In-plane optimal qubit operation with control fidelities exceeding 99 %

AI summary

Overview

Research area: Semiconductor quantum computing — specifically hole spin qubits confined in planar germanium quantum dots, and how the direction of an in-plane magnetic field governs coherence and gate fidelity.

Technical level: Advanced. The paper assumes familiarity with spin qubits, Larmor frequencies, Ramsey/Hahn-echo sequences, gate set tomography, and spin–orbit coupling.

Scope (one sentence): The paper measures how the in-plane magnetic-field angle changes a germanium hole spin qubit's sensitivity to charge noise, and shows that choosing the field direction that minimizes this sensitivity simultaneously extends the Hahn-echo coherence time roughly fivefold and raises single-qubit gate fidelities to 99.82 %.

What This Paper Is About

Holes confined in germanium quantum wells are attractive spin qubits because they can be driven electrically and manipulated quickly, but their strong spin–orbit coupling also makes their resonance frequency sensitive to electric-field (charge) noise. That sensitivity is anisotropic — it depends on crystalline direction and on the orientation of the applied magnetic field — so there may exist a "sweet spot" field direction where the qubit is unusually immune to noise. The authors set out to measure this sensitivity directly, across many field angles, for two hole spin qubits in a planar germanium double quantum dot, and to check whether the angle that minimizes sensitivity is also the angle that maximizes coherence time and control fidelity.

Key Contributions

1. Direct, angle-resolved measurement of longitudinal spin–electric susceptibility (LSES). The authors quantify LSES as the derivative of qubit resonance frequency with respect to plunger-gate voltage, ∂f_Q/∂V_Pi, for both qubits and for both plunger gates P1 and P2, over a range of in-plane magnetic-field angles.

2. Correlation of LSES with coherence across field angle. They show that the Hahn-echo coherence time T₂^Echo tracks the angular dependence of LSES, providing experimental evidence that charge-noise-induced frequency fluctuations dominate decoherence in this device.

3. Identification of an optimal operating point. A local field orientation is found where LSES is minimized, giving roughly a fivefold enhancement of T₂^Echo and a nearly tenfold suppression of control infidelity, with a maximum gate fidelity of 99.82 % measured by gate set tomography.

4. Gate-set-tomography benchmarking of single-qubit gates versus field angle. GST is used, rather than randomized benchmarking or standard process tomography, to separate gate errors from state-preparation-and-measurement (SPAM) errors, and the X/2 and Y/2 infidelities are mapped over in-plane angles within ±50° of θ_xy = 0°.

Main Findings

• Anisotropic g factors for both qubits. For qubit Q1 the in-plane g factor varies between 0.19 and 0.30; for Q2 it varies between 0.19 and 0.41. The maximum values are g_Q1 = 0.30 ± 1.6 × 10⁻⁵ and g_Q2 = 0.41 ± 2.3 × 10⁻⁵, reached when the field is close to the x axis (perpendicular to the interdot axis), and both minima are 0.19 near the y axis (parallel to the interdot axis).

• Misaligned principal axes. The principal axes of the in-plane g tensors of the two qubits are tilted by approximately 14° (Q1) and 12° (Q2) from the magnet axis, attributed to strain from device fabrication.

• LSES is parabolic in field angle. For Q1, ∂f_Q1/∂V_P1 and ∂f_Q1/∂V_P2 take both positive and negative values and show a pronounced parabolic dependence on θ_xy with a maximum at θ_xy = 0°. For Q2, LSES remains strictly positive over the whole angular range, with its maximum at θ_xy = −10°.

• Gate geometry matters. For Q2, ∂f_Q2/∂V_P1 exceeds ∂f_Q2/∂V_P2 even though P2 is the plunger gate directly above Q2. The authors attribute this to asymmetric tunnel rates and strong squeezing of the Q2 wave function under the enhanced latching readout configuration, which limits how much P2 can tune the confinement.

• T₂ is isotropic; T₂^Echo is not.* Ramsey measurements with a 10 MHz detuning give T₂* = 169.9 ns at θ_xy = 0° for Q1, and T₂* stays essentially constant versus angle for both qubits, which the authors ascribe to dominant low-frequency noise over long acquisition and averaging times. Hahn-echo gives T₂^Echo = 498.2 ns at θ_xy = 0° for Q1.

• Fivefold coherence enhancement. T₂^Echo for Q1 increases from approximately 500 ns to 2200 ns across the angular range (about a factor of five); for Q2 it increases from approximately 400 ns to 800 ns (about a factor of two). The minimum coherence time coincides with the maximum LSES for each qubit.

• Echo-based quality factor peaks at the sweet spot. For Q2, Q^E = 2 T₂^Echo f_Rabi reaches its maximum at the optimal operating point, where it is approximately five times larger than in the least favorable configuration.

• Gate fidelities above 99 %. GST on the gate set {X/2, Y/2, I} yields a maximum control fidelity of 99.82 % for the X/2 gate and 99.78 % for the Y/2 gate. For Q1, the infidelity at the optimal point is reduced by approximately one order of magnitude relative to the worst case; for Q2, by approximately a factor of five.

• Idle-gate errors behave differently. The X/2 and Y/2 infidelities follow the angular dependence of T₂^Echo, but the identity gate (implemented as a wait of the same duration as t_π/2) instead follows f_Rabi, since T₂* is nearly constant across the measured range. For Q2 the I-gate infidelity saturates, indicating that further increases in f_Rabi give diminishing returns and that T₂* and T₂^Rabi become the limiting factors.

• Extracted noise exponent. Analysis of the coherence decay gives a noise spectral exponent close to 1/f^0.907, consistent with the noise spectrum characterized by Ramsey and CPMG experiments (details in Appendix B).

• Model limitations. Simulated T₂* agrees well with experiment when only plunger-gate charge noise is included, but simulated T₂^Echo deviates from the measured values. The authors attribute the mismatch to barrier gates B1–B3, whose LSES could not be measured because fast voltage pulses can only be applied to P1 and P2.

Methodology in Plain English

The device is a double quantum dot defined by three overlapping layers of aluminum gates on an undoped SiGe/Ge heterostructure, with ohmic contacts made by ion implantation, and it is cooled in a dilution refrigerator to 15 mK. A separate quantum dot on the opposite side acts as a charge sensor. A 900 mT in-plane magnetic field creates the Zeeman splitting that defines the two spin qubits, and three-step voltage pulse sequences combined with microwave pulses initialize, manipulate, and read out the qubit states in the (1,1)–(2,0) charge transition regime using latched Pauli spin blockade readout.

To measure how sensitive each qubit is to electric fields, the researchers vary the voltage on a plunger gate during the "load" stage and record how far the spin resonance frequency shifts; the slope of that shift versus gate voltage is the LSES. They repeat this for both plunger gates and at many magnetic-field angles. They then measure coherence with Ramsey (two π/2 pulses separated by a variable idle time) and Hahn-echo (with a π pulse inserted halfway) sequences, keeping the qubit resonance frequency fixed while rotating the field so that the comparison is fair. Finally, they run gate set tomography on the gate set {X/2, Y/2, I} to extract gate fidelities while explicitly modeling and removing state-preparation-and-measurement errors.

Why This Matters

For research, the paper supplies a direct experimental link between a measurable materials/device parameter — the longitudinal spin–electric susceptibility — and the two quantities that matter for any qubit technology: how long coherence survives and how accurate gates are. It reframes magnetic-field engineering as a practical optimization knob for spin–orbit-coupled qubits, rather than treating anisotropy purely as a nuisance.

• Scalable quantum processors: Field-direction engineering could be applied across a multi-qubit array to reduce gate errors without changing the device or the fabrication process.
• Semiconductor manufacturing: Germanium hole qubits are compatible with established semiconductor processing, so improvements that come from field orientation rather than new materials lower the barrier to industry adoption.
• Qubit addressability and crosstalk: The measured site-dependent and anisotropic g factors are relevant to selecting qubits individually in arrays, and the work shows how that same anisotropy can be exploited rather than merely tolerated.
• Benchmarking practice: The use of gate set tomography, which separates SPAM errors from gate errors, provides a template for reporting fidelities that is more informative than randomized benchmarking alone.

Industry relevance: The authors are affiliated with the Beijing Superstring Academy of Memory Technology, the Hefei National Laboratory, and Origin Quantum Computing Company Limited, indicating direct interest from the semiconductor memory and quantum computing industries in germanium-based spin qubit architectures.

Future Directions

• Extending LSES measurements to barrier gates. Fast pulses could currently be applied only to the plunger gates, so LSES for B1–B3 is not reported; measuring it would likely close the remaining gap between simulated and measured T₂^Echo.
• Testing whether magnetic-field engineering scales. The conclusions rest on two qubits in one device; whether a single field direction optimizes a larger array, where each dot may have differently tilted g-tensor principal axes, is an open question the authors flag for future large-scale arrays.
• Combining field engineering with electrostatic tuning. The paper notes that tuning the electrostatic confinement is an alternative route to sweet spots; how the two approaches combine is not resolved here.
• Improving the I-gate limit. Since the identity-gate infidelity saturates with Rabi frequency and becomes limited by T₂* and T₂^Rabi, better low-frequency noise mitigation will be needed to push idle-gate fidelity further.

Target Audience

Experimental quantum computing researchers working on semiconductor spin qubits, especially hole spin qubits in germanium and silicon; device physicists interested in spin–orbit coupling, g-tensor anisotropy, and sweet-spot engineering; and quantum engineers who benchmark gates with gate set tomography. The paper is also useful for graduate students entering the field who want a concrete example of how a physical device parameter is quantitatively tied to coherence and gate fidelity.

Authors’ abstract

Hole spin qubits based on semiconductor quantum dots are promising for building future large-scale quantum computers owing to their all-electrical manipulation. However, abundant physical mechanisms of valence band holes lead to anisotropic qubit properties. There is an opportunity to prolong the coherence time and achieve high-fidelity qubit manipulations. Here, we report a single-hole spin qubit in a planar germanium quantum dot and investigate its anisotropic susceptibility to charge noise under an in-plane magnetic field. By correlating the longitudinal spin-electric susceptibility with qubit coherence and control performance, we identify an optimal operating point where the sensitivity to charge noise is minimized. We find that optimizing the magnetic-field orientation reduces the spin-electric susceptibility, resulting in a five-fold enhancement of the Hahn-echo coherence time and a nearly tenfold suppression of control infidelity. At the optimal operating point, gate set tomography demonstrates the maximum gate fidelity of 99.82 %. Our finding enhances the prospects of hole spin qubits for scalable quantum information processing.