A dynamic circuit for the honeycomb Floquet code

TL;DR

This work demonstrates that dynamic stabilizer measurement circuits, which remove ancillas and shrink gauge measurements to single-qubit operations, substantially improve the honeycomb Floquet code. By alternating measurement targets to remove leakage and boosting timelike distance, the dynamic approach yields higher thresholds and far lower logical error rates than the standard ancilla-based circuit, translating to a potential order-of-magnitude reduction in qubit overhead at practical error rates such as $p=10^{-3}$. The authors provide a detailed circuit design, compare resource requirements against the standard scheme, and validate performance with circuit-level simulations using MWPM and correlated matching decoders. The results suggest significant practical gains for fault-tolerant quantum computation with Floquet codes, while outlining boundaries for future work and extensions to other code families.

Abstract

In the typical implementation of a quantum error-correcting code, each stabilizer is measured by entangling one or more ancilla qubits with the data qubits and measuring the ancilla qubits to deduce the value of the stabilizer. Recently, the dynamic circuit approach has been introduced, in which stabilizers are measured without ancilla qubits. Here, we demonstrate that dynamic circuits are particularly useful for the Floquet code. Our dynamic circuit increases the timelike distance of the code, automatically removes leakage, and both significantly increases the threshold and lowers the logical error rate compared to the standard ancilla-based circuit. At a physical error rate of $10^{-3}$, we estimate a nearly $3\times$ reduction in the number of qubits required to reach a $10^{-12}$ logical error rate.

Figures (5)

• Figure 1: The dynamic circuit for measuring the stabilizers of the unrotated surface code. Here, $M$ ($M_X$) and $R$ ($R_x$) represent measurement and reset in the $Z$ ($X$) basis.We illustrate four neighboring stabilizers of the surface code in the first panel, with $Z$ ($X$) stabilizers in blue (red), and track their evolution through the circuit in subsequent panels. Half the stabilizers shrink and are measured, while the neighboring stabilizers grow.
• Figure 2: An illustration of an $L_1\times L_2$ patch of honeycomb Floquet code on a torus. The rightmost panel illustrates the gauge operators and stabilizers of the Floquet code. The stabilizers are supported on hexagons, with red/green/blue hexagons denoting $\otimes^6X$/$\otimes^6Y$/$\otimes^6Z$ stabilizers. Each bond supports two-qubit gauge operators that commute with all stabilizers, with red/green/blue bonds denoting $XX$/$YY$/$ZZ$ gauge operators. Rather than directly measure the stabilizers, we sequentially measure the $XX$/$YY$/$ZZ$ gauge operators, with the product of the gauge operators around each hexagon giving the value of the corresponding stabilizer. No single representative of the logical operators commutes with all gauge operators; instead, at each step, we multiply the current representative by some previous gauge operators to create a new representative that commutes with the next gauge operators. We show a horizontal (yellow) and vertical (gray) observable as they evolve through time. In the last panel, we illustrate pairs of stabilizers that can be flipped by a single physical error for the standard (purple) and dynamic (orange) circuits. In the standard circuit, errors flip neighboring pairs of stabilizers, while in the dynamic circuit they may flip next-nearest-neighbors.
• Figure 3: Top: Standard circuit for performing $XX$ gauge measurements, using one ancilla per gauge operator. The circuits for $YY$ and $ZZ$ are similar. Bottom: Gauge measurements in the dynamic circuit. We measure $XX$ gauge operators by shrinking them to a single qubit, which we then measure. In the $YY$ and $ZZ$ gauge measurements we alternate the measured qubits to remove leakage. In the next three gauge measurements (not shown), we reverse the target and control of each gate and continue the alternating measurement pattern. The two circuits have the same number of two-qubit gates, resets, and measurements.
• Figure 4: Logical error rates for a single pair of logical observables in the dynamic circuit (bold) and standard circuit (faded) for various circuit-level distances, using MWPM. Estimated thresholds are shown with horizontal lines. The dynamic circuit demonstrates significant improvement in both threshold and logical error rate. Circuits simulated with STIM gidney2021stim and decoded with PyMatching higgott2025sparse.
• Figure 5: Our extrapolation to the teraquop regime for $p=10^{-3}$, and the logical error rates separated into horizontal and vertical error rates, for (left) MWPM and (right) correlated matching. Circuits simulated with STIM gidney2021stim and decoded with PyMatching higgott2025sparse.