n$8. One may either deform the stabilizers directly or keep the CSS stabilizers fixed and transform the local noise model in the Heisenberg picture (Dua et al., 2022). The same paper introduces a bias-dependent effective distance $\mathsf S\subset \mathsf P_n$9 and half-distance $\mathsf S' = U \mathsf S U^\dagger$0,

$\mathsf S' = U \mathsf S U^\dagger$1

with

$\mathsf S' = U \mathsf S U^\dagger$2

and notes that for unbiased depolarizing noise $\mathsf S' = U \mathsf S U^\dagger$3 these reduce to the usual code parameters $\mathsf S' = U \mathsf S U^\dagger$4 and $\mathsf S' = U \mathsf S U^\dagger$5 (Dua et al., 2022).

For elongated compass codes, Clifford deformation preserves the support of stabilizers, their weights, and the code dimension, but can change the Pauli content and thereby make the code non-CSS. The paper defines two tailored deformations. In the $\mathsf S' = U \mathsf S U^\dagger$6 deformation, Hadamards are applied to the top-right and bottom-left qubits of each weight-4 $\mathsf S' = U \mathsf S U^\dagger$7-stabilizer. In the $\mathsf S' = U \mathsf S U^\dagger$8 deformation, Hadamards are applied to the top-left and bottom-right qubits of each weight-4 $\mathsf S' = U \mathsf S U^\dagger$9-stabilizer (Campos et al., 2024). Both are designed to preserve many weight-2 $[[n,k,d]]$0-stabilizers while introducing XZZX-like symmetry.

The Romanesco construction uses a still more structured version of the same principle. Starting from a self-dual bivariate bicycle code built from two reflected cellular-automaton codes, Hadamards are applied to all gray qubits and the CSS parity-check matrix

$[[n,k,d]]$1

is transformed into

$[[n,k,d]]$2

The result is a non-CSS code in which each stabilizer is obtained by multiplying an all-$[[n,k,d]]$3 stabilizer from one classical code with an all-$[[n,k,d]]$4 stabilizer from the other, or conversely (Leroux et al., 30 May 2025).

3. Bias-tailored families and lattice architectures

Clifford-deformed surface codes, or CDSCs, are obtained by applying single-qubit Clifford operators to the ordinary surface code on the square lattice. On the $[[n,k,d]]$5 lattice there are $[[n,k,d]]$6 such codes, since each qubit independently receives one of $[[n,k,d]]$7, $[[n,k,d]]$8, or $[[n,k,d]]$9, and the paper reports that under strong bias their logical error rates can differ by orders of magnitude even though all have standard distance $U=\bigotimes_i U_i$0 (Dua et al., 2022). In the thermodynamic study of random CDSCs, the parameters $U=\bigotimes_i U_i$1 interpolate between the CSS surface code $U=\bigotimes_i U_i$2, the XZZX code $U=\bigotimes_i U_i$3, and the XY code $U=\bigotimes_i U_i$4. The same work identifies a region in the $U=\bigotimes_i U_i$5 plane with a $U=\bigotimes_i U_i$6 threshold at infinite bias and constructs a translation-invariant code in the nearby rational family $U=\bigotimes_i U_i$7, which outperforms XZZX and XY in the tested regime (Dua et al., 2022).

Elongated compass codes begin as CSS gauge-fixings of the square-lattice quantum compass model, parameterized by an elongation $U=\bigotimes_i U_i$8. As $U=\bigotimes_i U_i$9 grows, weight-2 $\mathsf S$0-stabilizers proliferate while $\mathsf S$1-stabilizers become weight $\mathsf S$2, producing an asymmetric code whose optimal bias $\mathsf S$3 balances $\mathsf S$4- and $\mathsf S$5-error performance (Campos et al., 2024). After Clifford deformation, the paper reports threshold tables in the format CSS $\mathsf S$6 $\mathsf S$7 $\mathsf S$8 $\mathsf S$9. Representative entries include, at $U\mathsf S U^\dagger$0, $U\mathsf S U^\dagger$1: $U\mathsf S U^\dagger$2; at $U\mathsf S U^\dagger$3, $U\mathsf S U^\dagger$4: $U\mathsf S U^\dagger$5; and at $U\mathsf S U^\dagger$6, $U\mathsf S U^\dagger$7: $U\mathsf S U^\dagger$8. The paper further states that the $U\mathsf S U^\dagger$9-deformed compass codes surpass the XZZX surface code threshold at moderate bias, specifically between $\mathsf S\subset \mathsf P_n$0 and $\mathsf S\subset \mathsf P_n$1, and remain better up to $\mathsf S\subset \mathsf P_n$2 (Campos et al., 2024).

Clifford-deformed tile codes extend the same bias-tailoring program to planar, geometrically local qLDPC codes with bounded-weight checks, typically weight 6 in the bulk. The open-boundary family has fixed $\mathsf S\subset \mathsf P_n$3, while the periodic family used for phase-diagram studies has $\mathsf S\subset \mathsf P_n$4 (Das et al., 14 May 2026). Random deformations independently apply $\mathsf S\subset \mathsf P_n$5, $\mathsf S\subset \mathsf P_n$6, or $\mathsf S\subset \mathsf P_n$7 with probabilities $\mathsf S\subset \mathsf P_n$8, $\mathsf S\subset \mathsf P_n$9, and $U=\bigotimes_{i=1}^n U_i$0. At infinite bias, the paper finds an extended region with threshold $U=\bigotimes_{i=1}^n U_i$1, approximately

$U=\bigotimes_{i=1}^n U_i$2

It also gives three translationally invariant examples—linear deformation, XY deformation, and a realization associated with $U=\bigotimes_{i=1}^n U_i$3—for which a $U=\bigotimes_{i=1}^n U_i$4 threshold is reported at infinite bias for the sizes studied (Das et al., 14 May 2026).

Romanesco codes move the same idea into qLDPC codes derived from fractal or cellular-automaton input codes. They are defined on a bipartite honeycomb or hexagonal lattice, are non-CSS, and have stabilizer generators that are each half $U=\bigotimes_{i=1}^n U_i$5 and half $U=\bigotimes_{i=1}^n U_i$6. In the main families discussed, bulk stabilizer weight is typically 8, while boundaries can reduce to 6 or 4 (Leroux et al., 30 May 2025). The paper reports toric examples such as $U=\bigotimes_{i=1}^n U_i$7 with $U=\bigotimes_{i=1}^n U_i$8, and planar examples such as $U=\bigotimes_{i=1}^n U_i$9 with $U_i$0, and argues that under strong bias the effective distance approaches the classical distance of the input codes (Leroux et al., 30 May 2025).

4. Threshold theory, effective distance, and decoding

In the infinite-bias limit $U_i$1, $U_i$2, decoding failure is governed by pure-$U_i$3 logical operators

$U_i$4

For a pure-$U_i$5 logical $U_i$6, a $U_i$7-error $U_i$8 causes ambiguity if

$U_i$9

A general sufficient condition proved for zero-rate LDPC codes is that if every nontrivial pure-$\mathsf S\subset \mathsf P_n$00 logical operator satisfies $\mathsf S\subset \mathsf P_n$01 and the total number of pure-$\mathsf S\subset \mathsf P_n$02 logical operators is subexponential, $\mathsf S\subset \mathsf P_n$03, then for any $\mathsf S\subset \mathsf P_n$04,

$\mathsf S\subset \mathsf P_n$05

so the family has a $\mathsf S\subset \mathsf P_n$06 infinite-bias threshold (Das et al., 14 May 2026). The same paper reformulates this in terms of a basis of logical operators and overlap-load conditions: few, long, weakly overlapping pure-$\mathsf S\subset \mathsf P_n$07 logicals are the favorable regime.

Surface-code work emphasizes a complementary, bias-weighted notion of distance. The effective distance $\mathsf S\subset \mathsf P_n$08 and half-distance $\mathsf S\subset \mathsf P_n$09 are defined through the most likely logical and uncorrectable events, and explain why two Clifford-deformed codes with the same ordinary distance $\mathsf S\subset \mathsf P_n$10 can differ by orders of magnitude in logical error rate under biased noise (Dua et al., 2022). Romanesco codes introduce a related modified weight for a logical with $\mathsf S\subset \mathsf P_n$11 $\mathsf S\subset \mathsf P_n$12-type operators and total Hamming weight $\mathsf S\subset \mathsf P_n$13,

$\mathsf S\subset \mathsf P_n$14

and define $\mathsf S\subset \mathsf P_n$15 as the minimum of this modified weight over logical operators. In the strong-bias limit, $\mathsf S\subset \mathsf P_n$16 (Leroux et al., 30 May 2025).

For compass-code deformations, decoding can be reduced to the undeformed CSS decoding graph with an inhomogeneous noise model. The paper writes

$\mathsf S\subset \mathsf P_n$17

and therefore the effective local error probabilities become qubit dependent: $\mathsf S\subset \mathsf P_n$18 Minimum-weight perfect matching then uses inhomogeneous edge weights

$\mathsf S\subset \mathsf P_n$19

on the original CSS graph (Campos et al., 2024).

Circuit-level decoding adds another layer. For Clifford-deformed elongated compass codes, correlated MWPM improves thresholds for all tested biases relative to standard MWPM under circuit-level noise (Meinking et al., 26 May 2026). For tile codes, the relevant figure of merit is the residual bias after a full syndrome-extraction cycle. The paper estimates effective single-qubit marginals $\mathsf S\subset \mathsf P_n$20 and defines

$\mathsf S\subset \mathsf P_n$21

This links hardware-specific microscopic circuits to phenomenological threshold curves (Das et al., 14 May 2026).

5. Logical Clifford gates by deformation, folding, and doubling

A distinct branch of the literature uses code deformation to implement logical Clifford operations directly. In planar topological subsystem color codes, code deformation means changing which check operators are measured so that twists are moved, braided, created, or annihilated while logical operators are dragged with the geometry (Bombin, 2010). The paper introduces colored Majorana operators $\mathsf S\subset \mathsf P_n$22 attached to twists, with braid rules such as

$\mathsf S\subset \mathsf P_n$23

and a color-dependent transformation for $\mathsf S\subset \mathsf P_n$24, and derives from this the full Clifford group on encoded qubits. The same work emphasizes that TSCCs retain 2-local gauge measurements in a 2D setting while achieving all Clifford gates by deformation (Bombin, 2010).

Folded surface codes realize a related objective through mirror-dual geometry rather than moving twists. By folding square or diamond-derived surface codes, the paper constructs two families with transversal logical $\mathsf S\subset \mathsf P_n$25 and $\mathsf S\subset \mathsf P_n$26 gates for arbitrary qudit dimension $\mathsf S\subset \mathsf P_n$27 (Moussa, 2016). The square family uses $\mathsf S\subset \mathsf P_n$28 data qudits for distance $\mathsf S\subset \mathsf P_n$29, while two cone variants for the diamond family use either $\mathsf S\subset \mathsf P_n$30 or $\mathsf S\subset \mathsf P_n$31 data qudits (Moussa, 2016). The construction is presented as a logical identity that changes physical layout but preserves code space.

Short-length transversal-gate constructions use a different mechanism again. One paper constructs smallest known doubly even CSS codes, all of which admit a transversal implementation of the Clifford group, then applies a doubling procedure to obtain smallest known weak triply even codes for the same distances and one encoded qubit (Jain et al., 2024). The comparison is explicit: DE codes provide transversal Clifford support; TE* codes provide transversal logical $\mathsf S\subset \mathsf P_n$32 with no Clifford corrections; triorthogonal codes implement logical $\mathsf S\subset \mathsf P_n$33 via transversal $\mathsf S\subset \mathsf P_n$34 plus extra physical Clifford corrections $\mathsf S\subset \mathsf P_n$35 and $\mathsf S\subset \mathsf P_n$36. The paper lists QR-based DE examples $\mathsf S\subset \mathsf P_n$37, $\mathsf S\subset \mathsf P_n$38, $\mathsf S\subset \mathsf P_n$39, up to $\mathsf S\subset \mathsf P_n$40, and QR-based TE* examples from $\mathsf S\subset \mathsf P_n$41 through $\mathsf S\subset \mathsf P_n$42 (Jain et al., 2024).

Hypergraph product codes supply an LDPC analogue of fault-tolerant Clifford gates by deformation. The construction uses generalized punctures and wormholes inside a single code block. Punctures are created by removing connected subgraphs and measuring interior qubits in $\mathsf S\subset \mathsf P_n$43 or $\mathsf S\subset \mathsf P_n$44 basis, while wormholes pair smooth and rough punctures through boundary measurements such as

$\mathsf S\subset \mathsf P_n$45

and

$\mathsf S\subset \mathsf P_n$46

These hybrid stabilizers mix $\mathsf S\subset \mathsf P_n$47 and $\mathsf S\subset \mathsf P_n$48 at the logical level and enable single-qubit Clifford gates; together with state injection, the paper proposes universal computation within a single HGP block (Krishna et al., 2019).

6. Representation-theoretic, anyonic, and nonstandard generalizations

The term “Clifford code” also has a precise representation-theoretic meaning that is not identical to stabilizer conjugation by local Clifford gates. In the Ising-anyon or Majorana setting, commuting families of even Clifford operators are classified by $\mathsf S\subset \mathsf P_n$49-isotropic subspaces of $\mathsf S\subset \mathsf P_n$50, where

$\mathsf S\subset \mathsf P_n$51

If $\mathsf S\subset \mathsf P_n$52, the associated even Clifford operators commute, and $\mathsf S\subset \mathsf P_n$53-isotropic subspaces are in one-to-one correspondence with punctured all-even self-orthogonal codes in one higher dimension (Dutta, 11 Mar 2025). The resulting code space is denoted $\mathsf S\subset \mathsf P_n$54, and detectable even Clifford errors are governed by the $\mathsf S\subset \mathsf P_n$55-orthogonal complement (Dutta, 11 Mar 2025).

Projective representation theory broadens the notion further. A projective error model is a finite group $\mathsf S\subset \mathsf P_n$56 with a projectively faithful irreducible projective representation $\mathsf S\subset \mathsf P_n$57, and a Clifford subspace code is defined by a normal subgroup $\mathsf S\subset \mathsf P_n$58 and a projective representation $\mathsf S\subset \mathsf P_n$59 such that $\mathsf S\subset \mathsf P_n$60 (Eidesen et al., 1 Jun 2026). Hybridization is obtained by taking orthogonal translates $\mathsf S\subset \mathsf P_n$61 over a coset transversal, and subsystem structure is introduced through a factorization $\mathsf S\subset \mathsf P_n$62, leading to hybrid Clifford subsystem codes $\mathsf S\subset \mathsf P_n$63 and an associated operator algebra

$\mathsf S\subset \mathsf P_n$64

(Eidesen et al., 1 Jun 2026).

A closely related paper reformulates stabilizer codes, Clifford codes, and weak stabilizer codes uniformly inside projective error models (Eidesen, 2 Jun 2025). For a Clifford code $\mathsf S\subset \mathsf P_n$65, it gives

$\mathsf S\subset \mathsf P_n$66

The same source proves that every stabilizer code is a Clifford code, while also giving infinite families of non-stabilizer Clifford codes and non-Clifford weak stabilizer codes (Eidesen, 2 Jun 2025). A common misconception is therefore that “Clifford code” is synonymous with “stabilizer code”; in this representation-theoretic sense, it is not.

Outside qubit Pauli and Majorana settings, Clifford deformation also organizes homological rotor and oscillator codes. For planar rotors, the $\mathsf S\subset \mathsf P_n$67-rotor Clifford group is

$\mathsf S\subset \mathsf P_n$68

and homological rotor CSS codes are classified up to CSS-preserving rotor Clifford transformations by the Smith normal form of the $\mathsf S\subset \mathsf P_n$69-check matrix $\mathsf S\subset \mathsf P_n$70 (Xu et al., 2023). Reversing the correspondence maps rotor Clifford operations into oscillator number-phase settings and yields multimode bosonic codes protecting against occupation-number changes and dephasing (Xu et al., 2023).

A still broader, non-quantum use appears in distributed space-time coding. There, extended Clifford algebras $\mathsf S\subset \mathsf P_n$71, with Clifford generators $\mathsf S\subset \mathsf P_n$72 and commuting involutions $\mathsf S\subset \mathsf P_n$73, are used to construct full-diversity $\mathsf S\subset \mathsf P_n$74-group ML decodable distributed space-time codes for any power-of-two number of relays. The resulting codes permit lattice decoding on a lattice of four times lesser dimension than in the general case (0704.2505). This is not a quantum error-correcting construction, but it shows that Clifford deformation language also appears in coding theory more broadly.