Abstract
Experimental platforms routinely access Hilbert spaces far beyond qubit subspaces, including harmonic-oscillator (bosonic) and rotor modes that mediate gates or anharmonic modes providing effective logical encodings. While qubit stabilizer codes have received the most attention due to their mature theory, incorporating all the experimentally accessible systems into a unified framework for error correction may lead to lower resource overheads. Toward this end, we define a formalism that generalizes stabilizer quantum error correction to systems with abelian configuration spaces. Our approach unifies qubit/qudit stabilizer codes with bosonic and rotor codes, thus encompassing multi-qubit and multi-qudit stabilizer codes, multi-mode GKP codes, homological rotor codes, and rotation-symmetric bosonic codes (cat, binomial). Within our framework, we introduce a generalized notion of code distance that specializes to Pauli weight in the qubit setting and to Euclidean phase-space distance in the bosonic setting. The formalism makes explicit which experimentally accessible systems can be leveraged for protection, clarifying trade-offs in resource overheads when combining discrete and continuous variables.
*Supported by ARO grant W911NF2310376 and the NSA-LPS Qubit Collaboratory (LQC) National Quantum Fellowship administered by the Oak Ridge Institute for Science and Education (ORISE) through an interagency agreement between the U.S. Department of Energy (DOE) and the Department of Defense (DOD). ORISE is managed by ORAU under DOE contract number DE-SC0014664. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the DOD, DOE, ORAU/ORISE, or the U.S. Government. This work includes contributions of the National Institute of Standards and Technology, which are not subject to U.S. copyright.
