The role of higher-order terms in trapped-ion quantum computing with magnetic gradient induced coupling

TL;DR

This paper analyzes higher-order effects in trapped-ion quantum computing with magnetic gradient induced coupling, identifying key error sources and mitigation strategies to advance large-scale, error-tolerant quantum technologies.

Contribution

It provides a detailed analysis of higher-order terms in the MAGIC scheme, highlighting which effects are negligible and which require careful control for scalable quantum computing.

Findings

Most higher-order effects are negligible in realistic conditions.

Longitudinal fields increase with chain length and phonon occupation, affecting resonance.

Cooling phonons to the ground state mitigates key error sources.

Abstract

Trapped-ion hardware based on the Magnetic Gradient Induced Coupling (MAGIC) scheme is emerging as a promising platform for quantum computing. Nevertheless, in this -- as in any other -- quantum-computing platform, many technical questions still have to be resolved before large-scale and error-tolerant applications are possible. In this work, we present a thorough discussion of the structure and effects of higher-order terms in the MAGIC setup, which can occur due to anharmonicities in the external potential of the ion crystal (e.g., through Coulomb repulsion) or through curvature of the applied magnetic field. These terms generate systematic shifts in the leading-order interactions and take the form of three-spin couplings, two-spin couplings, local fields, as well as diverse phonon-phonon conversion mechanisms. We find that most of these are negligible in realistic situations, with only two contributions that need careful attention. First, there are undesired longitudinal fields contributing shifts to the resonance frequency, whose strength increases with chain length and phonon occupation numbers; while their mean effect can easily be compensated by additional $Z$ rotations, phonon number fluctuations need to be avoided for precise gate operations. Second, anharmonicities of the Coulomb interaction can lead to well-known two-to-one conversions of phonon excitations. Both of these error terms can be mitigated by sufficiently cooling the phonons to the ground-state. Our detailed analysis constitutes an important contribution on the way of making magnetic-gradient trapped-ion quantum technology fit for large-scale applications, and it may inspire new ways to purposefully design interaction terms.