Dynamical quantum codes and logic gates on a lattice with sparse connectivity

TL;DR

The paper addresses fault-tolerant quantum computation on planar, low-connectivity hardware by introducing dynamical codes on a hexagonal honeycomb lattice that exploit mid-circuit measurements and nearest-neighbor gates. It develops Floquet, dancing Floquet, and double Floquet codes, and introduces adaptive switching between the color code and Floquet variants to enable transversal Clifford operations on encoded qubits. Key findings include finite thresholds under circuit-level noise for several dynamical schemes, the ability to interleave multiple codes on a single lattice, and practical mechanisms for transversal gates via patch boundaries and code deformation. These results point toward modular, low-depth fault-tolerant architectures with potential for universal quantum computation when combined with magic-state techniques, leveraging lattice-surgery concepts on a sparse, scalable platform.

Abstract

We introduce several dynamical schemes that take advantage of mid-circuit measurement and nearest-neighbor gates on a lattice with maximum vertex degree three to implement topological codes and perform logic gates between them. We first review examples of Floquet codes and their implementation with nearest-neighbor gates and ancillary qubits. Next, we describe implementations of these Floquet codes that make use of the ancillary qubits to reset all qubits every measurement cycle. We then show how switching the role of data and ancilla qubits allows a pair of Floquet codes to be implemented simultaneously. We describe how to perform a logical Clifford gate to entangle a pair of Floquet codes that are implemented in this way. Finally, we show how switching between the color code and a pair of Floquet codes, via a depth-two circuit followed by mid-circuit measurement, can be used to perform syndrome extraction for the color code.

Figures (6)

• Figure 1: Threshold plot of the Floquet code under circuit-level noise. The logical $\ket{0}$ state is prepared followed by $T=1,2,3$ full (6-round) cycles of stabilizer measurements for $d=4,7,10$, respectively.
• Figure 2: Detector slice diagrams of the Floquet code with leakage-reducing parity measurements along every link. Red (blue) regions correspond to X (Z) regions. The first diagram represents the detectors after the expansion (Eq.\ref{['eq:ResetOut']}) of blue ZZ links. Then the red XX, and green ZZ measurements follow, and finally the contraction (Eq.\ref{['eq:MeasureIn']}) of the blue XX. The second half of the cycle is not shown. Snapshots on the right column correspond to the later three steps in Eqs. \ref{['eq:CSSFC']} which were depicted in the section above (here the left and right boundaries include additional complete links that are part of the lattice). Single nodes represent single-qubit measurements on incomplete links.
• Figure 3: Threshold plots of the dancing Floquet code under circuit-level noise. Top left: distance preserving version resetting every qubit at least once (on average 4/3 times) per 6-round cycle; Top right: resetting data qubits at least twice twice (on average 8/3 times) per cycle; Bottom left: resetting data qubits four times per cycle; Bottom right: distance-reducing version resetting every data qubit in every round. The colors correspond to the same physical system size (as in Fig. \ref{['fig:Floquet_threshold']}). The logical $\ket{0}$ state is prepared followed by $T=1,2,3$ full cycles of stabilizer measurements for increasing system size.
• Figure 4: Threshold plot of the Double Floquet code under circuit-level noise. The logical $\ket{00}$ state is prepared followed by $T=1,2,3$ full cycle of stabilizer measurements for $d=3,5,7$, respectively.
• Figure 5: Threshold of the triangle-patch switching color code under circuit-level noise using a belief-propagation decision-tree decoder of Ref. ott2025decision. Here the logical error rate refers to the transversal version of the logical Z observable. We find a threshold value of $0.25\%$.