Protocols for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:msqrt><mml:mi>i</mml:mi><mml:mi>SWAP</mml:mi></mml:msqrt></mml:math> gates using a fixed coupler driven by two microwave pulses

Achieving high-fidelity two-qubit gates is a key requirement for the realization of large-scale quantum computation and simulation. In this paper, we propose a microwave-control protocol for the implementation of a two-qubit gate employing two transmon qubits coupled via a fixed-frequency transmon coupler. This protocol entails applying two microwave pulses exclusively to the coupler, thereby inducing interaction between fixed-frequency transmon qubits. This interaction facilitates the realization of $\sqrt{i\mathrm{SWAP}}$ gates. Additionally, we explore the implementation of the gate scheme in two distinct qubit architectures (ABA and ABC). Without considering decoherence, the ABA architecture $\sqrt{i\mathrm{SWAP}}$ gate has a fidelity of 94.60% in 346 ns, while the ABC architecture achieves 99.63% in 378 ns. When considering the implementation of the $\sqrt{i\mathrm{SWAP}}$-like gate, we show that gate fidelity can exceed 99.9% in both architectures.