Bosonic quantum Fourier codes

Quantum 10, 2000 (2026).

https://doi.org/10.22331/q-2026-02-09-2000

While 2-level systems, aka qubits, are a natural choice to perform a logical quantum computation, the situation is less clear at the physical level. Encoding information in higher-dimensional physical systems can indeed provide a first level of redundancy and error correction that simplifies the overall fault-tolerant architecture. A challenge then is to ensure universal control over the encoded qubits. Here, we explore an approach where information is encoded in an irreducible representation of a finite subgroup of $U(2)$ through an inverse quantum Fourier transform. We illustrate this idea by applying it to the real Pauli group $langle X, Zrangle$ in the bosonic setting. The resulting two-mode Fourier cat code displays good error correction properties and admits an experimentally-friendly universal gate set that we discuss in detail.