# Theory of Hole-Spin Qubits in Strained Germanium Quantum Dots

## TL;DR

This paper provides a theoretical analysis of hole-spin qubits in strained germanium quantum wells, highlighting their advantageous properties such as large band splitting, light effective mass, and strong spin-orbit coupling, which facilitate fast quantum operations.

## Contribution

It offers a detailed theoretical framework for understanding and optimizing hole-spin qubits in Ge quantum wells, emphasizing the role of spin-orbit coupling and crystal field effects.

## Key findings

- Large intrinsic band splitting (>100 meV) in Ge quantum wells.
- Light in-plane effective mass (~0.05 m_0) enhances mobility.
- Strong spin-orbit coupling enables fast electric-dipole spin resonance.

## Abstract

We theoretically investigate the properties of holes in a Si$_{x}$Ge$_{1-x}$/Ge/ Si$_{x}$Ge$_{1-x}$ quantum well in a perpendicular magnetic field that make them advantageous as qubits, including a large ($>$100~meV) intrinsic splitting between the light and heavy hole bands, a very light ($\sim$0.05$\, m_0$) in-plane effective mass, consistent with higher mobilities and tunnel rates, and larger dot sizes that could ameliorate constraints on device fabrication. Compared to electrons in quantum dots, hole qubits do not suffer from the presence of nearby quantum levels (e.g., valley states) that can compete with spins as qubits. The strong spin-orbit coupling in Ge quantum wells may be harnessed to implement electric-dipole spin resonance, leading to gate times of several nanoseconds for single-qubit rotations. The microscopic mechanism of this spin-orbit coupling is discussed, along with its implications for quantum gates based on electric-dipole spin resonance, stressing the importance of coupling terms that arise from the underlying cubic crystal field. Our results provide a theoretical foundation for recent experimental advances in Ge hole-spin qubits.