Robust composite two-qubit gates for silicon-based spin qubits

Abstract

We propose a universal approach based on Hamiltonian inverse engineering to realize a set of parameterized two-qubit gates. This method possesses unique advantages to simultaneous control of transitions among four energy levels, providing a simpler and effective way to construct composite two-qubit gates with fewer operations than traditional methods. Applied to silicon double quantum dots (DQDs), one can realize a one-step fSim gate and a B gate with only one pulse switch. Of note, the method can be further integrated with various optimization theories to enhance gate performance. Based on quantum optimal control theory, we develop a high-fidelity fSim gate scheme with experimentally feasible pulse shapes, featuring an average gate time of 50 ns and a theoretical fidelity of 99.95% in the presence of decoherence and approximation error. By incorporating geometric quantum gate principles, we propose a combined geometric and dynamic fSim gate scheme. Numerical simulations demonstrate that this hybrid scheme exhibits stronger robustness against systematic errors compared to the purely dynamic approach. Our method is generalizable to arbitrary two-qubit physical systems, offering a feasible pathway for rapidly and robustly constructing composite two-qubit gates.

Figures (9)

• Figure 1: Spin qubits in the silicon DQDs. The single-qubit gate can be obtained by generating oscillating magnetic fields through the periodic modulation of gate voltages ($V_L$ and $V_R$), while two-qubit gate are realized by precisely regulating the exchange interaction $J$ between spins via the electrostatic barrier gate $V_M$.
• Figure 2: (a) The maximum value of pulse amplitude $|JT|_{max}$ with different fSim gate parameters $\{\vartheta, \Xi\}$ under the rectangular pulse scheme. Where the parameter ranges selected are $|\vartheta|\in[0,\pi/2]$ and $|\Xi|\in[0,\pi]$. (b) The $jT$ under the rectangular pulse scheme of fSim gate, where fSim gate parameters are $\vartheta=\pi/4,\Xi=\pi/2$, gate time $T\approx45$ ns, $\delta E_z \approx 2\pi\times 22$ MHz and $E_z\approx 2\pi\times 5.6$ MHz.
• Figure 3: A general construction scheme for B gate, combining a CNOT gate and a controlled-$e^{(\pi/4)i\sigma_x}$ gate.
• Figure 4: (a) The $jT, B^1_yT$ under the rectangular pulse scheme of B gate. (b) Population transfer during the B gate operation. The initial state is chosen as $(|00\rangle+|01\rangle)/\sqrt{2}$, where the horizontal axes of both subfigures represent the gate operation duration, and the gate time $T\approx 76$ ns.
• Figure 5: (a) The fSim gate fidelity for 1600 different initial states under the rectangular pulses scheme, the red curve denotes the average fidelity, where gate time $T\approx 45$ ns and set $\delta E_z=2\pi/T\approx 2\pi\times 22$ MHz. (b) The $jT$ under the rectangular pulse scheme of fSim gate, where gate time $T\approx 45$ ns and set $\delta E_z=6\pi/T\approx 2\pi\times 66$ MHz. (c) The variation curve of error sensitivity $q_s$ with tunable parameter $\eta$, where the upper-left inset is a magnification of the boxed region. (d) The variation curve of fSim gate fidelity $\mathcal{F}$ with tunable parameter $\eta$, where the upper-left inset showing a magnification of the boxed region. Here, each average fidelity value is obtained by averaging the results of numerical simulations performed over 100 distinct initial states. (e) The maximum value of pulse amplitude $|JT|_{max}$ with different fSim gate parameters $\{\vartheta, \Xi\}$ under the optimal parameter pulse scheme. Where the parameter ranges selected are $|\vartheta|\in[0,\pi/2]$ and $|\Xi|\in[0,\pi]$. (f) The $jT$ under the optimal parameter pulse scheme of fSim gate, where gate time $T\approx 50$ ns and set $\delta E_z=2N\pi/T\approx 2N\pi\times 20$ MHz. The red curve shows the result when $N=1$, and the blue curve shows the result when $N=3$. Where all the fSim gates parameters are chosen as $\vartheta=\pi/4,\Xi=\pi/2$.