The code distance of Floquet codes

TL;DR

This work analyzes Floquet and other dynamical quantum codes defined by periodic Pauli measurements, addressing the challenge of undetectable spacetime errors. It introduces benign spacetime errors—compositions of vacuous and sandwiching errors—and proves that any undetectable error in the steady stage is equivalent to a logical operator of some instantaneous stabilizer code at a future time, provided the code has bounded-inference (finite time window) for inferring stabilizers. Consequently, the code distance is the minimal weight of a nonbenign undetectable spacetime error, enabling efficient distance computation by pushing undetectable errors to a single time step within the inference window; this generalizes the stabilizer-code distance to dynamical codes. The paper substantiates the framework with three concrete examples—the ladder code, planar honeycomb code, and Floquet Bacon–Shor code—demonstrating how spacetime distance can differ from instantaneous distances and how benign errors govern correctability. These insights enable intrinsic comparisons of Floquet codes and lay groundwork for extending fault tolerance to a broader class of dynamical codes without relying on fixed time boundaries.

Abstract

For fault-tolerant quantum memory defined by periodic Pauli measurements, called Floquet codes, we prove that every correctable, undetectable spacetime error occurring during the steady stage is a product of (i) measurement operators inserted at the time of the measurement and (ii) pairs of identical Pauli operators sandwiching a measurement that commutes with the operator. We call such errors benign; they define a binary vector subspace of spacetime errors which properly generalize stabilizers of static Pauli stabilizer codes. Hence, the code distance of a Floquet code is the minimal weight of an undetectable spacetime Pauli error that is not benign. Our results apply more generally to families of dynamical codes for which every instantaneous stabilizer is inferred from measurements in a time interval of bounded length.

Figures (5)

• Figure 1: Ladder code for $m=3$. Qubits are associated to vertices, enumerated as shown. Measurement operators on vertical legs are $ZZ$ on the two qubits. Measurement operators on horizontal legs are alternately $XX$ or $YY$; a couple are shown. Persistent stabilizers of the parent subsystem code are weight-4 plaquettes of type $Y$ (green) or $X$ (red) which are not measured directly, but rather inferred through the measurement schedule.
• Figure 2: Measurement schedule of planar-boundary honeycomb variant. Measurement operators at each time step are two-body measurements, with a pair of single-qubit measurements at opposite corners. Red, blue, and green measurement operators are supported in $X$, $Z$, and $Y$, respectively. Thin black lines represent the equivalent hexagonal surface code describing the state at each time step. Illustration provided by C. Vuillot and used with permission.
• Figure 3: Compression of weight-2 spacetime error supported on time window $\{1,2,3\}$ to a single-time error at time $t=3$, at which point the error is a logical of ${\mathsf{ISG}}(3)$. Past supports of the spacetime error are commuted forward by multiplying the error by measurement operators in the dashed boxes, following the procedure of \ref{['thm:PushingUndetectable']}. Teardrop arrows indicate the introduction of spacetime error. Illustration provided by C. Vuillot and used with permission.
• Figure 4: 4-periodic measurement schedule of the Floquet Bacon--Shor code. At times $t=1,3 \bmod 4$, $XX$ is measured on the horizontal line segments. When $t=2,4 \bmod 4$, $ZZ$ is measured on the vertical line segments. We denote each set of measurements by $M_i$, $i \in [4]$. Reproduced from Figure 3 of Alam and Rieffel Alam2024, licensed under CC-BY 4.0.
• Figure 5: Left: up to vacuous errors from $M_1$, $X$-type errors are generated by single-qubit $X$ operators supported on the far left, right vertices of the lattice. Any error component lying on the $x$-axis is translated to the left. Center: the only element of $\langle M_1 \rangle$ supported on the far left, right vertices. Right: the basis of all $X$-type errors commuting with $M_2$; the far left, right elements are labeled as shown. Modified from Figure 3 of Alam and Rieffel Alam2024, licensed under CC-BY 4.0.

Theorems & Definitions (72)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4: Error timing convention
• Proposition 1.5
• proof
• Definition 2.1
• Definition 2.1
• Lemma 2.2
• proof