Fault-Tolerant Encoding of Logical Qudits in Spin Systems

TL;DR

This paper introduces a resource-efficient framework for encoding fault-tolerant logical qudits in spin systems, enabling scalable quantum error correction with fewer physical resources.

Contribution

It presents a general method for fault-tolerant logical qudit encoding in spin systems, including code constructions and implementation strategies.

Findings

Constructed distance-3 and distance-5 codewords for logical qudits.

Achieved encoding with fewer Hilbert space dimensions than traditional qubit-based methods.

Proposed implementation protocols compatible with current spin qudit platforms.

Abstract

Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical qudits in finite-dimensional spin systems. We construct distance-$3$, distance-$5$ codewords, and general $2t+1$-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. We further discuss schematic designs for physical implementation and required single-gate fidelities, which are compatible with current spin qudit platforms. This strategy provides a resource-efficient path toward realizing fault-tolerant logical qudits in finite multi-level physical systems.