# Locating Two-Level Systems in a Superconducting Xmon Qubit
## 5 Conclusions  
In this study, we first observed the modulation of qubit energy levels with \(Z\) bias and did not find any obvious anti-crossings in the spectroscopy in Figure 1b. Then, we measured the long-term stability of the qubit relaxation time \(T_{1}\) at different frequencies and found that there exists an obvious time-varying reduction in \(T_{1}\). To elucidate the causation of the observed reduction in \(T_{1}\), we used a time-multiplexed protocol to measure the stability in \(T_{1}\), \(T_{2}^{R}\), and the qubit frequency shift of a tunable Xmon qubit during a time exceeding \(85\) h. By time-domain Allan deviation and frequency-domain power spectral density analyses, we conclude that there exist two spectrally unstable TLSs which are mainly responsible for the qubit parameters' fluctuations. Additionally, we calculated the coupling strengths and energy differences between the qubit and the TLSs, and we located these two TLSs within the range of \(2\) um in the junction area.  
The analytical methods used in this paper are effective tools to analyze the long-term stability of the quantum system parameters, such as decoherence times of quantum dots [46], optical gains of laser diodes using quantum wires [47; 48], and spontaneous currents in superconducting rings [49; 50], and are not limited to the superconducting qubit system discussed here. The presence of fluctuations in \(T_{1}\), \(T_{2}^{R}\) and qubit frequency highlights the importance of recalibrating qubits frequently, as these fluctuations contribute to errors in quantum gate fidelities and quantum teleportation fidelities [51; 52], and this study provides an approach for selecting the ideal working points of tunable superconducting Xmon qubits to avoid the unstable area. Basically, this emphasizes that in order to accurately evaluate the quality of a qubit, not only the exceptional coherence time, but also the long-term average value should be estimated.  
Additionally, the methods provide a way to locate the main source of system noise via the Allan deviation and frequency-domain power spectral density. The observed coherent qubit-TLS couplings (Figure 4) are an unambiguous indication that there exist near-resonant TLSs. The occurring frequency of revivals is close to the frequency of \(T_{1}\) decay, indicating the instability of the spectrum, which is consistent with the interacting TLS model in Figure 2c. Therefore, we attribute the reduction in \(T_{1}\) in Figure 2b to the near-resonant TLSs. However, the Allan deviation and frequency-domain power spectral density analyses require a vast amount of data, which is time consuming. Other fast and accurate noise analysis methods still need to be developed.  
Conceptualization, X.-X.Y.; Data curation, X.-Y.Y. and L.-L.G.; Formal analysis, X.-X.Y. and X.-Y.Y.; Funding acquisition, H.-O.L. and G.-P.G.; Resources, L.D. and Z.-L.J.; Writing--original draft, X.-X.Y.; Writing--review & editing, P.D. and H.-O.L. All authors have read and agreed to the published version of the manuscript.  
Not applicable.  
We thank [PERSON] for helpful discussions and improving the paper.  
The authors declare no conflict of interest.