This article presents a 4-to-5 GHz LC oscillator operating at 4.2K for quantum computing applications. The phase noise (PN) specification of the oscillator is derived based on the control fidelity for a single-qubit operation. To reveal the substantial gap between the theoretical predictions and measurement results at cryogenic temperatures, a new PN expression for an oscillator is derived by considering the shot-noise effect. To reach the optimum performance of an LC oscillator, a common-mode (CM) resonance technique is implemented. Additionally, this work presents a digital calibration loop to adjust the CM frequency automatically at 4.2K, reducing the oscillator's PN and thus improving the control fidelity. The calibration technique reduces the flicker corner of the oscillator over a wide temperature range (10 x and 8 x reduction at 300K and 4.2K, respectively). At 4.2K, our 0.15-mm(2) oscillator consumes a 5-mW power and achieves a PN of -153.8dBc/Hz at a 10MHz offset, corresponding to a 200-dB FOM. The calibration circuits consume only a 0.4-mW power and 0.01-mm(2) area.