Encoding a qubit in a spin

TL;DR

This paper introduces a novel quantum error-correcting code encoding a qubit in a large spin, enabling Clifford operations through spatial rotations, with practical implementations and noise correction criteria.

Contribution

It presents a new scheme for encoding a qubit in a spin system with natural Clifford operations and provides comprehensive code constructions and performance analysis.

Findings

Constructed all possible spin-based qubit codes.

Demonstrated universal gate implementation with quadratic Hamiltonians.

Derived criteria for noise correction and showed numerical performance.

Abstract

I present a new approach for designing quantum error-correcting codes that guarantees a physically natural implementation of Clifford operations. Inspired by the scheme put forward by Gottesman, Kitaev, and Preskill for encoding a qubit in an oscillator, in which Clifford operations may be performed via Gaussian unitaries, this approach yields new schemes for encoding a qubit in a large spin in which single-qubit Clifford operations may be performed via spatial rotations. I construct all possible examples of such codes, provide universal-gate-set implementations using Hamiltonians that are at most quadratic in angular-momentum operators, and derive criteria for when these codes exactly correct physically relevant noise channels to lowest order, illustrating their performance numerically for specific low-dimensional examples.