Instantaneous Stabilizer Group in Floquet Codes

• Instantaneous Stabilizer Group (ISG) is the set of Pauli operators stabilizing the Floquet-evolved quantum state after sequential measurements.
• ISG is updated by incorporating new two-qubit check measurements and discarding incompatible stabilizers, ensuring dynamic protection against errors.
• The accompanying Steady Stabilizer Group (SSG) emerges as the invariant subset of the ISG, enabling repeated syndrome extraction and reliable error correction.

The Instantaneous Stabilizer Group (ISG) is a central concept in the theory of Floquet codes, particularly those constructed from coupled spin chains and related quantum error-correcting codes. The ISG at any given time step captures the set of Pauli operators that stabilize the Floquet-evolved quantum state after the sequential measurement of predetermined local checks. This temporal stabilizer structure informs the characterization of dynamically protected quantum information and underpins the connection between periodic measurement protocols, error detection, and the emergence of classical code structures within quantum codes (Yan et al., 2024).

1. Formal Definition and Temporal Dynamics

Consider a system of $N$ qubits, with the Pauli group denoted as $\mathcal{P}_N$. At discrete time step $t$, the code protocol measures a prescribed layer of two-qubit commuting checks $C(t) \subset \mathcal{P}_N$. Each check $O_c$ corresponds to a projector $P_c = \frac{1}{2} (1 + O_c)$ onto the $+1$ eigenspace. Evolving from an initial state $|\psi_0\rangle$, the Floquet state after $t$ layers is

$$|\psi(t)\rangle \propto \left(\prod_{s=0}^{t} \prod_{c\in C(s)} P_c\right)|\psi_0\rangle.$$

The Instantaneous Stabilizer Group at time $t$ is

$$\mathrm{ISG}(t) := \{\,g \in \mathcal{P}_N\;|\; g|\psi(t)\rangle = |\psi(t)\rangle\,\}$$

which is equivalently generated by the union of newly measured checks $C(t)$ and all previously surviving stabilizers from earlier time steps. The update rule is

$$\mathrm{ISG}(t) = \langle\,\mathrm{ISG}(t-1) \cap \mathrm{Comm}(C(t)) \cup C(t)\,\rangle,$$

with $\mathrm{Comm}(C(t))$ denoting the centralizer of $C(t)$ in $\mathcal{P}_N$. Each measurement round, stabilizers that anticommute with newly measured checks are eliminated, and the new checks are incorporated.

2. Relationship to the Steady Stabilizer Group (SSG)

Over a Floquet cycle of $m$ measurement rounds, the ISG traverses a sequence of subgroups:

$$\mathrm{ISG}(0) \rightarrow \mathrm{ISG}(1) \rightarrow \dots \rightarrow \mathrm{ISG}(m-1) \rightarrow\mathrm{ISG}(m)=\mathrm{ISG}(0).$$

The Steady Stabilizer Group (SSG) is defined as the intersection:

$$\mathrm{SSG} := \bigcap_{t=0}^{m-1} \mathrm{ISG}(t),$$

comprising Pauli operators that commute with every measurement check in every layer and therefore persist through every stage. These elements never leave the ISG; they generate a time-invariant stabilizer subgroup. In specific models: in 2D, SSG comprises three-color plaquette operators; in 3D Floquet toric codes, SSG includes octahedron and loop-chain operators.

3. Error Correction Paradigm and Classical Code Structure

A foundational property of Floquet codes is that not all ISG elements participate equivalently in error correction. Only SSG elements, which survive through every measurement stage, can be reliably "referred to" at least twice per cycle. In the code's execution, SSG generators are not directly measured; rather, their eigenvalues are inferred via products of outcomes from two-qubit check measurements. If a stabilizer appears in only a single round (i.e., is not in SSG), its syndrome can be referenced only once, preventing detection of errors occurring between references. In contrast, SSG elements are inferred multiple times per cycle. This periodic referencing ensures that single-qubit errors result in a detectable syndrome change, effectively rendering SSG a classical parity-check code on effective qubits. Every syndrome generator must be referenced at least twice—this repetition is the crux of single-error detectability in Floquet codes.

4. 2-Step Bacon–Shor Floquet Code: Concrete Realization

An explicit example is provided by the 2-step Bacon–Shor Floquet code on an $L \times L$ square lattice with qubits on vertices:

• Even rounds ($2r$): measure all horizontal $Z\otimes Z$ checks on edges.
• Odd rounds ($2r+1$): measure all vertical $X\otimes X$ checks.

The SSG is generated by:

• $Z$-type line parities:

$$S^Z_i = \prod_{x\;\text{on vertical lines }i\text{ and }i+1} Z_x, \quad i=1,\ldots,L-1$$

• $X$-type line parities:

$$S^X_j = \prod_{y\;\text{on horizontal rows }j\text{ and }j+1} X_y, \quad j=1,\ldots,L-1$$

All SSG generators commute with both horizontal and vertical two-qubit checks, thus remain in the ISG at all times. For even rounds, $S^X_j$ form a classical $L$-bit repetition code on horizontal lines and can be inferred from $X\otimes X$ check outcomes in at least two rounds per cycle; similarly, $S^Z_i$ on vertical lines in odd rounds. This satisfies the requirement that SSG generators form a classical error-correcting code with the needed repetition to detect single-qubit errors.

A summary of ISG and SSG structure in the 2-step Bacon–Shor code is organized as:

Round	Measured Checks	ISG Generators
Even ($2r$)	Horizontal $Z\otimes Z$	$\{Z\otimes Z$, $S^Z_i$, $S^X_j\}$
Odd ($2r+1$)	Vertical $X\otimes X$	$\{X\otimes X$, $S^Z_i$, $S^X_j\}$

5. Group-Theoretic Properties and Update Rules

From a group-theoretic viewpoint,

$$\mathrm{SSG} = \{g\in \mathrm{ISG}(t): \text{g never gets discarded}\} = \{g\in \mathcal{P}_N: [g, c]=0\;\forall\;\text{checks }c\;\text{in every layer}\}.$$

At each time step,

$$\mathrm{ISG}(t) = \langle\,\mathrm{SSG} \cup C(t)\,\rangle.$$

This formalism clarifies the dynamics: ISG evolves by centralizing the newly measured checks, discarding old incompatible stabilizers, while SSG is preserved throughout. Only SSG elements are suitable for reliably extracting error syndromes as their corresponding checks are inferable at all rounds of the cycle.

6. Implications for Floquet Code Design and Generalizations

The ISG formalism reveals a sharp criterion for correctability in periodically measured stabilizer codes: only stabilizers persisting through all rounds (SSG) provide reliable error-detecting information. In broader constructions, such as the generalized Floquet 3D toric code and Floquet $X$-cube code on locally cubic manifolds, the ISG/SSG distinction enables the extension of error-correcting properties to higher-dimensional and more complex spatial lattices. The unified ISG-SSG framework underlies the extension of these codes to $n$-dimensional Floquet $(n,1)$ toric codes and generalized $n$-dimensional Floquet $X$-cube codes (Yan et al., 2024).