The present article primarily focuses on the design of an ultra-low-noise amplifier specifically tailored for quantum applications. The circuit design places a significant emphasis on improving the noise figure, as quantum-associated applications require the circuit's noise temperature to be around 0.4 K. This requirement aims to achieve performance comparable to the Josephson Junction amplifier. Although this task presents considerable challenges, the work concentrates on engineering the circuit to minimize mismatch and reflection coefficients, while simultaneously enhancing circuit transconductance. These efforts aim to improve the noise figure as efficiently as possible. The results of this study indicate the possibility of achieving a noise figure of approximately 0.009 dB for a unique circuit design operating at 10 K. In a departure from traditional approaches, this study employs quantum mechanical theory to analyze the circuit comprehensively. By employing quantum theory, the researchers derive relationships that highlight the crucial quantities upon which the circuit design should focus to optimize the noise figure. For example, the circuit's gain power, which depends on the circuit's photonic modes, is theoretically derived and found to affect the noise figure directly. Ultimately, by merging quantum theory with engineering approaches, this study successfully designs a highly efficient circuit that significantly minimizes the noise figure in a quantum application setting.