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arxiv: 2604.28081 · v1 · submitted 2026-04-30 · ❄️ cond-mat.mes-hall · quant-ph

Recognition: unknown

g-tensor Optimization in Ge/SiGe Quantum Dots

Aram Shojaei , Edmondo Valvo , Maximilian Rimbach-Russ , Eliska Greplova , Ana Silva
Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:32 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords g-tensorGe/SiGe heterostructuresquantum dotshole-spin qubitspotential engineeringsingle-spin encodingquantum computing
0
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The pith

Adjusting silicon concentration in a SiGe-Ge-SiGe quantum well reshapes the out-of-plane potential to suppress in-plane g-tensor components and realize gapless single-spin qubit encoding.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a flexible optimization framework for controlling g-tensor properties in germanium heterostructures used for hole-spin qubits. By numerically optimizing the silicon concentration in the quantum well, the confining potential can be reshaped to minimize the in-plane components of the g-tensor. This approach benchmarks the realization of a gapless single-spin qubit encoding that reduces sensitivity to certain variations. It addresses key challenges like qubit energy uncertainty and random spin axis orientations in scalable quantum computing. The framework can be extended to other optimization targets in device design.

Core claim

We introduce a flexible optimization framework for engineering g-tensor properties in planar germanium heterostructures hosting hole-spin qubits. As a benchmark, we numerically obtain the optimal reshaping of the out-of-plane potential in a SiGe-Ge-SiGe quantum well to suppress the in-plane g-tensor components and realize the recently proposed gapless single-spin qubit encoding. This reshaping is achieved through heterostructure engineering, specifically by adjusting the silicon concentration within the quantum well.

What carries the argument

A numerical optimization framework that engineers the g-tensor by varying the silicon concentration to reshape the quantum well potential profile.

Load-bearing premise

The numerical model of hole states and g-tensor calculation accurately reflects the physics in real fabricated devices, and silicon concentration variations achieve the target potential without introducing disorder that cancels the benefits.

What would settle it

An experiment fabricating SiGe-Ge-SiGe quantum dot devices with the optimized silicon concentration profile and measuring the g-tensor to verify suppression of the in-plane components.

Figures

Figures reproduced from arXiv: 2604.28081 by Ana Silva, Aram Shojaei, Edmondo Valvo, Eliska Greplova, Maximilian Rimbach-Russ.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 3
Figure 3. Figure 3: , where panel (a) shows the evolution of the seven Si segments. Initially, the segment values fluctuate ran￾domly across their full [5, 20]% range. After a sufficient number of generations, the optimizer learns from feed￾back, and a clear pattern emerges: the Si concentration decreases near both interfaces, while the middle segments rise and flatten to form a uniform, high-Si plateau. This “double-well” pa… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 view at source ↗
Figure 10
Figure 10. Figure 10: Panel (a) shows the best-achieved gxx val￾ues, labeled g ∗ xx, for different Ez values. Panel (b) provides structural insight by showing the optimized HH potentials V HH0 ⊥ (z; sopt) (solid curves), together with the corresponding HH ground-state densities (dashed curves), |ϕ HH 0 (z)| 2 , for fields ranging from 0.19 MV/m to 0.50 MV/m. As the electric field increases, the Stark tilt pushes the HH envelop… view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 view at source ↗
read the original abstract

Planar germanium heterostructures hosting hole-spin qubits are among the leading platforms for scalable semiconductor-based quantum computing. Yet, device performance is hindered by significant quantum dot variability, which leads to uncertainty in qubit energy levels and random orientations of the spin quantization axis. Tailored control of the g-tensor offers a strategy to overcome these limitations and achieve more reliable qubit operations. Here, we introduce a flexible optimization framework for engineering g-tensor properties. As a benchmark, we numerically obtain the optimal reshaping of the out-of-plane potential in a SiGe-Ge-SiGe quantum well to suppress the in-plane g-tensor components and realize the recently proposed gapless single-spin qubit encoding. This reshaping is achieved through heterostructure engineering, specifically by adjusting the silicon concentration within the quantum well, though the framework remains readily adaptable to alternative design objectives. Our results provide practical design principles for improving the tunability of the spin response, paving the way towards large-scale germanium-based quantum computers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a flexible numerical optimization framework for engineering g-tensor properties in planar Ge/SiGe heterostructures hosting hole-spin qubits. As a benchmark, it claims to obtain an optimal reshaping of the out-of-plane potential profile in a SiGe-Ge-SiGe quantum well by varying the silicon concentration, thereby suppressing the in-plane g-tensor components to enable the gapless single-spin qubit encoding proposed in prior literature. The framework is presented as adaptable to other design goals.

Significance. If the numerical results are shown to be robust, the work offers practical heterostructure design guidelines that could improve g-tensor tunability and reduce qubit variability in Ge-based platforms, supporting scalability of semiconductor spin qubits. The optimization approach itself, if implemented with full material-parameter updating and validated against experiment, would constitute a useful engineering tool.

major comments (3)
  1. [§2 and §3] §2 (Numerical Model) and §3 (Optimization Results): The manuscript provides no explicit description of the Hamiltonian (k·p or envelope-function form), the method for computing the g-tensor from the hole wavefunctions, or any convergence tests with respect to basis size or discretization. Without these, the central claim that an optimal potential profile suppresses in-plane g-components cannot be evaluated.
  2. [§3] §3 (Heterostructure Engineering): When silicon concentration x is varied inside the Ge well to reshape the confining potential, the model must update the Luttinger parameters, hole effective mass, and band offsets self-consistently. If these quantities are instead held fixed at pure-Ge (x=0) values while only the electrostatic potential is altered, the reported g-tensor suppression is an artifact; the abstract presents concentration adjustment as the engineering knob without indicating parameter updating.
  3. [§4] §4 (Discussion): No comparison is made to existing experimental g-tensor measurements in Ge/SiGe quantum dots, nor is alloy scattering or interface disorder from Si incorporation quantified. These omissions leave open whether the predicted suppression survives realistic material imperfections.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'numerically obtain the optimal reshaping' should be accompanied by a reference to the specific figure or table that displays the optimized profile and the resulting g-tensor values.
  2. [Throughout] Notation: Define all symbols appearing in the g-tensor expressions (e.g., g_xx, g_zz) at first use and ensure consistent use of subscripts throughout.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of clarity, model completeness, and experimental context that we will address in a revised manuscript. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [§2 and §3] §2 (Numerical Model) and §3 (Optimization Results): The manuscript provides no explicit description of the Hamiltonian (k·p or envelope-function form), the method for computing the g-tensor from the hole wavefunctions, or any convergence tests with respect to basis size or discretization. Without these, the central claim that an optimal potential profile suppresses in-plane g-components cannot be evaluated.

    Authors: We agree that additional technical details are required for full reproducibility and evaluation. The model employs a 4-band Luttinger-Kohn k·p Hamiltonian for the heavy- and light-hole states, discretized on a finite-element mesh in the growth direction with envelope-function approximation. The g-tensor components are obtained by computing the Zeeman splitting under a small in-plane magnetic field via first-order perturbation theory on the hole envelope functions, incorporating the spin-orbit contributions. Convergence with respect to mesh density (typically 0.1 nm spacing) and basis truncation was verified internally but not reported. In the revised manuscript we will expand §2 with the explicit Hamiltonian form, the g-tensor extraction procedure, and a dedicated convergence subsection (or appendix) showing that the in-plane g-components stabilize to <1% variation beyond the chosen discretization. revision: yes

  2. Referee: [§3] §3 (Heterostructure Engineering): When silicon concentration x is varied inside the Ge well to reshape the confining potential, the model must update the Luttinger parameters, hole effective mass, and band offsets self-consistently. If these quantities are instead held fixed at pure-Ge (x=0) values while only the electrostatic potential is altered, the reported g-tensor suppression is an artifact; the abstract presents concentration adjustment as the engineering knob without indicating parameter updating.

    Authors: We confirm that all material parameters are updated with silicon concentration x. The Luttinger parameters (γ1, γ2, γ3), hole effective masses, and valence-band offsets are interpolated linearly (Vegard’s law) between pure-Ge and Si values using established literature parametrizations for Si1−xGex alloys. These x-dependent parameters enter both the k·p Hamiltonian and the self-consistent Poisson–Schrödinger solution that determines the confining potential. The optimization therefore accounts for both the electrostatic reshaping and the composition-induced changes in band structure. We will add an explicit statement and the interpolation formulas in the revised §2 and §3 to eliminate any ambiguity. revision: yes

  3. Referee: [§4] §4 (Discussion): No comparison is made to existing experimental g-tensor measurements in Ge/SiGe quantum dots, nor is alloy scattering or interface disorder from Si incorporation quantified. These omissions leave open whether the predicted suppression survives realistic material imperfections.

    Authors: We acknowledge the value of experimental anchoring. In the revised §4 we will include a direct comparison of our optimized in-plane g-components (targeting near-zero values) against published experimental g-tensor data for Ge/SiGe hole quantum dots, citing representative works on g-factor anisotropy in planar Ge heterostructures. Regarding alloy scattering and interface disorder, we will add a qualitative assessment noting that typical SiGe interface roughness (∼0.5–1 nm) and alloy fluctuations introduce a small residual in-plane g-component; however, the optimization still provides a useful target for heterostructure design. A quantitative disorder-averaged calculation lies beyond the present scope but will be flagged as future work. These additions will clarify the robustness of the predicted suppression under realistic conditions. revision: partial

Circularity Check

0 steps flagged

No circularity: numerical optimization uses external target and independent physical parameter

full rationale

The derivation consists of a numerical optimization framework that takes an externally defined target (gapless single-spin qubit encoding from prior literature) and varies silicon concentration as a growth parameter to reshape the confining potential. The g-tensor components are computed from the heterostructure model rather than fitted or defined in terms of the optimization objective. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the chain. The approach remains self-contained against external benchmarks because the target encoding and the material parameter are independent of the computed result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard semiconductor models for hole states in Ge/SiGe wells and on the assumption that the optimization target corresponds to a physically realizable heterostructure.

free parameters (1)
  • silicon concentration profile
    Adjusted within the quantum well to reshape the confining potential; the specific values that achieve the reported optimum are outputs of the numerical search.
axioms (1)
  • domain assumption Effective-mass or k·p theory accurately computes the hole g-tensor from the electrostatic potential in planar Ge/SiGe heterostructures
    Invoked implicitly when the optimization framework maps potential shape to g-tensor components.

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Reference graph

Works this paper leans on

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