Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions

Qudits can be described by a state vector in a $q$-dimensional Hilbert space, enabling a more extensive encoding and manipulation of information compared to qubits. This implies that conducting fault-tolerant quantum computations using qudits rather than qubits might entail less overhead. In this work, we investigate the viability of qudits in error correction codes by creating and simulating the quantum circuitry for the smallest qudit error correction code with a multidimensional circuit-level noise model and specifically adapted decoders. After introducing a flag qudit to protect the code from hook errors, comparable error thresholds of the order of $10^{-4}$ are obtained for qudits of dimensions $2$, $3$ and $5$.