Dynamical Codes in Quantum Error Correction

Updated 4 July 2026

Dynamical codes are coding schemes where the constraints evolve over time, enabling protection through time-ordered measurement sequences.
They utilize periodic or aperiodic schedules—such as Floquet codes—to transition between instantaneous stabilizer groups while preserving logical information.
The concept extends beyond quantum error correction to applications in holographic networks, dynamical decoupling, and polar coding, highlighting its versatility.

Searching arXiv for papers on dynamical/Floquet codes to ground the article in published work. Dynamical codes are coding schemes in which the constraints that protect information evolve in time rather than remaining fixed. In contemporary quantum error-correction literature, the term usually denotes dynamical stabilizer codes, including Floquet codes, where a periodic or otherwise prescribed sequence of local Pauli measurements replaces a static stabilizer group, and logical information is carried by evolving equivalence classes of operators across instantaneous stabilizer groups (Alam et al., 2024). More broadly, the same term also appears in holographic tensor-network dynamics, coding-theoretic constructions for dynamical decoupling, algebraic-geometry decoding, and polar subcodes with dynamic frozen symbols (Osborne et al., 2017).

1. Dynamical stabilizer formalism

A dynamical stabilizer code is specified by a time-ordered family of commuting measurement sets rather than by one fixed commuting stabilizer group. In the Floquet subclass, the schedule is periodic, so the measurement pattern repeats after a fixed number of rounds, but the instantaneous stabilizer group can still change nontrivially inside the period (Fu et al., 2024). Logical preservation is therefore a relation between successive code spaces, not a property of a single static subspace.

A central criterion for preserving logical information is the equivalence of logical representatives across consecutive rounds. If $S^A$ and $S^B$ are the stabilizer groups in consecutive rounds with logical representatives $O^A$ and $O^B$ , logical information is preserved provided there exist $s^A\in S^A$ and $s^B\in S^B$ such that

$s^A O^A = s^B O^B.$

This condition underlies the notion of a dynamical logical qubit: a logical degree of freedom carried by an evolving equivalence class rather than by a fixed code space (Alam et al., 2024).

Several formalisms make this precise. Fu and Gottesman formulate measurement-only dynamical codes through sequences of Pauli measurements and the induced instantaneous stabilizer groups, while the strategic-code framework represents a general dynamical code by an encoder, an interrogator consisting of one or more rounds of channels or instruments, and an outcome-dependent decoder, with the overall logical channel written as a comb product $E\ast Q$ (Fu et al., 2024). A complementary qudit formulation treats a dynamical stabilizer code as an ordered product of measurement projectors, so that the effective constraint is an ordered projection $P_TP_{T-1}\cdots P_1$ ; this makes explicit that noncommuting rounds must be understood as time-ordered rather than simultaneously imposed (Radhakrishnan et al., 10 Oct 2025).

This formal breadth is significant because it places periodic Floquet schedules, adaptive approximate codes, and spacetime decoding constructions within one conceptual class: time-dependent coding protocols whose protection mechanism is intrinsically dynamical rather than merely a circuit implementation of a static code.

2. Detectors, distance, and approximate correction

Distance in a dynamical code is subtler than in a static stabilizer code because not every syndrome that exists instantaneously is necessarily accessible through the measurement schedule. Fu and Gottesman therefore distinguish the distance of an instantaneous stabilizer group, the distance of the subsystem code formed by adjacent rounds, and the unmasked distance, which tracks only those stabilizers whose syndromes are actually available through the protocol (Fu et al., 2024). Their unmasked-distance framework formalizes the fact that schedule-induced masking can reduce effective protection even when each instantaneous code space has large static distance.

Later work sharpened this into a spacetime definition. For Floquet codes, every correctable undetectable spacetime error occurring during the steady stage is a benign error: a product of measurement operators inserted at the time of measurement and pairs of identical Pauli operators sandwiching a commuting measurement. The code distance is therefore the minimal weight of an undetectable spacetime Pauli error that is not benign (Blackwell et al., 7 Oct 2025). This is the dynamical analogue of the static statement that distance is the minimum weight of an undetectable non-stabilizer logical operator.

Detectors arise from redundant measurement relations across time. In the non-invertible-symmetry formulation, a sequence of qudit Pauli measurements is mapped to a fusion of non-invertible symmetry operators in a $4+1$-dimensional $S^B$ 0-form gauge theory. Error detectors correspond to endable surface operators whose endpoints define line operators, and detectable errors are precisely those surface operators that braid non-trivially with these lines (Radhakrishnan et al., 10 Oct 2025). The same framework recovers the associated spacetime stabilizer code, making detector construction a topological statement about braiding and endability.

Approximate dynamical codes replace exact correction conditions by fidelity-based optimization. In the strategic-code setting, one optimizes the entanglement fidelity of a time-dependent encoder–interrogator–decoder triple through semidefinite programming, and the optimal encoding, decoding, and check measurements are proved unique and robust (Basak et al., 13 Feb 2025). The same work derives an approximate information-theoretic condition: if

$S^B$ 1

for all measurement transcripts $S^B$ 2, then the decoded state satisfies

$S^B$ 3

It also introduces a temporal Petz recovery map, which reduces to the usual Petz map in the static limit and coincides with the perfect recovery map when the generalized dynamical Knill–Laflamme condition holds (Basak et al., 13 Feb 2025).

3. Major code families and constructions

The best-studied dynamical codes are lattice-based quantum codes built from low-weight measurement schedules, but the family is broader than periodic Floquet constructions alone.

Family	Defining schedule or geometry	Reported property
Floquet–Bacon–Shor	Period-4 gauge-defect schedule on a $S^B$ 4 square lattice	$S^B$ 5 and can saturate $S^B$ 6
CSS honeycomb	Period-6 schedule $S^B$ 7	No explicit connection to a parent subsystem code
Qudit Floquet codes	Period-3 schedule on three-colorable lattices	Encodes $S^B$ 8 logical qudits and achieves $S^B$ 9 on hyperbolic families
Dynamic tree codes	Aperiodic schedules generated by a probabilistic finite automaton	Correct any single-qubit Pauli errors
Dynamically generated concatenated codes	Identical Clifford gates on a binary tree	Encode one logical qubit with distance growing exponentially in tree depth

Floquet–Bacon–Shor codes are constructed by viewing the Bacon–Shor subsystem code as a Floquet code with a deliberately moving gauge defect. A period-4 measurement schedule preserves the original static logical qubit and introduces dynamical logical qubits whose operators are transported around the defect. On an $O^A$ 0 lattice with $O^A$ 1 dynamical logical qubits, the distance scales as $O^A$ 2, and the family can saturate the subsystem bound $O^A$ 3 while using only weight-2 nearest-neighbor checks (Alam et al., 2024). The same schedule also exhibits autonomous self-correction of certain errors by the measurement dynamics itself.

The CSS honeycomb code and its three-dimensional extension show that Floquet codes need not descend from parent subsystem codes. The two-dimensional construction uses the period-6 schedule $O^A$ 4, dynamically embeds an instantaneous toric code, and has no explicit connection to any parent subsystem code because the group generated by all checks has only trivial global symmetries in its center (Davydova et al., 2022). The same paper generalizes the idea to three dimensions, where the schedule alternates between realizing checkerboard and X-cube fracton models, and then extends both constructions to aperiodic dynamic tree codes generated by a probabilistic finite automaton.

Qudit Floquet codes extend the honeycomb-style picture beyond qubits. On three-colorable lattices with degree-3 vertices, a period-3 schedule of two-qudit generalized Pauli measurements yields a family with persistent plaquette stabilizers after initialization, $O^A$ 5 logical qudits on a genus- $O^A$ 6 surface, and an asymptotic rate

$O^A$ 7

for the hyperbolic $O^A$ 8 family, so $O^A$ 9 as $O^B$ 0 and $O^B$ 1 (Tanggara et al., 2024).

A distinct non-Floquet branch is given by dynamically generated concatenated codes on expanding trees. There, a fixed Clifford gate applied at every node of a binary tree encodes one logical qubit into $O^B$ 2 leaves at depth $O^B$ 3. For certain gate classes the distance grows exponentially in $O^B$ 4, and under bulk or boundary noise the model exhibits coding and non-coding phases accessible through recursive tensor-enumerator methods (Sommers et al., 2024). This suggests that the dynamical-code concept is not restricted to periodic lattice schedules.

4. Circuit-level optimization, biased noise, and decoder structure

Recent work has emphasized that the practical performance of dynamical codes depends as much on syndrome-extraction circuits and decoder structure as on the underlying code family. The dynamic $O^B$ 5 Floquet code provides a particularly explicit circuit-level comparison on a torus (Capatos, 8 Jun 2026).

Circuit	Qubits / TICKs per period	Per-round threshold MWPM / BP+matching
Reset dynamic	$O^B$ 6, 24 TICKs	$O^B$ 7 / $O^B$ 8
No-reset dynamic	$O^B$ 9, 18 TICKs	$s^A\in S^A$ 0 / $s^A\in S^A$ 1
Standard ancilla-based	$s^A\in S^A$ 2, 24 TICKs	$s^A\in S^A$ 3 / $s^A\in S^A$ 4
Pipelined ancilla-based	$s^A\in S^A$ 5, 8 TICKs	$s^A\in S^A$ 6 / $s^A\in S^A$ 7

These data show that dynamic, ancilla-free syndrome extraction on the square–octagon lattice can preserve full spatial distance $s^A\in S^A$ 8 while substantially improving threshold and reducing qubit count relative to standard ancilla-based circuits (Capatos, 8 Jun 2026). The no-reset dynamic variant reaches the highest threshold of the four compared circuits, whereas the reset dynamic variant yields a faster-growing timelike distance.

The $s^A\in S^A$ 9 Floquet code illustrates a different optimization axis: tailoring the schedule to biased noise. It has no constant stabilizers, yet under infinitely biased noise it retains a persistent strip symmetry that simplifies decoding. Under code-capacity noise, its threshold increases from $s^B\in S^B$ 0 at depolarizing noise to $s^B\in S^B$ 1 at infinite bias; under SDEM3 it increases from $s^B\in S^B$ 2 to $s^B\in S^B$ 3 (Setiawan et al., 2024). The same work proves that a general dynamical code with two-qubit parity measurements cannot admit one-dimensional decoding graphs, so bias-tailored Floquet codes cannot inherit the repetition-code decoding structure that underlies the highest thresholds of some static biased-noise codes.

Dynamically condensed color codes under noisy readout sharpen the circuit-level resource picture further. Repeating commuting measurements improves performance in measurement-biased noise models, but offers little improvement in unbiased or $s^B\in S^B$ 4-biased regimes. To capture this, the teraquop footprint is generalized to the teraquop volume, the product of physical qubits and rounds of measurements needed to drive both spacelike and timelike logical-failure probabilities below $s^B\in S^B$ 5 (Derks et al., 12 May 2025). A recurring theme in that study is that decoder choice can dominate schedule choice: belief matching, by exploiting correlated hyperedge structure, can turn a worst-performing code under a given noise model into a best-performing code.

5. Logical gates, automorphisms, and fault-tolerant transformations

Dynamical codes are not only memory protocols; they also support logical computation by schedule design. Dynamic automorphism codes formalize this by viewing a computation as a sequence of low-weight measurements that simultaneously encode, correct, and apply logical automorphisms. The two-dimensional DA color code realizes all 72 automorphisms of the 2D color code, and on a stack of $s^B\in S^B$ 6 triangular patches it encodes $s^B\in S^B$ 7 logical qubits and implements the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements (Davydova et al., 2023). A three-dimensional DA color code then provides a route toward universality by realizing a non-Clifford logical gate with adaptive two-qubit measurements.

A closely related viewpoint comes from measurement-based code switching. In the ZX-calculus extension for Floquet and other dynamical stabilizer codes, a closed loop of gauge-fixing steps can return to the original codespace up to a logical Clifford gate. This produces dynamical automorphisms as “chutes and ladders” in the space of stabilizer codes, and the paper’s main worked example is a distance-preserving loop that implements a logical phase gate on the seven-qubit bare code, even though that code has no non-trivial logical Clifford gates based on single-qubit Clifford operations and qubit permutations alone (Frei et al., 1 Jun 2026).

Compilation frameworks now make such constructions systematic. Spacetime concatenation compiles syndrome-extraction circuits into dynamical codes through structured gadget layouts and encoding matrices, with sufficient conditions for fault tolerance stated as “not measuring logical operators” and “preserving the spacetime distance” (Xu et al., 11 Apr 2025). Explicit examples include a dynamical bivariate bicycle code and a dynamical Haah code. In parallel, the chain-complex framework for spacetime codes characterizes fault-tolerant transformations by maps that preserve the number of encoded qubits, fault distance, and the minimum-weight decoding problem; one application is an extension of foliated cluster-state constructions from static stabilizer codes to arbitrary spacetime codes, including Floquet and subsystem codes (Pesah et al., 11 Sep 2025).

Hardware-constrained implementations push these ideas toward architectural design. On a lattice of maximum vertex degree three, one can implement Floquet codes with nearest-neighbor gates and mid-circuit measurement, reset all qubits every cycle by “dancing” schedules, run a pair of Floquet codes simultaneously by switching the roles of data and ancilla qubits, and entangle the two codes by a logical Clifford gate. The same sparse-connectivity setting supports switching between a color code and a pair of Floquet codes to perform color-code syndrome extraction (Williamson et al., 6 Oct 2025). At the same time, geometric locality still imposes strong limits: with only a limited amount of long-range connectivity, non-Clifford gates cannot be implemented by finite-depth local circuits in two dimensions (Fu et al., 2024). This suggests that dynamical scheduling enlarges the design space without eliminating the fundamental constraints of locality.

6. Other technical meanings of “dynamical codes”

Outside quantum error correction, “dynamical codes” appears in several technically distinct literatures. In holographic tensor networks, dynamics for holographic codes is introduced through the semicontinuous limit of Jones and a unitary representation of Thompson’s group $s^B\in S^B$ 8 on the boundary Hilbert space. The resulting bulk subspace is generated from a vacuum by the action of $s^B\in S^B$ 9, while bulk large diffeomorphisms are represented by the Ptolemy group, yielding the Pt/T correspondence as a toy AdS/CFT model (Osborne et al., 2017).

In Hamiltonian control, the term denotes dynamical decoupling sequences synthesized from classical additive codes and interaction-hypergraph colorings. There the relevant objects are orthogonal arrays and additive codes over $s^A O^A = s^B O^B.$ 0 or $s^A O^A = s^B O^B.$ 1, and the sequence-length guarantees scale as

$s^A O^A = s^B O^B.$ 2

with $s^A O^A = s^B O^B.$ 3 the chromatic number of the colored interaction hypergraph (Nguyen et al., 11 Aug 2025).

In algebraic-geometry coding theory, a “dynamical system-based key equation” is derived for decoding one-point AG codes. The syndrome array is treated as a linear recurring sequence, a polynomial-matrix kernel defines a dynamical system in Oberst’s sense, and the Berlekamp–Massey–Sakata algorithm is interpreted as solving Cauchy’s homogeneous equations for that system (Andriamifidisoa et al., 2019).

In polar coding, dynamic frozen symbols give a further, unrelated use of the term. Certain frozen inputs are made data-dependent through linear relations

$s^A O^A = s^B O^B.$ 4

so that a polar code becomes a subcode of an extended BCH code while remaining decodable by successive cancellation and its extensions (Trifonov et al., 2015). Here “dynamic” refers to causal dependence in the decoding order, not to time-periodic evolution.

Across these usages, the common feature is not a single mathematical formalism but the replacement of fixed constraints by ordered, evolving, or context-dependent ones. In the quantum-error-correction literature, that replacement leads from static stabilizer groups to time-dependent measurement schedules, spacetime detectors, and logical information carried by dynamical rather than static code spaces.