The Small Stellated Dodecahedron Code and Friends

TL;DR

This paper investigates the small stellated dodecahedron quantum code, a 3D connectivity code with high qubit density, comparing its performance to the surface code through fault-tolerant circuits and decoding strategies.

Contribution

It introduces the small stellated dodecahedron code, develops fault-tolerant circuits and a decoder, and assesses its performance relative to the surface code.

Findings

Encodes 8 logical qubits into 30 physical qubits.

Suitable for 3D connectivity with data and ancilla qubits on edges and vertices.

Provides numerical assessment of pseudo-threshold for the code.

Abstract

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 resp. vertices of a small stellated dodecahedron, making this code suitable for 3D connectivity. This code encodes 8 logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) as compared to 1 logical qubit into 9 physical qubits (plus 8 ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit-based pseudo-threshold.