Fast, Universal Scheme for Calibrating Microwave Crosstalk in Superconducting Circuits

1. Introduction

Quantum computing is promising for solving complex problems beyond the reach of traditional computing methods [1,2,3]. Despite recent demonstrations of quantum supremacy [4,5,6,7,8,9,10], utility [11], and quantum logical error correction [12,13,14,15,16,17,18], achieving scalable quantum computing is still obstructed by numerous challenges [19]. Among these, crosstalk is well-recognized in various quantum computing systems [20,21,22,23,24,25,26,27,28]. Specifically, microwave crosstalk in superconducting circuits significantly impacts system performance by inducing unwanted quantum state transitions and gate errors [29,30,31,32]. Figure 1(a) shows such an example for transmon qubit, where the first three levels are included [33]. A microwave drive signal intended for bias qubit $Q_{\mathrm{b}}$ with Rabi strength ${\Omega }_{\mathrm{b}}$ and frequency ${\omega }_{\mathrm{cr}}$ couples unintentionally to target qubit, inducing an effective Rabi strength ${\Omega }_{\mathrm{cr}}$ on the target qubit. $r{e}^{i{\varphi }_{\mathrm{cr}}}={\Omega }_{\mathrm{cr}}/{\Omega }_{\mathrm{b}}$ is the complex transfer amplitude or the so-called crosstalk coefficient. Leakage occurs when the frequency ${\omega }_{\mathrm{cr}}$ of the spurious microwave signal is near resonance with the transition frequency ${\omega }_{\mathrm{ef}}$ between the first excited ($\left|e\right.〉$) and second excited ($\left|f\right.〉$) states. Gate errors manifest when ${\omega }_{\mathrm{cr}}$ is near resonance with the transition frequency ${\omega }_{\mathrm{ge}}$ between the ground ($\left|g\right.〉$) and the first excited ($\left|f\right.〉$) state. This form of interference disrupts the precise control over qubit states which is essential for accurate quantum computation and error correction. As such, the characterization and mitigation of microwave crosstalk within superconducting circuits are critical for robust and scalable quantum computing architectures.

As microwave crosstalk can always be compensated with an additional out-of-phase signal, as shown in Figure 1(a), the task translates to how one can characterize and calibrate the complex amplitude $r{e}^{i{\varphi }_{\mathrm{cr}}}$ of the crosstalk signal, so a compensation signal of the same frequency and amplitude but out-of-phase can be applied to null the crosstalk signal. Some previous works have addressed this issue, each in a particular condition. When ${\omega }_{\mathrm{cr}}={\omega }_{\mathrm{ge}}$, a clear constructive and destructive dynamic Rabi oscillation pattern is observed when ${\varphi }_{\mathrm{cr}}$ changes periodically [34,35], which can be used to extract r and ${\varphi }_{\mathrm{cr}}$. Refs. [36,37] extended this method to off-resonant case (${\omega }_{\mathrm{cr}}\ne {\omega }_{\mathrm{ge}}$) and Ref. [36] further provided an analytical formula to fit the oscillation pattern. For far-off resonance, the interference pattern of Rabi oscillation becomes invisible, and the spin-echo sequence is proposed [38] to calibrate microwave crosstalk utilizing the AC-Stark effect. Ref. [15] used the Ramsey error filter sequence to suppress leakage when ${\omega }_{\mathrm{cr}}$ is in the vicinity of ${\omega }_{\mathrm{ef}}$. Although these schemes can cover a majority of crosstalk scenarios in the three-level subspace of transmon qubit, they are not universal and lack a unified model that governs the basic principle of the calibration procedure. Most of these schemes require at least one two-dimensional scan to obtain the final results, which makes it rather time-consuming to bring up a large quantum processor. Moreover, the sensitivity of these schemes is challenged when $|{\omega }_{\mathrm{cr}}-{\omega }_{\mathrm{ge}}|\gg 0\mathrm{or}|\eta |$, where $\eta ={\omega }_{\mathrm{ef}}-{\omega }_{\mathrm{ge}}$ is the anharmonicity of the superconducting qubit.

Here, we propose and demonstrate a fast and universal scheme using Ramsey-like sequences for calibrating microwave crosstalk. Given the abovementioned limitations, this work contributes to the literature mainly from three aspects. First, we propose and experimentally validate a simple two-level off-resonant driving model; this theoretical model unifies the calibrating principle for all crosstalk scenarios and provides explicit formulas to fit the experimental data. Second, instead of requiring a two-dimensional parameter scanning procedure, only multiple one-dimensional sweeps are sufficient for accurately retrieving target parameters. Third, we introduce error-amplifying sequences to improve accuracy, especially in large detuning regions. With these advancements in terms of universality, efficiency, and accuracy, our method can be readily applied to a fully automated calibration of microwave crosstalk in a large-scale quantum processor. This approach helps to utilize the full potential of contemporary noisy-intermediate-scale quantum processors.

2. Methods

2.1. Theoretical Model

While superconducting qubits have multiple energy levels, a two-level-system (TLS) model is adequately comprehensive for addressing the issue of microwave crosstalk. In the presence of leakage, our focus is primarily on the $\{\left|e\right.〉,\left|f\right.〉\}$ subspace. Conversely, when gate errors occur, attention is directed toward the computational subspace, $\{\left|g\right.〉,\left|e\right.〉\}$. This simplification allows for a focused analysis of the phenomena critical to understanding and calibrating microwave crosstalk. Considering both the crosstalk and its compensation signals applied on a TLS, the Hamiltonian in the laboratory frame ($\hslash =1$) is

$$\begin{array}{c}\hfill H=-\frac{{\omega }_{\mathrm{TLS}}}{2}{\sigma }_z+\Omega cos\left({\omega }_{\mathrm{cr}}t+\phi \right){\sigma }_x\end{array}$$

(1)

where ${\sigma }_{x,y,z}$ are Pauli matrixes, ${\omega }_{\mathrm{TLS}}$ takes the value of ${\omega }_{\mathrm{ge}}$ in the computational subpace, and ${\omega }_{\mathrm{ef}}$ in the $\{\left|e\right.〉,\left|f\right.〉\}$ subspace. $\Omega$ ($\phi$) is the effective drive amplitude (phase) under the action of ${\Omega }_{\mathrm{cr}}$ and ${\Omega }_{\mathrm{com}}$. The simple trigonometric formula gives

$$\begin{array}{c}\hfill \Omega =\sqrt{{\Omega }_{\mathrm{com}}^2+{\Omega }_{\mathrm{cr}}^2+2{\Omega }_{\mathrm{com}}{\Omega }_{\mathrm{cr}}cos\left({\varphi }_{\mathrm{com}}-{\varphi }_{\mathrm{cr}}\right)}\end{array}$$

(2a)

$$\begin{array}{c}\hfill \phi =arctan\frac{{\Omega }_{\mathrm{com}}sin{\varphi }_{\mathrm{com}}+{\Omega }_{\mathrm{cr}}sin{\varphi }_{\mathrm{cr}}}{{\Omega }_{\mathrm{com}}cos{\varphi }_{\mathrm{com}}+{\Omega }_{\mathrm{cr}}cos{\varphi }_{\mathrm{cr}}}\end{array}$$

(2b)

Moving to the rotation frame and with the rotation wave approximation, we have

$$\begin{array}{c}\hfill H_R=\frac{\Delta }{2}{\sigma }_z+\frac{\Omega }{2}\left[cos\left(\phi \right){\sigma }_x-sin\left(\phi \right){\sigma }_y\right]\end{array}$$

(3)

where, $\Delta ={\omega }_{\mathrm{cr}}-{\omega }_{\mathrm{TLS}}$ is the detuning between TLS and crosstalk signal. Eq. (3) is simply an off-resonant Rabi model, which we used to model all kinds of crosstalk scenarios. The effective Rabi strength ${\Omega }_R=\sqrt{{\Delta }^2+{\Omega }^2}$, and the corresponding rotation axis is $\left(sin\theta cos\phi ,sin\theta sin\phi ,cos\theta \right)$ with $\theta =arctan\left(\Omega /\Delta \right)$, see Figure 1(b).

We classify microwave crosstalk into two different types. The first type is the far-off-resonant case, $\Delta \gg \Omega$, $\theta \approx 0$; the crosstalk mainly induces phase error that is better observed in the $\mathbf{XY}$ plane. We thus use the $X_{\pi /2}$ Ramsey-like sequence to prepare the qubit in the equatorial plane and measure it on the basis of ${\sigma }_y$, as shown in Figure 2(a). The excited state probability $P_1$ of the final state is

$$\begin{array}{c}\hfill P_1^{X/2}=A+\left(1-A\right)cos\alpha ,\end{array}$$

(4)

where $A=\left(1+{sin}^2\theta {sin}^2\phi \right)/2,\alpha ={\Omega }_R\tau$. Note that a simplification $\Delta \tau =2n\pi$ is used to obtain the above equation so that the phase results from frame rotation can be ignored. For more general results, see Appendix A.

The second type is the resonant or near-resonant case, $\Delta \ll \Omega$, $\theta \approx \pi /2$; we use the $X_{\pi }$ Ramsey-like sequence (also shown in Figure 2(a)) to calibrate the crosstalk effect where amplitude or leakage error manifests. The probability $P_1$ in this case is

$$\begin{array}{c}\hfill P_1^X=B\left(1-cos\alpha \right),\end{array}$$

(5)

where $B={sin}^2\theta /2,\alpha ={\Omega }_R\tau$. We note that in principle, the $X_{\pi /2}$ Ramsey-like sequence can also be used for the second type, see Figure 2(b), but the $X_{\pi }$ Ramsey-like sequence provides greater contrast and sensitivity which may lead to better accuracy.

2.2. Crosstalk Calibration Procedure

In the experiment, applying a fixed amplitude drive on the bias qubit for a duration of $\tau$, our goal is to search for the best compensation parameters on the target qubit. To avoid two-dimensional parameter scanning and improve the experimental efficiency, we fix $\tau$ and only need two sets of data of scaning ${\Omega }_{\mathrm{com}}$ and ${\varphi }_{\mathrm{com}}$ successively.

For the first crosstalk type, first, the compensation signal on the target qubit is fixed to ${\Omega }_{\mathrm{com}}=r{\Omega }_{\mathrm{b}}$ (r is arbitrary in this step), and the experimental data of scanning its phase are fitted with Eq. (4), as shown in Figure 2(c). We obtain a crosstalk phase ${\varphi }_{\mathrm{cr}}$ of 3.54; however, this phase is not necessarily accurate, especially when the difference between the two Rabi frequencies ${\Omega }_{\mathrm{com}}$ and ${\Omega }_{\mathrm{cr}}$ is large. At this time, the difference between ${({\Omega }_{\mathrm{com}}+{\Omega }_{\mathrm{cr}})}^2$ and ${({\Omega }_{\mathrm{com}}-{\Omega }_{\mathrm{cr}})}^2$ is negligible, and corresponds to a pair of opposite phases. Next, we set the compensation phase ${\varphi }_{\mathrm{com}}=3.54-\pi$, and scan r to fit with ${\varphi }_{\mathrm{cr}}=3.54$ and ${\varphi }_{\mathrm{cr}}=3.54-\pi$, which are shown in Figure 2(d). The phase that best fits the data is the true crosstalk signal phase, and the crosstalk coefficient is extracted to 0.22 at the same time.

For the second crosstalk type, the process aligns with that shown in Figure 2(c) and (d) except for the $X_{\pi }$ Ramsey-like sequence. Figure 2(e) and (f) show an example in the leakage subspace, where ${\omega }_{\mathrm{cr}}$ is close to ${\omega }_{\mathrm{ef}}$. Initially, we fix the compensation amplitude, and the experimental results (blue points) along with the fitting results (black solid line) for scanning the phase are depicted in Figure 2(e), where the fitted phase with Eq. (5) is 3.04. Subsequently, similar to the previous procedure, we fix the compensation phase at ${\varphi }_{\mathrm{com}}=3.04-\pi$ and fit the experimental results with ${\varphi }_{\mathrm{cr}}=3.04$ and ${\varphi }_{\mathrm{cr}}=3.04-\pi$, respectively, based on Eq.(5), as shown in Figure 2(f). Ultimately, we determine that the phase of the crosstalk signal is 3.04, and the crosstalk coefficient is 0.15. Finally, we note that when the crosstalk signal is exactly in resonance with the qubit frequency, the above procedure is applicable, but there is a simple method worth noting. The crosstalk coefficient r can be directly obtained by measuring Rabi oscillations as shown in Figure 2(g) and the crosstalk phase is fitted with Eq. (5) in Figure 2(h), applying the $\pi$ Ramsey-like sequence as before.

In short, we use the Ramsey-like sequence $X_{\pi }$ or $X_{\pi /2}$ to scan the phase and the compensation amplitude separately to obtain the correct crosstalk parameters. In the case of resonance, the crosstalk coefficient is first obtained through Rabi oscillation and then the phase is fit with the sequence $X_{\pi }$. This method is simple, efficient, and universal, so it is beneficial for fully automated calibration of microwave crosstalk in a large-scale quantum processor.

2.3. Error Amplifying Sequence

However, when the detuning is sufficiently large and not close to the anharmonicity, we observe that the experiment of sweeping the crosstalk coefficient exhibits a flat-top phenomenon near full compensation, which can also be seen in previous works. This indicates the insensitivity of the Ramsey-like experiment in the large detuning case. Therefore, we supplement the srror amplifing sequence (EAS) to fine-tune the crosstalk parameters. The pulse is depicted in Figure 3(a), where $X_{\pi }{X}_{-\pi }$ is particularly sensitive to phase error, and $X_{\pi }{X}_{\pi }$ can amplify amplitude error and leakage error [39]. Therefore, different pulse combinations can amplify the dominant error under different detuning conditions.

In the experiment, the frequency of the target qubit is 4799.0 MHz, and that of the bias qubit is 4699.0 MHz, with the anharmonicity of the target qubit being -232 MHz. Because the phase error is the dominant error, repeated pulse sequences $X_{\pi }{X}_{-\pi }$ are applied to the target qubit, and the corresponding crosstalk and compensation signals are applied in a similar fashion. Following this, the excited state population $P_1$ of the EAS experimental result and ground state population $P_0$ in the Ramsey-like experiment are compared in Figure 3(b), indicating that the EAS is more sensitive to minor errors. In our experiments, due to the existence of distortion and other nonideal factors on the control lines, the parameters obtained from the EAS experiment perform better in multiqubit experiments.

3. Results

Finally, to verify the effectiveness of our method, we perform individual single-qubit-gate randomized benchmarking (RB) and simultaneous RB [40,41] with and without microwave crosstalk compensation for five qubits in a 1D chain. The frequencies and anharmonicities of the five qubits are shown in Table 1. The resulting fidelities are shown in Figure 4(a), and the corresponding average standard deviation of multiple sequences repeated 30 times per sequence is shown in Figure 4(b). Microwave crosstalk causes the fidelities of some qubits in the simultaneous RB to decrease, and the average standard deviation also increases significantly. Following the above experimental process, after crosstalk compensation, the fidelities and average standard deviations of the simultaneous RBs are basically close to the levels of the individual RB, confirming the effectiveness of our method for microwave crosstalk calibration.

4. Conclusion

In summary, we introduced a rapid and universally applicable calibration method for microwave crosstalk, leveraging Ramsey-like sequences that significantly enhance calibration processes in superconducting circuits. This method circumvents the complexities of traditional calibration by implementing a simplified two-level off-resonant driving model, which standardizes calibration across various crosstalk scenarios and simplifies the extraction of target parameters through one-dimensional sweeps rather than complex two-dimensional scans. Additionally, our incorporation of error-amplifying sequences notably enhances measurement accuracy, particularly in the case of significant detuning. The validity of our method has been verified through simultaneous RB, where the fidelities are recovered to the level of individual RB. Our methodology represents a significant step forward in automating the calibration process for large-scale quantum processors. This advancement is pivotal for harnessing the capabilities of current noisy-intermediate-scale quantum processors, paving the way for more reliable and efficient quantum computing operations.