Using different configurations of applied strong driving and weak probe fields, we find that only a single three-level superconducting quantum circuit (SQC) is enough to realize amplification, attenuation, and frequency conversion of microwave fields. Such a three-level SQC has to possess $\mathrm{\ensuremath{\Delta}}$-type cyclic transitions. Different from the parametric amplification (attenuation) and frequency conversion in nonlinear optical media, the real energy levels of the three-level SQC are involved in the energy exchange when these processes are completed. We quantitatively discuss the effects of amplification (attenuation) and the frequency conversion for different types of driving fields. The values and conditions are obtained for the optimal gains for amplification (attenuation) and the optimal conversion efficiencies. Our study provides a method to amplify (attenuate) microwave, realize frequency conversion, and also lay a foundation for generating single or entangled microwave photon states using a single three-level SQC.
Adaptive tomography has been widely investigated with the aim of achieving faster state tomography processing of quantum systems. Infidelity of the nearly pure states in a quantum information process generally scales as O(1/N), which requires a large number of statistical ensembles in comparison to the infidelity scaling of O(1/N) for mixed states. One previous report optimized the measurement basis in a photonic qubit system, whose state tomography uses projective measurements, to obtain an infidelity scaling of O(1/N). However, this dramatic improvement cannot be applied to indistinct measurement systems in which two quantum states cannot be distinguished with perfect measurement fidelity. We introduce in this work an optimal measurement basis to achieve fast adaptive quantum state tomography and a minimum magnitude of infidelity in an indistinct measurement system. We expect that the adaptive quantum state tomography protocol can lead to a reduction in the number of required measurements of approximately 33.74% via analytical expectation without changing the O(1/N) scaling. Experimentally, we find a 15.06% measurement number reduction in a superconducting circuit system.