Robust spin-qubit control in a natural Si-MOS quantum dot using phase modulation

TL;DR

Phase-modulated concatenated continuous driving (CCD) creates a double-dressed spin basis that dynamically decouples a Si-MOS quantum dot qubit from low-frequency detuning and Rabi-noise. The authors demonstrate two-axis control, energy-selective readout, and high-fidelity single-qubit gates in a natural silicon device, achieving a Clifford gate fidelity of $99.1\%$ and extending coherence times by over two orders of magnitude. This showcases robust, feedback-free qubit control suitable for scalable silicon quantum processors, with potential extensions to two-qubit operations and global-control schemes. The work highlights a practical pathway to high-fidelity quantum computation in isotopically natural silicon using phase modulation rather than amplitude modulation, mitigating issues from MW instability and heating.

Abstract

Silicon quantum dots are one of the most promising candidates for practical quantum computers because of their scalability and compatibility with the well-established complementary metal-oxide-semiconductor technology. However, the coherence time is limited in industry-standard natural silicon because of the $^{29}$Si isotopes, which have non-zero nuclear spin. Here, we protect an isotopically natural silicon metal-oxide-semiconductor (Si-MOS) quantum dot spin qubit from environmental noise via electron spin resonance with a phase-modulated microwave (MW) drive. This concatenated continuous drive (CCD) method extends the decay time of Rabi oscillations from 1.2 $\mathrm{μs}$ to over 200 $\mathrm{μs}$. Furthermore, we define a protected qubit basis and propose robust gate operations. We find the coherence time measured by Ramsey sequence is improved from 143 ns to 40.7 $μ$s compared to that of the bare spin qubit. The single qubit gate fidelity measured with randomized benchmarking is improved from 95% to 99%, underscoring the effectiveness of the CCD method. The method shows promise for improving control fidelity of noisy qubits, overcoming the qubit variability for global control, and maintaining qubit coherence while idling.

Figures (9)

• Figure 1: CCD-protected spin qubit in a natural Si-MOS quantum dot. a False-color scanning electron microscope image of the device. The device is designed to host an array of three quantum dots under gates SG1, SG2, and SG3 in a Si/SiO2 channel along with an SET. We manipulate the electron spin under gate SG3. b Schematic of the Si-MOS device. An external magnetic field $B_{\rm{ext}}= 825$ mT is applied to the device. c Charge stability diagram measured as a function of the gate voltages $V_{\rm{SGS}}$ and $V_{\rm{SG3}}$. The white dashed line shows the charge transition line and $N$ denotes the number of electrons in the quantum dot under the SG3 gate. Colored dots show the SG3 and the SGS gate biases for control and readout of the spin state. d Gate-voltage pulse cycle for energy-selective initialization and readout. e Principle of coherence protection by phase-modulated MW. In the laboratory frame, the bare qubit is protected from spin-flips by an energy-gap $\hbar\omega_{\rm{L}}$, but detuning errors in $\omega_{\rm{L}}$ can cause phase-drift. In the first rotating frame of a resonant MW drive, the qubit is protected from detuning errors due to an energy-gap $\hbar\Omega$ between the dressed states, and the phase-modulation appears as an a.c. drive. In the second rotating frame with the phase-modulation, the qubit is protected from fluctuations in the MW-drive by an energy-gap $\hbar\epsilon_{\rm{m}}$, and the MW-drive appears as a d.c. B-field. In other words, the qubit is dynamically decoupled from noise by a rotation about two axes. f Comparison of the bare qubit Rabi oscillations and the CCD-protected Rabi oscillations in the case with $\theta_{\rm{m}}=0, \pi/2$, and $\pi/4$. For $\theta_{\rm{m}}=0$, we use the same data as ref kuno2024concatenated. The inset shows the Fourier transform spectrum of these Rabi oscillations. When we set $\theta_{\rm{m}}=\pi/4$, we can observe the Mollow triplet. When $\theta_{\rm{m}}=0$, which we call the idle pulse state, only the center peak appears, corresponding to the stabilized Rabi oscillations.
• Figure 2: Protected qubit control and readout protocol.a Schematic of the Bloch sphere along with the Hamiltonian vector for idle and operation. When executing idle, $\theta_{\rm{m}}$ is set to 0 and the Hamiltonian vector aligned to the $z$-axis. The idle pulse is used to maintain coherence between operations and after the operation pulses to match the laboratory and the second rotating frame bases for readout. To perform gate operations, we set $\theta_{\rm{m}}=\pi/2$ and change the azimuth angle of the Hamiltonian vector in the $x$-$y$ plane, enabling spin rotation around an arbitrary axis perpendicular to the $z$-axis. b Indicator representing the periodic readout matching between the laboratory and second rotating frames. The color of the ring shows the inner product of the laboratory frame basis state and the second rotating frame basis state, $\left\lvert \left\langle 1 \lvert 1" \right\rangle \right\rvert^2$. To achieve readout matching the total pulse length satisfies $\omega_{\rm{m}} t=2\pi n$. c Example drive conditions for diffrent single protected qubit operations. $t_c$ is the pulse duration.
• Figure 3: Demonstration of manipulation and readout of the CCD-protected qubit. All measurements are conducted under $\epsilon_{\rm{m}} = \omega_{\rm{m}}/4 =2\pi\times235$ kHz. The top panel of each figure shows time evolutions of the CCD-protected qubit using the Bloch sphere representation of the effective field $\epsilon_{\rm{m}}$ and the readout indicators. a CCD-Rabi experiment. After initialization, the operation pulse of the duratoin of $t_c$ is applied, followed by the idle pulse for readout matching. b Two-axis control experiment. Two consecutive $\pi/2$ pulses, with MW phases 0 and $\phi_{\rm{var}}$, are applied. The measured spin-up probability as a function of $\phi_{\rm{var}}$ oscillates with a period of $2\pi$, demonstrating the control of the rotation axis in the CCD qubit basis. c CCD-Ramsey experiment. Two $\pi/2$ pulses and an idle pulse between them are applied. The spin-up probability is plotted as a function of the duration of the idle pulse $t_c$, and the coherence time is evaluated from the decay.
• Figure 4: Gate operations and randomized benchmarking measurement.a Schematic illustration of co-rotating (CCD drive) term and counter-rotating term on the Bloch sphere. The Hamiltonian vector of the co-rotating term is constant in time, while that of the counter-rotating term rotates at angular frequency $2\omega_{\rm{m}}$. b Numerical calculation of the probability $\vert 1"\rangle$ state due to the co-rotating and the counter-rotating term for $\epsilon_{\rm{m}} = \omega_{\rm{m}}/4$. The readout indicators are shown on the top. The rotation caused by the counter-rotating term is approximately zero at the period of $\pi/\omega_{\rm{m}}$. c$\pi/2$ and $\pi$ rotations constituting Clifford gates. As we choose $\epsilon_{\rm{m}} = \omega_{\rm{m}}/4$, the indicator rotates $2\pi$ and $4\pi$ during the $\pi/2$ and $\pi$ rotation respectively, indicating that the readout condition is satisfied. d Randomized benchmarking results for the bare and CCD-protected qubits. The sequence is repeated for $K=15$ random Clifford gate sets. The Clifford gates are composed of $\pi/2$ and $\pi$ pulses with durations of 1.06 $\mu$s and 2.13 $\mu$s, respectively. The sequence fidelities decay exponentially as a function of the number of Clifford gates. The gate infidelity is improved by a factor of 5 for the CCD-protected qubit. The average gate fidelity of the bare qubit is 95$\pm1\%$. The CCD spin control achieves a higher fidelity of 99.1$\pm 0.1\%$.
• Figure 5: Bare qubit properties.a Micorwave amplitude $A_{\rm{mw}}$ dependence of Rabi frequency. The dotted line is a linear fitting for $A_{\rm{mw}}\leq 2.4$. Rabi oscillations above 3 MHz cannot be obtained due to the strong damping. b$A_{\rm{mw}}$ dependence of Rabi decay time $T_2^{\rm{Rabi}}$. c$A_{\rm{mw}}$ dependence of quality factor. Since $T_2^{\rm{Rabi}}$ rapidly decreases as $A_{\rm{mw}}$ is increased, the quality factor is limited to 2.7. d Ramsey sequence. The relative phase of the second pulse is artificially modulated at 25 MHz to improve the fit. The coherence time $T_2^{\rm{*}}$ is 143 ns. e Spin echo result. A spin echo sequence improves the coherence time and we obtain $T_2^{\rm{echo}} = 230$$\mu$s. $T_{\rm{2}}^{\rm{echo}}$ is more than 1000 times longer than $T_{\rm{2}}^{\rm{*}}$, suggesting that the source of dephasing is low frequency noise compared with the timescale of the electron spin dynamics. These results are consistent with those of previous measurements kuno2024concatenated.