Theory of single and two-qubit operations with donor-bound electron spins in germanium

TL;DR

This paper develops a comprehensive theory of donor electron states in germanium, demonstrating advantages over silicon for quantum computing due to stronger spin-orbit interaction and larger wave functions, enabling faster and more robust qubit operations.

Contribution

It provides the first complete theoretical analysis of Ge donor electrons and shows their potential advantages over Si for scalable, high-fidelity quantum computing architectures.

Findings

Ge donors allow greater tuning of single qubit energies.

Exchange coupling in Ge donors is significantly larger than in Si.

Ge donors exhibit a higher probability of strong coupling within an order of magnitude.

Abstract

The possibility of quantum computing with spins in germanium nanoscale transistors has recently attracted interest since it promises highly tuneable qubits that have encouraging coherence times. We here present the first complete theory of the orbital states of Ge donor electrons, and use it to show that Ge could have significant advantages over silicon in the implementation of a donor-based quantum processor architecture. We show that the stronger spin-orbit interaction and the larger electron donor wave functions for Ge donors allow for greater tuning of the single qubit energy than for those in Si crystals, thus enabling a large speedup of selective (local) quantum gates. Further, exchange coupling between neighboring donor qubits is shown to be much larger in Ge than in Si, and we show that this greatly relaxes the precision in donor placement needed for robust two-qubit gates. To do this we compare two statistical distributions for Ge:P and Si:P pair couplings, corresponding to realistic donor implantation misplacement, and find that the spin couplings in Ge:P have a $33\%$ chance of being within an order of magnitude of the largest coupling, compared with only $10\%$ for the Si:P donors. This allows fast, parallel and robust architectures for quantum computing with donors in Ge.