The small stellated dodecahedron code and friends

We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code. The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges respectively vertices of a small stellated dodecahedron, making this code suitable for threedimensional connectivity. This code encodes eight logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity checkmeasurements) in contrast with one logical qubit into nine physical qubits (plus eight ancilla qubits) for the surface code. We develop faulttolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuitbased pseudo-threshold. This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.