How do I generate the microwave pulses for single qubit gate operations?

Generating microwave pulses for single-qubit gates requires modulating a continuous-wave (CW) microwave carrier at the qubit transition frequency with precisely shaped envelope pulses. The pulse rotates the qubit state on the Bloch sphere: an X gate (pi rotation about X-axis) uses a pulse with real-valued envelope, a Y gate uses an imaginary-valued envelope (90° phase shift), and arbitrary rotations use combined I and Q modulation. Pulse generation chain: (1) A microwave signal generator or synthesizer produces a CW tone at the qubit frequency (4-8 GHz) with phase noise < -120 dBc/Hz at 1 MHz offset for coherent control. (2) An IQ mixer modulates the carrier with baseband I(t) and Q(t) waveforms from arbitrary waveform generators (AWGs). The AWG sample rate must be at least 2× the maximum modulation bandwidth: for 20 ns Gaussian pulses, the bandwidth is ~200 MHz, requiring AWG sample rates ≥ 500 MSa/s (practically, 1-2 GSa/s is standard). (3) The modulated pulse passes through attenuators, filters, and the cryogenic cable chain to reach the qubit. Typical pulse parameters: pi-pulse duration 10-40 ns, Gaussian or DRAG (Derivative Removal by Adiabatic Gate) pulse shape with sigma = 4-10 ns, peak amplitude corresponding to Rabi frequency Omega = pi/t_gate (for a pi pulse in time t_gate). DRAG pulses add a quadrature component proportional to the derivative of the Gaussian envelope to suppress leakage to the |2⟩ state, improving gate fidelity from ~99.5% to >99.9%.