Increasing the distance of topological codes with time vortex defects

TL;DR

This work introduces time vortex defects, spatially varying delays in the measurement schedule of Floquet topological codes, to effectively increase code distance and reduce the qubit count needed for a target logical error rate. Focusing on the Floquet color code on a torus, the authors optimize torus embeddings and vortex counts, showing asymptotic qubit savings (less than half the qubits) and concrete gains (a 30-qubit vortexed code outperforming a 42-qubit vortex-free code) at the same distance. Using circuit-level EM3 noise in Monte Carlo simulations, they report a distance boost by a factor of about $1.46$ in the large-qubit limit and demonstrate strong performance advantages for vortexed codes at fixed qubit budgets. The study also extends the construction to the toric code, planar architectures with punctures, and discusses implications for other good quantum LDPC codes, highlighting a practical path to more resource-efficient quantum error correction.

Abstract

We propose modifying topological quantum error correcting codes by incorporating space-time defects, termed ``time vortices,'' to reduce the number of physical qubits required to achieve a desired logical error rate. A time vortex is inserted by adding a spatially varying delay to the periodic measurement sequence defining the code such that the delay accumulated on a homologically non-trivial cycle is an integer multiple of the period. We analyze this construction within the framework of the Floquet color code and optimize the embedding of the code on a torus along with the choice of the number of time vortices inserted in each direction. Asymptotically, the vortexed code requires less than half the number of qubits as the vortex-free code to reach a given code distance. We benchmark the performance of the vortexed Floquet color code by Monte Carlo simulations with a circuit-level noise model and demonstrate that the smallest vortexed code (with $30$ qubits) outperforms the vortex-free code with $42$ qubits.

Figures (8)

• Figure 1: Inserting a time vortex into the measurement sequence of the repetition code with periodic boundary conditions. (a) Without the time vortex, native two-body Pauli $ZZ$ measurements (green) are applied in a "brick wall" circuit. The height of the measurement boxes represents the minimal time required to perform a measurement. The matching graph of the code is shown for a phenomenological noise model. Detectors given by the product of two consecutive parity measurements are drawn as yellow square markers. Errors represented by edges trigger the detectors at their endpoints. Vertical (time-like) edges are measurement errors while other (space-like) edges correspond to bit flips between rounds of syndrome measurement (along the dashed orange lines). The code distance $D$ is determined by the length of the minimal non-trivial homotopy (red edges). (b) A single time vortex is inserted by introducing delays between consecutive measurements. The distance is increased by $1$ as compared with the vortex-free code. This comes at the expense of an increased circuit depth.
• Figure 2: The Floquet color code with and without a time vortex. (a) The qubits of the Floquet color code are arranged on the vertices of a honeycomb lattice on a torus. The plaquettes are colored red, green, and blue and the bonds are colored according to the plaquettes at their endpoints. (b) The code is defined by a periodic sequence of 2-body Pauli $XX$ and $ZZ$ measurements on the bonds with period $T=6$. The time of each measurement is given by the schedule indicated on the black circle. (c) A time vortex is inserted along the $\mathbf{L}_1$ lattice vector by adding a spatially dependent delay to the time at which each measurement is applied within the cycle. The delay accumulated on a path going around the torus in the $\mathbf{L}_1$ direction is equal to the period $T$.
• Figure 3: The space-time matching graph of the Floquet color code for the EM3 error model. Vertices correspond to volume-like detector cells and edges correspond to errors that trigger pairs of detectors. The dotted edges belong to the set $E_2$ in Eq. \ref{['eq: edges']}. The shortest path between distant points can be chosen such that it contains at most one edge of this set.
• Figure 4: Lowest value of $R=N/D^2$ reached by choosing the optimal embedding of the Floquet color code on the torus with and without time vortices as a function of the code distance $D$.
• Figure 5: The logical error rate vs. the physical error rate for optimal embeddings of the Floquet color code on the torus with different code distances (determined by the basis vectors $\mathbf{L}_1,\mathbf{L}_2$). Configurations without time vortices are drawn with circular markers and solid lines and those with vortices are drawn with x markers and dotted lines. The color indicates the number of physical qubits in the code block. The inset shows the logical error rate vs. the number of qubits at a fixed physical error rate $p=10^{-2.5}$.