# Locating Two-Level Systems in a Superconducting Xmon Qubit
## 2 Device and Methods  
We use the endmost qubit of a tunable six-Xmon-qubit-chain device to probe the TLS defects [38; 39] as shown in Figure 1a. The qubit is made of aluminum on silicon, with alarge shunt capacitor to decouple the charge noise. The qubit is equipped with a microwave XY control line, a flux bias Z line for tuning the frequency, and an individual \(\lambda/4\) readout resonator. The maximum transition frequency of the qubit is \(\omega_{\text{max}}/2\pi=5423.5\) MHz, and the qubit anharmonicity is \(\alpha/2\pi=-240\) MHz. The bare frequency of the readout resonator is \(\omega_{r}/2\pi=6424.3\) MHz, which lies in the dispersive regime. The other five qubits are modulated to the lowest frequencies with significant frequency differences from the operating qubit, leaving them fully uncoupled.  
By adding a Z bias pulse behind the \(X_{\pi}\) driving, we demonstrate the frequency spectroscopy of the qubit. The result is shown in Figure 1b. The frequency of the qubit versus Z bias amplitude is extracted and fitted by  
\[\omega=(\omega_{\text{max}}+\omega_{c})\sqrt{\sqrt{1+d^{2}\text{tan}^{2}[M(V-V_ {0})]}\ |\cos[M(V-V_{0})]|}-\omega_{c}, \tag{1}\]  
where \(\omega_{\text{max}}/2\pi=\ 5423.5\) MHz, \(\omega_{c}/2\pi=\ 23.7\) MHz, \(d=-0.2447\), \(M=\ 2.062\), and \(V_{0}=-0.0043\) V, respectively. Due to the large junction area (\(\approx\)1 um\({}^{2}\)), previous experiments have revealed characteristic level avoidance or anti-crossing on superconducting phase qubits [18; 19; 20; 21; 22], flux qubits [23; 24], and Quantronium [25]. However, the junction area of the Xmon qubit used in this experiment is \(\approx\)200  nm\({}^{2}\), and the spectroscopy in Figure 1b shows no obvious anti-crossing. The result implies that the TLS density per junction area is too low to be detected with a 1 MHz frequency resolution and 1 min timescale.  
Then, we evaluate the fluctuations of the qubit relaxation time \(T_{1}\) with different qubit frequencies, both by sweeping the Z bias and by consecutive measurements. The experimental pulse sequence that we use to measure \(T_{1}\) at a single frequency is shown in Figure 2a with odd pulse sequences. The qubit is driven to the first-excited state by a calibrated \(X_{\pi}\) pulse. After waiting for a variable free evolution time \(\Delta t_{i}\), the population of the excited state is measured. \(T_{1}\) is determined by fitting the population to a single-exponential decay as a function of \(\Delta t_{i}\). We sweep a frequency range across 230 MHz and a spanning time of up to 21 h with 0.005 V and 16 min step sizes. There is an obvious time-varying reduction in \(T_{1}\) between \(-0.05\) V and 0.05 V as shown in Figure 2b.  
Figure 1: (a) An electron microscope image of a single cross-shaped Xmon qubit and the corresponding simplified measurement circuit. (b) Dependence of the qubit frequency on the Z bias. The peaks are extracted and fitted to Equation (1) (red dotted line). The frequency resolution is 1 MHz. No anti-crossings can be found in these spectroscopy results.  
To elucidate the origin of the observed time-varying reduction of \(T_{1}\), we tune the qubit to \(-0.033\) V, where the qubit frequency is \(\omega/2\pi=5419\) MHz. Using the interleaved sequences shown in Figure 2a, we acquire the qubit parameters at once, including the lifetime \(T_{1}\), decoherence time \(T_{2}^{R}\), and qubit frequency shift \(\Delta\omega/2\pi\). The whole sequence takes approximately 65 s. While the interleaved sequences prolong the time to obtain a single parameter, which leads to an increased noise window, it ensures that each data point of \(T_{1}\), \(T_{2}^{R}\), and \(\Delta\omega/2\pi\) is exposed to the same noise environment. By repeatedly running the sequences shown in Figure 2a, we monitored the stability of the parameters. Figure 2c shows the results of 4700 sequence repetitions over 85 h.