Floquet Code in Quantum Error Correction

• Floquet code is a dynamical quantum error-correcting protocol that uses a periodic schedule of low-weight Pauli measurements to create a time-dependent stabilizer group for logical qubit protection.
• It leverages evolving instantaneous stabilizer groups and detector-based spacetime decoding, offering practical benefits over static stabilizer codes under hardware constraints.
• Canonical realizations, such as the honeycomb Floquet code, demonstrate efficient logical gate operations, syndrome extraction strategies, and competitive fault-tolerance thresholds.

Searching arXiv for recent and foundational Floquet-code papers to ground the article. {"query": "\"Floquet code\" quantum error correction honeycomb Stairway logical gates", "max_results": 10} Floquet codes are dynamical quantum error-correcting codes in which quantum information is protected not by a single static stabilizer group, but by a periodic schedule of low-weight Pauli measurements that generates a time-dependent instantaneous stabilizer group (ISG). In contrast to conventional stabilizer codes, where one repeatedly measures a fixed commuting set of checks, Floquet protocols allow the active stabilizers and even the representatives of logical Pauli operators to evolve over the cycle. This framework is especially natural when the hardware natively supports fast two-qubit parity measurements, while multi-qubit entangling unitaries are costly or unavailable (Townsend-Teague et al., 2023, Jacoby et al., 27 Feb 2026).

1. Dynamical stabilizers, detectors, and code distance

In a static stabilizer code one fixes commuting checks $\{S_1,\dots,S_m\}$ and repeatedly measures them. In a Floquet code one instead specifies a periodic schedule $M=[M_0,M_1,\dots,M_{T-1}]$, where each $M_t$ is an Abelian subgroup of the $n$-qubit Pauli group, and the schedule repeats with period $T$. After each step the ISG is updated by the usual stabilizer-update rules: if a newly measured Pauli already lies in the previous ISG there is no change, if it commutes with the ISG it is adjoined, and if it anticommutes with some stabilizer it replaces one anticommuting generator. After a finite warm-up stage, the rank of the ISG stabilizes, so the code encodes a fixed number $k=n-\mathrm{rank}(S_t)$ of logical qubits in the steady stage (Townsend-Teague et al., 2023, Blackwell et al., 7 Oct 2025).

Because different rounds need not commute with one another, deterministic syndrome information is generally not attached to a single measurement outcome. Instead, deterministic quantities are extracted from products of outcomes over an appropriate time window; these are the detectors. In the spacetime formulation, a Floquet code is therefore naturally analyzed in terms of measurement histories, detector flips, and undetectable spacetime Pauli errors rather than only instantaneous code states (Blackwell et al., 7 Oct 2025).

Recent formal work sharpens the notion of distance in this setting. For periodic Pauli measurements in the steady stage, every correctable, undetectable spacetime error is a product of measurement operators inserted at the time of measurement and “sandwiching” pairs of identical Pauli operators placed before and after a commuting measurement. These generate the benign subspace. The code distance is then the minimal weight of an undetectable spacetime Pauli error that is not benign, rather than the minimum weight of a nontrivial logical of any single instantaneous code. This distinction is substantive: a Floquet protocol can have large instantaneous-code distance while admitting a much smaller spacetime logical fault (Blackwell et al., 7 Oct 2025).

A related refinement is the Steady Stabilizer Group (SSG), defined as the subgroup of Pauli operators that appears in every ISG of a period. The SSG captures the part of the dynamics that survives the entire cycle and can therefore be referred to at least twice before removal. In that formulation, only SSG elements function as reliable syndromes, and correctable Floquet codes naturally require the SSG to form a classical error-correcting code in the relevant basis (Yan et al., 2024).

2. Canonical two-dimensional realizations

The paradigmatic two-dimensional realization is the honeycomb Floquet code and its CSS variants. On a trivalent, three-colorable lattice with qubits on vertices, one measures weight-2 Pauli checks on edges in a periodic sequence. In the original honeycomb form the cycle is $XX$ on one edge color, $YY$ on a second, and $ZZ$ on a third. In CSS color-code-lattice variants, a common period-six schedule is

$$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$

with one QEC round defined as one full six-step cycle. In each subround the ISG consists of the measurements just performed together with earlier ISG generators that commute with them. On the torus, the honeycomb CSS construction with periodic boundary conditions encodes two logical qubits (Moylett et al., 19 Dec 2025).

One important line of development uses ZX-calculus to “Floquetify” static codes. In "Floquetifying the Colour Code" (Townsend-Teague et al., 2023), a static color-code stabilizer measurement is rewritten as a chain of weight-2 spiders, and the time direction is reinterpreted so that the resulting protocol uses only weight-1 and weight-2 measurements. The resulting Floquet color code can be laid out on a square lattice and has period $M=[M_0,M_1,\dots,M_{T-1}]$0, while preserving the linear-distance behavior associated with the underlying static color code.

Another two-dimensional branch tailors the schedule to biased noise. The $M=[M_0,M_1,\dots,M_{T-1}]$1 Floquet code deforms the CSS honeycomb schedule by applying Hadamards on every second vertical strip of qubits, producing “A-type” and “B-type” plaquettes and a period-6 measurement cycle. Under pure $M=[M_0,M_1,\dots,M_{T-1}]$2-noise it exhibits persistent strip symmetries, so odd-parity syndrome events must occur in pairs within each strip. This simplifies decoding relative to generic Floquet schedules, even though a no-go theorem shows that no dynamical code built solely from two-qubit measurements can have a decoding graph that is a disjoint union of repetition-code chains (Setiawan et al., 2024).

These constructions already display a central structural feature of Floquet coding: the protected logical subspace is not tied to a fixed set of measured generators. Instead, the code is defined by a controlled orbit through a family of ISGs, often all locally equivalent to familiar static topological codes, but connected by measurement-induced transformations that are not available in a static protocol (Townsend-Teague et al., 2023, Moylett et al., 19 Dec 2025).

3. Construction paradigms and code families

A major theme in the subject is the systematic conversion of static LDPC-style constructions into periodic measurement protocols. "Stairway Codes: Floquetifying Bivariate Bicycle Codes and Beyond" (Jacoby et al., 27 Feb 2026) gives an explicit example. It begins with Abelian two-block group algebra (2BGA) codes, a family that includes the bivariate bicycle codes, with CSS parity checks

$M=[M_0,M_1,\dots,M_{T-1}]$3

acting on $M=[M_0,M_1,\dots,M_{T-1}]$4 physical qubits and uniform stabilizer weight $M=[M_0,M_1,\dots,M_{T-1}]$5. The syndrome-extraction circuit is represented as a foliated ZX-calculus network in a $M=[M_0,M_1,\dots,M_{T-1}]$6-dimensional space-time lattice, the time axis is rotated using the covector $M=[M_0,M_1,\dots,M_{T-1}]$7, and each weight-8 spider is decomposed into three pairwise $M=[M_0,M_1,\dots,M_{T-1}]$8 or $M=[M_0,M_1,\dots,M_{T-1}]$9 measurements plus Pauli-frame updates. The resulting protocol has period $M_t$0, with eight full time steps each divided into three sub-steps.

In this construction, different choices of periodic boundary conditions give different finite memories with different $M_t$1 and $M_t$2, and no closed-form classification is known; instances are found by numerical search over sublattices orthogonal to $M_t$3. The paper reports, among others, $M_t$4, $M_t$5, $M_t$6, and $M_t$7 examples. It also states that the $M_t$8 Stairway code achieves $M_t$9, surpassing semi-hyperbolic codes of comparable size such as $n$0 with $n$1, and that fewer than 300 physical qubits can match the distance and encoding rate of semi-hyperbolic Floquet codes using over 1300 qubits (Jacoby et al., 27 Feb 2026).

Hyperbolic constructions pursue a different tradeoff. The hyperbolic Floquet color (HCF) code on the regular $n$2 tiling uses only $n$3 and $n$4 measurements in a six-step schedule

$n$5

For this family, $n$6, the distance scales as $n$7 on the regular hyperbolic tiling, and each single fault affects at most two detectors, yielding graph-edge syndromes rather than hypergraph syndromes (Ozawa et al., 28 Sep 2025). Closely related work on derived semi-regular hyperbolic tessellations extends Floquet-code construction to compact orientable and non-orientable surfaces, again with constant-rate behavior and $n$8 scaling (Copatti et al., 31 Mar 2026).

Qudit generalizations broaden the algebraic setting further. On three-colorable $n$9 lattices with prime qudit dimension $T$0, a period-three schedule $T$1 satisfying explicit local commutation and global-identity constraints yields a Floquet code with $T$2 logical qudits on a genus-$T$3 surface and asymptotic rate

$T$4

as $T$5. This construction includes earlier qubit and qudit Floquet codes as special cases (Tanggara et al., 2024).

4. Distance, decoding, and noise thresholds

Decoding Floquet codes is a spacetime problem because a single fault can influence several rounds of inferred stabilizers. Accordingly, most analyses define detectors over windows, construct detection graphs or hypergraphs, and decode histories rather than single snapshots.

For Stairway codes, two distinct distances are emphasized. The embedded distance $T$6 is the minimum weight of an operator commuting with all ISGs when measured pairs are merged after each measurement, while the circuit distance $T$7 is the minimum number of EM3 faults yielding an undetected logical error. Under the EM3 noise model of Gidney and Newman, each two-qubit parity measurement faults with probability $T$8 by applying a full depolarizing channel to the two qubits and, in half of those events, also flipping the reported outcome. Monte Carlo $T$9-round memory experiments decoded with the Tesseract beam-search decoder yield pseudo-thresholds $k=n-\mathrm{rank}(S_t)$0, and at $k=n-\mathrm{rank}(S_t)$1 Stairway codes outperform hyperbolic and compiled bivariate bicycle benchmarks by up to $k=n-\mathrm{rank}(S_t)$2 in logical error rate (Jacoby et al., 27 Feb 2026).

The $k=n-\mathrm{rank}(S_t)$3 code was designed specifically for biased noise. Under circuit-level SDEM3 noise and a three-QEC-round memory experiment, the reported threshold rises from $k=n-\mathrm{rank}(S_t)$4 at $k=n-\mathrm{rank}(S_t)$5 to $k=n-\mathrm{rank}(S_t)$6 at $k=n-\mathrm{rank}(S_t)$7, whereas the CSS honeycomb benchmark decreases from $k=n-\mathrm{rank}(S_t)$8 to $k=n-\mathrm{rank}(S_t)$9. Under code-capacity noise, the threshold for $XX$0 rises from $XX$1 to $XX$2 as the bias increases, while the honeycomb code remains near $XX$3 (Setiawan et al., 2024).

The syndrome structure itself can dominate decoder complexity. In the HCF code, each single fault flips at most two detectors, so the syndrome lives on a sparse graph and can be decoded efficiently by minimum-weight perfect matching. The reported circuit-level threshold under EM3 is $XX$4, and the mean decoding time is fit to $XX$5 with $XX$6 for MWPM (Ozawa et al., 28 Sep 2025). By contrast, Floquet codes involving $XX$7, $XX$8, and $XX$9 schedules often produce hypergraph-structured syndromes, which motivates belief-propagation, OSD-type, or beam-search decoders.

Logical gates can also be benchmarked as decoding problems. In fold-transversal and Dehn-twist constructions for CSS Floquet color-code lattices, circuit-level depolarizing simulations using Stim, PyMatching, and BP + LSD-0 yield logical-gate thresholds in the range $YY$0, with subthreshold scaling

$YY$1

and gate performance within a factor $YY$2 of the corresponding memory experiment for the same QEC-round budget (Moylett et al., 19 Dec 2025).

A separate line of work derives a statistical-mechanics formulation for local decoherence in the Hastings–Haah Floquet code. In a “simple-error” regime the full three-dimensional maximum-likelihood decoding problem factorizes into independent two-dimensional random-bond Ising models, with threshold $YY$3. In that regime a diagnostic built from relative entropy detects the persistence or breakdown of the defining $YY$4 anyon automorphism at the same threshold (Tang et al., 26 Apr 2025).

5. Logical operations, defects, boundaries, and no-go structure

A central question for Floquet codes is whether the measurement dynamics itself can implement useful logical operations. For CSS Floquet codes on color-code lattices, two such mechanisms are explicit. First, fold-transversal gates exploit a lattice reflection implementing a ZX-duality. A fold-transversal Hadamard-type operator

$YY$5

maps $YY$6 and $YY$7 on both encoded qubits. Second, a fold-transversal $YY$8-type gate built from $YY$9 gates across the fold implements $ZZ$0 and $ZZ$1. Logical CNOT can be realized by Dehn twists along nontrivial cycles, either in linear time using $ZZ$2 QEC rounds or in $ZZ$3 depth with long-range qubit permutations when the hardware permits them (Moylett et al., 19 Dec 2025).

Twist defects provide another route to logical structure. In the honeycomb Floquet code, one can create condensation defects by condensing emergent fermions along one-dimensional paths while preserving the hexagonal connectivity, using only 2-body measurements and retaining the three-round period. The endpoints of such paths are twist defects. In the $ZZ$4 case, $ZZ$5 twists encode $ZZ$6 dynamical qubits beyond the two global torus logicals, with logical operators given by long fermion strings connecting different endpoints. Boundary versions of the same construction produce a planar $ZZ$7 Floquet code with a single logical qubit stored in a pair of boundary twists (Ellison et al., 2023).

Floquet protocols have also been adapted to defective hardware. A qubit-removal algorithm for two-dimensional Floquet codes deletes a defective data-qubit pair, splits and merges nearby plaquettes, preserves trivalence and three-face-colorability, requires no additional connectivity, and leaves the original measurement schedule unchanged. For the planar honeycomb code, the reported fault-tolerant regime extends up to a fabrication defect probability of $ZZ$8, with performance described as competitive with the surface code despite sparser connectivity (McLauchlan et al., 2024).

At the same time, Floquet dynamics does not evade locality-based limitations on transversal or shallow logical gates. A Bravyi–König theorem has been established for Floquet codes generated by locally conjugate ISGs, and extended to a class of generalized logical unitaries that need not preserve the codespace at each intermediate step. In this setting, any finite-depth, finite-range Floquet circuit in $ZZ$9 dimensions effects a logical operator in the $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$0-th level of the Clifford hierarchy. A plausible implication is that universality still requires ingredients such as magic-state injection, higher-depth circuits, or nonlocal code deformation, even when logical information is transported by a measurement schedule rather than a static stabilizer group (Mackeprang et al., 29 Jan 2026).

6. Higher dimensions, physical phases, and experimental realizations

Floquet coding extends beyond two-dimensional topological memories. The X-Cube Floquet code is defined on a three-dimensional lattice built from intersecting $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$1, $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$2, and $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$3 layers and uses a six-step sequence of two-qubit measurements. Over one cycle the codespace switches between that of the X-Cube fracton order and stacks of entangled two-dimensional toric codes. On an $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$4 periodic system it encodes $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$5 logical qubits, with logical operators of length $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$6, and the construction argues for a non-zero error threshold (Zhang et al., 2022). More generally, coupled-spin-chain constructions recover Floquet versions of the 3D toric code and X-Cube code, and extend to $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$7-dimensional Floquet $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$8 toric codes and generalized $$\mathrm{rXX}\rightarrow \mathrm{gZZ}\rightarrow \mathrm{bXX}\rightarrow \mathrm{rZZ}\rightarrow \mathrm{gXX}\rightarrow \mathrm{bZZ},$$9-dimensional Floquet $M=[M_0,M_1,\dots,M_{T-1}]$00-cube codes, with the SSG providing a unified error-correction criterion (Yan et al., 2024).

Three-dimensional Floquet topological order can also be engineered by “rewinding” measurement schedules. One construction produces a Floquet code whose ISGs have the same topological order as the 3D toric code and exhibits a splitting in which a single copy of 3D toric code transforms into two copies up to nonlocal stabilizers. The same work describes boundaries and argues that stacking the code with two copies of 3D subsystem toric code allows a transversal logical $M=[M_0,M_1,\dots,M_{T-1}]$01 gate (Dua et al., 2023). A distinct 3D construction realizes a fermionic toric code throughout the cycle while preserving all three logical qubits at every step; with periodic sizes $M=[M_0,M_1,\dots,M_{T-1}]$02, it uses $M=[M_0,M_1,\dots,M_{T-1}]$03 qubits, encodes $M=[M_0,M_1,\dots,M_{T-1}]$04, and has distance $M=[M_0,M_1,\dots,M_{T-1}]$05 (Watanabe et al., 13 Feb 2026).

Floquet codes are also studied as phases of monitored quantum matter. On the honeycomb lattice, the periodic measurement sequence can be interpreted as a measurement-only realization of a Floquet-enriched topological phase with $M=[M_0,M_1,\dots,M_{T-1}]$06 transmutation after every cycle. Along a continuous interpolation to a stationary toric-code phase obtained by randomly omitting green and blue measurements with probability $M=[M_0,M_1,\dots,M_{T-1}]$07, numerics find a critical point at $M=[M_0,M_1,\dots,M_{T-1}]$08 with correlation-length exponent $M=[M_0,M_1,\dots,M_{T-1}]$09 and $M=[M_0,M_1,\dots,M_{T-1}]$10 (Vu et al., 2023). In a complementary twist-defect-network formulation, the honeycomb Floquet code is associated with a unitary loop of irrational chiral Floquet index $M=[M_0,M_1,\dots,M_{T-1}]$11, implying a boundary anomaly: there is no simple gapped boundary with the same period as the bulk schedule (Sullivan et al., 2023). Under weak measurements and coherent errors, the same honeycomb code maps to random Gaussian fermionic circuits and exhibits three crossover regimes interpreted as qubit fractionalization, an emergent Majorana liquid, and Majorana pairing with gauge ordering (Zhu et al., 2023).

Finally, small-scale experimental realization has begun. A Floquet–Bacon–Shor code has been implemented on a superconducting quantum processor using a $M=[M_0,M_1,\dots,M_{T-1}]$12 lattice of data qubits. The period-4 schedule encodes one static logical qubit and one dynamical logical qubit, supports fault-tolerant encoding and repeated error detection, demonstrates universal single-qubit logical gates on the dynamical qubit, and realizes a logical CNOT between the static and dynamical qubits. The reported error-detected logical Bell-state fidelity is $M=[M_0,M_1,\dots,M_{T-1}]$13 (Sun et al., 5 Mar 2025).

Taken together, these developments define Floquet codes as a broad dynamical coding framework rather than a single lattice model. Their distinguishing ingredients are periodic low-weight measurements, time-dependent logical representatives, detector-based spacetime decoding, and the ability to interpolate between static-code behavior and genuinely dynamical phenomena such as anyon automorphisms, twist manipulations, and monitored topological phases.