Fold-Transversal Gadgets

• Fold-transversal gadgets are fault-tolerant logical gate constructions that interleave fixed single-qubit operations with disjoint two-qubit interactions governed by symmetries like mirror and ZX-duality.
• They extend ordinary transversality by allowing controlled pairwise interactions, which expand the logical gate set while introducing a trade-off in fault tolerance due to potential error propagation.
• These gadgets are applied in diverse settings such as CSS codes, surface codes, and qLDPC families, offering scalable paths for implementing logical operations in quantum error correction.

Fold-transversal gadgets are fault-tolerant logical-gate constructions in which physical operations are organized not as independent single-qubit factors alone, but as single-qubit gates on fixed points together with disjoint two-qubit gates on qubit pairs specified by a fold, mirror symmetry, or $ZX$-duality. In the quantum error-correction literature, they occupy an intermediate position between strict transversality and more general in-block circuits: they enlarge the available logical gate set by allowing controlled interaction within paired subsystems, while retaining constant-depth, symmetry-driven implementations on many CSS, surface-code, and qLDPC families (Breuckmann et al., 2022, Chakraborty et al., 13 Feb 2026).

1. Definition and conceptual scope

A standard formulation treats a stabilizer code with physical Clifford gadgets $U\in Cl_n$ that preserve the codespace, equivalently

$$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$

where $\mathcal S\subset \mathcal P_n$ is the stabilizer group (Chakraborty et al., 13 Feb 2026). Within that setting, an ordinary transversal gadget is a tensor product of one-qubit Cliffords,

$$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$

whereas a fold-transversal gadget allows certain two-qubit Clifford interactions inside a block, organized by a pairing map $\tau$ (Chakraborty et al., 13 Feb 2026).

For CSS codes, the fold is usually defined by a $ZX$-duality $\tau$. A unitary is fold-transversal if it is supported on the pairs

$\{i,\tau(i)\},\qquad i=1,2,\dots,n,$

so that the physical circuit decomposes into one-qubit gates on fixed points $i=\tau(i)$ and two-qubit gates on disjoint orbit pairs $U\in Cl_n$0 (Breuckmann et al., 2022, Chakraborty et al., 13 Feb 2026). In the CSS formulation of Breuckmann and Burton, a fold-transversal gate is explicitly distinguished from a strictly transversal gate: the former comprises single-qubit gates and disjoint two-qubit gates, whereas the latter uses only single-qubit gates (Breuckmann et al., 2022).

The literature consistently presents fold-transversality as more expressive than ordinary transversality but less fault tolerant. One paper ranks the architectures as

$U\in Cl_n$1

because a single fault can propagate to a two-qubit error under folding (Chakraborty et al., 13 Feb 2026). At the same time, fold-transversal gates are motivated by geometries in which those propagated two-qubit errors may still have distinguishable syndromes and remain correctable in practice (Chakraborty et al., 13 Feb 2026).

A broader topological variant appears in “quantum origami,” where folding manifolds converts modular transformations into local layer permutations implemented by products of onsite SWAPs. There the fold-transversal gadget is a product of independent local SWAPs between layers, realizing mapping-class-group operations such as $U\in Cl_n$2, $U\in Cl_n$3, and charge conjugation in constant depth (Zhu et al., 2017). This use is geometrically different from the CSS $U\in Cl_n$4-duality framework, but it shares the central idea that a nonlocal logical transformation can become local after an appropriate fold.

2. Algebraic and symplectic framework

The modern theory is formulated in binary symplectic language. A Pauli operator is written as

$U\in Cl_n$5

with symplectic form

$U\in Cl_n$6

and Clifford conjugation represented by symplectic matrices $U\in Cl_n$7 satisfying $U\in Cl_n$8 (Chakraborty et al., 13 Feb 2026). This is the natural language for proving whether a physical fold-transversal circuit induces a desired logical Clifford.

For CSS codes, the structural input is a $U\in Cl_n$9-duality: an automorphism of the underlying classical code that exchanges $$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$0- and $$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$1-check sectors. If a code is self-$$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$2-dual, the duality can be used to build fold-transversal Hadamard-type and phase-type logical gates (Breuckmann et al., 2022). The canonical Hadamard-type construction is

$$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$3

which is always an encoded logical gate for a CSS code admitting a $$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$4-duality $$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$5 (Breuckmann et al., 2022). The phase-type construction is

$$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$6

subject to explicit parity constraints on fixed points and on the support of each $$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$7-check (Breuckmann et al., 2022).

In group-algebra and two-block qLDPC codes, the same logic appears in different notation. For bivariate bicycle codes $$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$8, the standard $$U^\dagger S U \in \mathcal S \qquad \forall S\in\mathcal S,$$9-duality $\mathcal S\subset \mathcal P_n$0 exchanges horizontal and vertical sectors, leading to

$\mathcal S\subset \mathcal P_n$1

while a second duality $\mathcal S\subset \mathcal P_n$2 for symmetric BB codes gives the phase-/CZ-type gadget

$\mathcal S\subset \mathcal P_n$3

(Eberhardt et al., 2024). These formulas make explicit that the fold is simultaneously geometric and algebraic: it is a symmetry pairing of qubits, checks, or lattice directions that turns a logical Clifford into a layerwise or orbitwise local circuit.

3. Realizations in surface codes, CSS codes, and qLDPC families

Fold-transversal gadgets first became prominent in folded surface-code constructions, then generalized to algebraic CSS and qLDPC settings.

Setting	Representative gadget	Reported role
Folded square and cone surface codes	Fold-transversal $\mathcal S\subset \mathcal P_n$4, $\mathcal S\subset \mathcal P_n$5	Imports color-code-like transversal Cliffords into surface-code-derived architectures (Moussa, 2016)
Self-$\mathcal S\subset \mathcal P_n$6-dual CSS codes	$\mathcal S\subset \mathcal P_n$7, $\mathcal S\subset \mathcal P_n$8	General framework for fold-transversal Clifford gates from $\mathcal S\subset \mathcal P_n$9-dualities (Breuckmann et al., 2022)
Bring’s code $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$0	Fold-transversal $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$1-type and $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$2-type gates	Generates a large logical Clifford subgroup; full $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$3 on a restricted 4-qubit subspace (Breuckmann et al., 2022)
BB codes $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$4, $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$5	$$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$6, $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$7, $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$8	Fold-transversal Clifford gates without overhead in symmetric qLDPC codes (Eberhardt et al., 2024)
Unrotated surface-code cultivation	Measured $$U_{\mathrm{trans}}=\bigotimes_{i=1}^n U_i,\qquad U_i\in Cl_1,$$9	Central logical-check gadget for magic-state cultivation (Sahay et al., 5 Sep 2025)
Rotated surface code	Fold-transversal $\tau$0 interpreted as $\tau$1 preparation	Diagrammatic certification of $\tau$2 via Pauli webs (Wan et al., 2 Feb 2025)

Moussa’s folded surface-code construction is foundational because it identifies a mirror-dual pairing between $\tau$3-type and $\tau$4-type structure in a doubled-and-glued surface-code geometry. In the folded square and cone families, logical $\tau$5 and $\tau$6 become transversal in the folded sense: single-qudit gates on qudits lying on the fold and paired two-qudit gates across mirror-related qudits off the fold (Moussa, 2016). The mechanism is explicitly modeled on the color-code self-duality responsible for transversal Clifford gates, but realized through mirror duality rather than identical $\tau$7 stabilizers on the same plaquette (Moussa, 2016).

Breuckmann and Burton generalize this to arbitrary CSS codes through $\tau$8-dualities. Their worked example is Bring’s code $\tau$9, where the automorphism group and a chosen involutive duality yield explicit fold-transversal Hadamard-type and phase-type logical Cliffords (Breuckmann et al., 2022). On all eight logical qubits these operations generate $ZX$0, not the full $ZX$1, but on a four-qubit invariant symplectic subspace they realize the full Clifford group $ZX$2 (Breuckmann et al., 2022).

Bivariate bicycle codes provide a qLDPC realization with especially explicit logical-operator bases. For odd $ZX$3, the paper proves purity and principality, giving

$ZX$4

and then computes the induced symplectic action of $ZX$5, $ZX$6, $ZX$7, and $ZX$8 for codes such as $ZX$9 and $\tau$0 (Eberhardt et al., 2024). The fold-transversal $\tau$1-type gate in these codes is described geometrically by a reflection line: mirrored qubits receive $\tau$2, while qubits fixed by the fold receive $\tau$3 or $\tau$4 (Eberhardt et al., 2024).

Two later surface-code papers reposition fold-transversal gadgets as state-preparation and measurement primitives. One studies a rotated-surface-code folded $\tau$5-gate circuit and shows, using ZX-calculus and Pauli webs, that applying it to a transversally prepared logical $\tau$6 yields $\tau$7 by transforming the logical $\tau$8 correlator into a logical $\tau$9 correlator (Wan et al., 2 Feb 2025). Another uses a fold-transversal logical Hadamard-type check

$\{i,\tau(i)\},\qquad i=1,2,\dots,n,$0

on the unrotated surface code as the central measured gadget in a cultivation protocol, implemented via GHZ ancillas together with controlled-$\{i,\tau(i)\},\qquad i=1,2,\dots,n,$1 on diagonal qubits and CCZ on mirrored pairs (Sahay et al., 5 Sep 2025).

4. Addressability, high-rate codes, and compositional synthesis

A major limitation of many symmetry-derived transversal or fold-transversal gates is addressability: natural code automorphisms often act on many logical qubits simultaneously. Several recent constructions address this directly.

A prominent example is the family of self-dual quantum Reed–Muller codes

$\{i,\tau(i)\},\qquad i=1,2,\dots,n,$2

for even positive $\{i,\tau(i)\},\qquad i=1,2,\dots,n,$3, with asymptotics

$\{i,\tau(i)\},\qquad i=1,2,\dots,n,$4

(Tansuwannont et al., 10 Feb 2026). In this setting, the physical fold-transversal gadgets are built from involutive coordinate permutations $\{i,\tau(i)\},\qquad i=1,2,\dots,n,$5, giving swap-type gates $\{i,\tau(i)\},\qquad i=1,2,\dots,n,$6 and phase-type gates $\{i,\tau(i)\},\qquad i=1,2,\dots,n,$7. Their action on Pauli strings is

$\{i,\tau(i)\},\qquad i=1,2,\dots,n,$8

and

$\{i,\tau(i)\},\qquad i=1,2,\dots,n,$9

where $i=\tau(i)$0 (Tansuwannont et al., 10 Feb 2026). A single such gadget is generally not addressable, but the paper’s central combinatorial device is to take products over subsets,

$i=\tau(i)$1

which cancel unwanted logical action and isolate addressable $i=\tau(i)$2, then addressable $i=\tau(i)$3, $i=\tau(i)$4, and arbitrary two-qubit Clifford gates. The resulting theorem states that the full logical Clifford group $i=\tau(i)$5 is generated by $i=\tau(i)$6 together with the fold-transversal family $i=\tau(i)$7 (Tansuwannont et al., 10 Feb 2026).

A different compositional direction appears in tensor-network “quantum lego” constructions. There the basic operation is gluing small symmetry-carrying tensors by Bell or generalized hyperedge states, while tracking logical action through operator matching and operator flow (Cao et al., 3 Mar 2026). The fundamental rule is that if a symmetry acts as

$i=\tau(i)$8

then the code defined by the tensor $i=\tau(i)$9 implements logical operator $U\in Cl_n$00 physically as $U\in Cl_n$01 (Cao et al., 3 Mar 2026). Although this framework does not literally define fold-transversal gates, it supplies a gadgetized methodology for constructing localized and addressable transversal gates by composing small modules. It produces finite-rate families with strongly transversal $U\in Cl_n$02, $U\in Cl_n$03, $U\in Cl_n$04, and $U\in Cl_n$05, as well as holographic and fractal-like codes with addressable transversal inter-, meso-, and intra-block $U\in Cl_n$06, $U\in Cl_n$07, and $U\in Cl_n$08 gates (Cao et al., 3 Mar 2026). This suggests a broader synthesis viewpoint in which fold-transversal gadgets are one member of a larger class of symmetry-preserving modular constructions.

Addressability also appears in heterogeneous logical interfaces. Chain-map synthesis for CSS codes constructs sparse, low-depth logical CNOT and CZ gadgets between distinct code blocks by solving homological compatibility equations for a bipartite coupling matrix $U\in Cl_n$09 or $U\in Cl_n$10 (Benhemou et al., 2 Jul 2026). The paper does not define these as fold-transversal, but it explicitly recovers known transversal constructions and finds new low-depth solutions between heterogeneous code pairs, which is closely aligned with the general problem of replacing monolithic logical operations by shallow gadget interfaces (Benhemou et al., 2 Jul 2026).

5. Fundamental limits and no-go theorems

The strongest general limitations presently known are formulated in terms of $U\in Cl_n$11-fold transversality. A set of gadgets is $U\in Cl_n$12-fold transversal if there is a partition of physical qubits into blocks $U\in Cl_n$13 with $U\in Cl_n$14 such that every gadget decomposes as

$U\in Cl_n$15

with $U\in Cl_n$16 in the Clifford setting (Chakraborty et al., 13 Feb 2026). This unifies ordinary transversality ($U\in Cl_n$17), fold-transversal gadgets ($U\in Cl_n$18), and larger in-block support.

The central result is:

In order to implement the full Clifford group on stabilizer codes with $U\in Cl_n$19 logical qubits, we need at least Clifford $U\in Cl_n$20-fold transversal gadgets acting on the physical qubits (Chakraborty et al., 13 Feb 2026).

The immediate corollary is that no stabilizer code permits a Clifford transversal implementation of the full Clifford group on multiple logical qubits, and no stabilizer code permits a Clifford fold-transversal implementation of the full Clifford group on more than two logical qubits (Chakraborty et al., 13 Feb 2026). The proof is group-theoretic and symplectic. Its ingredients are an order-divisibility lemma, a primitive-prime-divisor argument using Zsigmondy’s theorem, explicit existence of Clifford elements of orders $U\in Cl_n$21 and $U\in Cl_n$22, and a bound on element orders in lower-dimensional Clifford groups via

$U\in Cl_n$23

(Chakraborty et al., 13 Feb 2026).

Several consequences are sharply delimited. For $U\in Cl_n$24, there is no obstruction: the $U\in Cl_n$25 Steane code remains the standard example admitting the full logical Clifford group transversally (Chakraborty et al., 13 Feb 2026). For $U\in Cl_n$26, the fold-transversal no-go does not apply, so two logical qubits remain an open small case in the fold-transversal setting (Chakraborty et al., 13 Feb 2026). For $U\in Cl_n$27, the impossibility is universal under the paper’s assumptions: single-block stabilizer codes, physical gadgets restricted to Clifford fold-transversal form, and target gate set equal to the full logical Clifford group (Chakraborty et al., 13 Feb 2026).

The same paper also proves a separate no-go theorem for code automorphisms: $U\in Cl_n$28 Its proof uses a two-qubit Bell gate of order $U\in Cl_n$29 and a permutation lemma showing that pure qubit permutations preserve a commutation property incompatible with the Bell-gate action on logical Paulis (Chakraborty et al., 13 Feb 2026). This establishes that merely adding qubit relabelings to transversal gadgets does not recover the full multiqubit logical Clifford group.

These no-go results do not imply that fold-transversal gadgets are useless. They rule out only the full logical Clifford group in the specified regimes. The same paper explicitly notes that partial Clifford subsets may still be realizable, and that the motivation for fold-transversal gates is precisely that they can implement a larger set of logical operations than strict transversality (Chakraborty et al., 13 Feb 2026). A plausible implication is that the research problem has shifted from asking whether folding alone suffices for complete logical control to asking which subsets, compositions, or hybridizations remain both useful and fault tolerant.

6. Architectures, performance, and open directions

Recent architectural work treats fold-transversal gadgets not only as abstract code symmetries but as deployable building blocks. In the transversal STAR architecture for neutral atoms, the logical Clifford gadget family includes $U\in Cl_n$30, $U\in Cl_n$31, $U\in Cl_n$32, and CNOT, with logical $U\in Cl_n$33 implemented by the fold-transversal surface-code protocol on rotated patches, specifically the distance-preserving “Intra-SE” construction (Ismail et al., 22 Sep 2025). These gadgets are executed with one syndrome-extraction round after each logical gate, and the paper develops an effective logical-noise model by comparing base and full circuits. At its projected limit, the architecture can simulate local Hamiltonians with total simulation volume exceeding $U\in Cl_n$34, using approximately $U\in Cl_n$35 physical qubits at physical error rate $U\in Cl_n$36, corresponding to a fully fault-tolerant computation requiring over $U\in Cl_n$37–$U\in Cl_n$38 $U\in Cl_n$39 gates (Ismail et al., 22 Sep 2025).

Fold-transversal measurement gadgets also play a central role in magic-state preparation. In “fold-transversal surface code cultivation,” the key gadget is a measured logical $U\in Cl_n$40 on the unrotated surface code, where the fold is reflection across the main diagonal and the controlled logical check uses GHZ ancillas, controlled-$U\in Cl_n$41 on diagonal qubits, and CCZ on mirrored pairs (Sahay et al., 5 Sep 2025). The paper reports that at physical error rate $U\in Cl_n$42 under standard depolarizing noise, the $U\in Cl_n$43 scheme reaches a lowest logical error rate of $U\in Cl_n$44 with average $U\in Cl_n$45 attempts, while the $U\in Cl_n$46 scheme reaches logical error rate $U\in Cl_n$47 in about $U\in Cl_n$48 attempts per kept shot; for target logical error rate $U\in Cl_n$49, the protocol gives an order-of-magnitude spacetime-volume improvement over the compared cultivation schemes (Sahay et al., 5 Sep 2025).

At the verification and analysis level, ZX-calculus and Pauli webs have become a compact language for certifying fold-transversal action. A dedicated analysis of rotated-surface-code $U\in Cl_n$50 preparation shows that when the fold-transversal $U\in Cl_n$51-gate circuit is applied to transversally initialized $U\in Cl_n$52 data, the logical $U\in Cl_n$53 correlator is converted into the logical $U\in Cl_n$54 correlator, i.e.

$U\in Cl_n$55

(Wan et al., 2 Feb 2025). The contribution is deliberately conceptual rather than threshold- or decoder-oriented, but it demonstrates how fold-transversal gadgets can be validated diagrammatically when full stabilizer proofs are cumbersome (Wan et al., 2 Feb 2025).

Several open problems are explicit in the literature. One is whether non-Clifford transversal or fold-transversal physical gates can help construct the full logical Clifford group, since the strongest no-go theorems assume the physical gadgets themselves lie in the Clifford group (Chakraborty et al., 13 Feb 2026). Another is the extension to hybrid architectures such as “$U\in Cl_n$56-fold transversal plus permutation” constructions, which later work suggests may access larger logical gate sets at the price of worse error propagation (Chakraborty et al., 13 Feb 2026). A third is the search for geometrically favorable codes where $U\in Cl_n$57-fold gadgets, though they spread a fault onto $U\in Cl_n$58 qubits, produce error patterns with unique syndromes and remain practically manageable (Chakraborty et al., 13 Feb 2026). More broadly, the coexistence of strong impossibility theorems for single-block Clifford fold-transversal architectures and explicit high-rate constructions based on code-specific symmetries suggests that the exact scope of “fold-transversal” remains an active locus of refinement rather than a settled taxonomy (Chakraborty et al., 13 Feb 2026, Tansuwannont et al., 10 Feb 2026).

Fold-transversal gadgets therefore occupy a distinct place in fault-tolerant quantum computation. They are neither mere variants of ordinary transversality nor generic constant-depth logical circuits. Their defining feature is the exploitation of a pairing structure—geometric, algebraic, or topological—that promotes selected logical operations to disjoint one- and two-body physical patterns. The resulting theory now spans folded surface codes, self-$U\in Cl_n$59-dual CSS constructions, qLDPC symmetries, cultivation and state-preparation gadgets, neutral-atom logical architectures, and topological origami, while remaining sharply constrained by multiqubit no-go theorems for the full logical Clifford group (Moussa, 2016, Breuckmann et al., 2022, Eberhardt et al., 2024, Zhu et al., 2017, Chakraborty et al., 13 Feb 2026).