High-fidelity and robust three-qubit Toffoli gates in silicon-based quantum-dot spin qubits

Electron spin qubits in semiconductor silicon quantum dots (QDs) are a promising solid-state system for quantum computing. To increase the reliable circuit depth on noisy intermediate-scale quantum (NISQ) computing machines, or achieve the ultimate goal of error-corrected fault-tolerant quantum computation, constructing high-fidelity and robust universal quantum gates to meet the stringent computing requirements is an important and timely issue. We aim to construct control pulses for quantum gates that are robust against both quantum and classical noises, as well as system parameter uncertainties, with fault-tolerant gate fidelities for the QD spin-qubit system using the optimal control method. Here, we report the results of optimized three-qubit Toffoli gates in the presence of relaxation and dephasing noise. The optimized three-qubit Toffoli gates in ideal cases (without any noise), regardless of which qubit serves as the target (Q1, Q2, or Q3), all achieved infidelities below 10^-5, limited primarily by rotating wave approximation errors. Without considering dephasing noise, the optimized Toffoli gate infidelity can be reduced below 10^-4, approaching the fundamental energy relaxation error limit (with T₁ = 16 ~ 34 μs). More importantly, under realistic experimental noise conditions, the optimized Toffoli gate with a gate time t_f = 460 ns can achieve an infidelity of approximately 4×10^-4, representing a two orders of magnitude improvement compared to the reported experimental i-Toffoli gate infidelity of ~ 4×10^-2. The optimized Toffoli gate with a longer gate time,  t_f  = 520 ns, can tolerate stronger dephasing noise (over 4 times) and still maintain an infidelity below 10^-3.