Baseband control of superconducting qubits with shared microwave drives

For a baseband controlled two-level system (TLS) subjected to an always-on global drive, the system Hamiltonian is (hereinafter, we set $\hbar=1$)

where $\omega_{q}$ is the bare qubit frequency and can change according to the Z control pulse, $\omega_{d}$ and $\Omega_{d}$ are the frequency and the amplitude of the drive, respectively. Moving into the rotating frame with respect to the global drive and after applying the rotating wave approximation (RWA), the Hamiltonian reads

where $\Delta_{d}=\omega_{q}-\omega_{d}$ denotes the drive detuning. Note that unless otherwise stated, the RWA is used throughout this work.

At the idle point, the drive detuning is far large than the drive strength, thus the Bloch vector is slightly tilted towards the X-axis, as shown in Fig. 1(b). Generally, the dressed eigenstates defined by this tilted Bloch vector are chosen to be the logical computational states. The tilted angle and the dressed states can be quantitatively obtained by diagonalization of the Hamiltonian in Eq. (2), giving rise to

where $\theta=\arctan(\Omega_{d}/\Delta_{d})$ is the tilted angle and $\Delta=\sqrt{\Delta_{d}^{2}+\Omega_{d}^{2}}$. From the above discussions, when $|\Delta_{d}|\gg\Omega_{d}$, one can neglect the angle, as well as the difference between the bare states and the dressed states.

As shown in Eq. (3), by biasing the qubit at the idle point, the Z rotations can be easily realized by choosing suitable delay times $\tau$ between Z pulses, i.e.,

Note here that compared with the traditional microwave control, Virtual-Z (VZ) gate scheme McKay2017 is not suitable for the present baseband control. However, similar to the VZ gate, besides time delay, here, no actual control pulses are needed for implementing Z rotations.

Generally, as shown in Fig. 1(c), by tuning the qubit on-resonance with the global drive, single-qubit X rotations can be achieved. However, we note that since the initial Bloch vector is slightly tilted, as shown in Fig. 1(b), a small drive detuning $\delta_{d}=|\Omega_{d}^{2}/\Delta_{d}|$ is needed for enabling ideal X rotations with respect to the initial Bloch vector defined by Eq. (3). Thus, according to Z control pulses, X rotations can be realized by tuning the qubit from the idle point to the working point with a small overshoot Barends2019 . This fact is further illustrated by the results shown in Fig. 2. By initializing the qubit in state $|0\rangle$ and using square pulses (results with cosine-decorated square pulses can be found in Appendix A), Figure 2(a) shows populations $P_{1}$, i.e., the population in state $|1\rangle$ at the end of the applied pulse, versus the drive detuning $\delta_{d}$ and the pulse length. Here, the drive amplitude is $10\,\rm MHz$ and the detuning at the idle point is $-100\,\rm MHz$. Similarly, given a fixed pulse length of $50\,\rm ns$, Figure 2(b) shows $P_{1}$ versus $\delta_{d}$ and $\Omega_{d}$. The optimal parameters for X rotations are indicated by the red stars. Indeed, we find that a small frequency overshoot is needed for X rotations.

In the present work, note that choosing $\sqrt{X}$ gates as the native gates could simplify the tune-up procedure of single-qubit gate operations. This is because:

(i) arbitrary single-qubit rotations can be generated by two $\sqrt{X}$ gates and three Z gates McKay2017 , i.e., $Z_{\phi 1}-\sqrt{X}-Z_{\phi 2}-\sqrt{X}-Z_{\phi 3}$, with $Z_{\phi}\equiv\exp[-i\phi Z/2]$;

(ii) compared with the native X gate, the implementation of $\sqrt{X}$ gates does not pose stringent requirements on the on-resonance condition, i.e., even the qubit is slightly off-resonance with the drive, $\sqrt{X}$ gates can still be achieved. This can be captured by the analytical expression of Rabi oscillations for the two-level system initialized in state $|0\rangle$, i.e., Rabi’s formula

From Eq. (5), implementing $\sqrt{X}$ gates requires $P_{1}=1/2$ at the end of the applied pulse, giving rise to the relations among the pulse length, the drive detuning $\Delta_{d}$, and the drive amplitude $\Omega_{d}$, as illustrated by the dotted lines of Fig. 2. Accordingly, the results based on numerical simulations are also presented, as indicated by the dashed line of Fig. 2. Note here that the derivation of the analytical equation ignores the slight tilt at the idle point, and this explains the discrepancy between the analytical and numerical results. Both the analytical and numerical results show that compared to X rotations, the available parameter ranges of $\sqrt{X}$ rotations can provide great flexibility in its tune-up procedure.

Generally, due to the flexibility of $\sqrt{X}$ rotations, for tuning-up $\sqrt{X}$ gates, the above-mentioned overshoot can be ignored. In the next subsection, we will show that following this way, given a fixed drive detuning, $\sqrt{X}$ gates can be realized by only optimizing the ramp times of control pulses, as suggested by Fig. 2(a). Meanwhile, in large-scale quantum systems with multiplexed control, this flexibility can be the most encouraging advantage as to mitigate single-qubit gate error due to stray coupling between qubits and to compensate for the non-uniformity of qubit parameters. This will be discussed in detail in Sec. IV.

In the above discussion, the single-qubit baseband control is discussed for an ideal two-level system. Nevertheless, for practical superconducting qubits, such as the transmon qubit, the weak qubit anharmonicity makes single-qubit gate operations particularly prone to leakage outside the qubit subspace. For one such baseband control transmon qubit, which is driven by an always-on global drive, the system Hamiltonian is (hereafter, transmon qubits are modeled as anharmonic oscillators Koch2007 )

Here, $a_{q}\,(a_{q}^{\dagger})$ is the annihilation (creation) operator. Figure 3(a) shows the energy spectrum of the driven qubit versus the drive detuning with qubit anharmonicity $\alpha_{q}/2\pi=-250\,\rm MHz$, drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$, and drive amplitude $\Omega_{d}/2\pi=10\,\rm MHz$. One can find that due to the global drive, there exits an off-resonance coupling between states $|2\rangle$ and $|1\rangle$, which can cause leakage to state $|2\rangle$ when performing X rotations, i.e., biasing the qubit from its idle point to the working point. Note that there exist two leakage channels, caused by the $|1\rangle\leftrightarrow|2\rangle$ and $|0\rangle\leftrightarrow|2\rangle$ interactions, as shown in Fig. 3(a). Since the $|0\rangle\leftrightarrow|2\rangle$ interaction involves second-order processes, generally, the induced leakage error is far smaller than that of the $|1\rangle\leftrightarrow|2\rangle$ interaction. Here, we thus mainly focus on the leakage error caused by the $|1\rangle\leftrightarrow|2\rangle$ interaction. However, we note that in our numerical analysis, all the leakage channels, including the $|0\rangle\leftrightarrow|2\rangle$ interaction, are taken into consideration. While this leakage issue can be addressed by using the DRAG scheme in the traditional microwave control setup, this scheme cannot be directly utilized for the baseband flux control setup, as here only Z control is available.

Additionally, in principle, at the idle point, qubits could be far detuned above or below the frequency of the always-on drive. However, to avoid possible leakage error during qubit control, such as gate operations, qubit initialization, and readout, we prefer to bias the qubit away from the harmful avoid crossing caused by coupling between states $|1\rangle$ and $|2\rangle$, as shown in Fig. 3(a). In this way, during the baseband-controlled gate operations, the qubit system will not sweep through or operate nearby this harmful avoid crossing, generally allowing the suppression of the leakage to state $|2\rangle$. Considering this fact, hereafter, we consider biasing the qubit below the drive frequency at the idle point. Even in the setting, during Z-controlled single-qubit gate operations, leakage error can still occur due to the non-adiabatic error, as shown in Fig. 3(b). In the following, we will consider using a fast-adiabatic control scheme for suppressing the leakage further.

As shown in Fig. 3(b), the leakage error occurs when one non-adiabatically varies the driving detuning. Considering that coherence times of superconducting qubits are still limited, our target is to find a good flux control pulse, thus the non-adiabatic error is suppressed while maintaining a fast operation speed. Fortunately, this issue has already been addressed successfully by using a fast-adiabatic scheme introduced in Ref. Martinis2014b . Within the scheme, optimal control pulses can be obtained for minimizing non-adiabatic errors for any pulse longer than the chosen pulse length. However, we note that the original scheme only addresses the non-adiabatic error in the pulse ramps. Thus, here, to address the leakage issue in our setting, we consider using a square control pulse with optimal fast-adiabatic ramps, which is obtained following the fast-adiabatic scheme Martinis2014b (see Appendix B for details). The inset of Fig. 4(a) shows the optimal ramp pulse (solid blue line), which is used for generating our target control pulse with a flat middle part and fast-adiabatic ramps (solid orange line). Hereafter, we refer to this pulse as the fast-adiabatic flat-top pulse.

Here, we turn to evaluate the efficiency of the proposed fast-adiabatic flat-top pulse. We consider that the qubit idle frequency is $6.0\,\rm GHz$ and during the implementation of single-qubit X rotations, the drive detuning $\Delta_{d}$ varies from the idle point at $-100\,\rm MHz$ to the work point at $0\,\rm MHz$ and then coming back, according to the fast-adiabatic flat-top pulse. By initializing the qubit in state $|1\rangle$, Figure 4(a) shows the population leakage to state $|2\rangle$ as a function of ramp times with the hold time of 20 ns and the qubit anharmonicities $\alpha_{q}/2\pi=\{-200,\,-250,\,-300\}\rm MHz$. For easy comparison, the results for applying only the fast-adiabatic pulse are also presented. One can find that by using the fast-adiabatic flat-top pulse, the leakage can be suppressed below $10^{-6}$ for ramp times longer than 10 ns, and inserting a square pulse in the fast-adiabatic pulse does not change the efficiency of the original fast-adiabatic scheme. In Fig. 4(b), we also show the populations in $|0\rangle$ and $|1\rangle$ versus the ramp times. Additionally, Appendix B presents further results for different drive strengths.

From the results shown in Fig. 4, one can find that $\sqrt{X}$ rotations can be realized with the ramp time at about 10 ns, giving rise to the total pulse length of about 30 ns. Meanwhile, same as the case for two-level systems (in Sec.II.1), here, single-qubit Z rotations can be easily implemented by controlling the time delay between Z pulses. Therefore, we could reasonably expect that with the help of the fast-adiabatic scheme, fast-speed single-qubit operations could be achieved with low leakage errors (we will evaluate the single-qubit gate performance in detail in the following section).

Within the introduced baseband control setup, at the idle point, the global always-on drive acts as an off-resonance drive and can induce ac-Stark frequency shifts on the qubits. For the two-level system studied in Sec.II.1, the shift is given as $\delta\omega=\Delta-\Delta_{d}\approx\Omega_{d}^{2}/(2\Delta_{d})$, while, taking the higher energy levels of the transmon qubit into consideration, the shift is Schneider2018

From Eq. (7), the shift has a quadratic-dependent on the drive amplitude, making the qubit frequency more susceptible to possible amplitude noise. Therefore, fluctuations in the drive amplitude can cause qubit dephasing, which has been recently observed in superconducting qubits Wei2022 .

Here, to numerically study the amplitude-fluctuation-induced qubit dephasing, we consider an amplitude-dependent noise, i.e., amplitude fluctuations are proportional to the amplitudes. By assuming the drive subject to zero-mean Gaussian noise, i.e., $\textsl{N}(0,\sigma)$, we numerically simulate the time evolution of the off-diagonal matrix element $\rho_{01}(t)$ for the qubit initialized in state $(|0\rangle+|1\rangle)/\sqrt{2}$. After averaging $\rho_{01}(t)$ over 2000 trajectories (i.e., realizations of noise), the magnitudes of the off-diagonal matrix element display a clear exponential decay, as shown in Fig. 5(a). Here, the evolution time is $10\,\mu s$, the other used parameters are: $\Delta_{d}/2\pi=-100\,\rm MHz$, $\Omega_{d}/2\pi=10\,\rm MHz$, and $\alpha_{q}/2\pi=-250\,\rm MHz$, giving rise to $\delta\omega/2\pi\approx-0.36\,\rm MHz$. By fitting the decay curves to $[1-\exp(-t/T_{\phi})]/2$, Figure 5(b) shows the dephasing time $T_{\phi}$ versus the noise variance $\sigma$. Here, we also show the results for $\Omega_{d}/2\pi=20\,\rm MHz$. Additionally, in the inset, we further show the dephasing times versus the drive amplitudes.

From the results shown in Fig. 5(b), and given the typical noise variance of $1\%$, we can conclude that the amplitude-noise induced dephasing can be safely neglected by detuning the qubit far from the drive frequency. Meanwhile, we note that to ensure high-fidelity gate operations within sub-100 ns, the drive amplitude itself should be larger than $10\,\rm MHz$. Finally, we note that besides the qubit dephasing, when the always-on drive is shared by multiple qubits, there are two additional potential issues related to the drive and its fluctuation:

(i) qubit decoherence due to the excess quasiparticles (QPs) Catelani2011 . Previous works have demonstrated that QPs can be injected into a qubit by applying a high-power microwave pulse resonance with its readout resonator Vool2014 ; Wang2014 . However, at the present setting, the strength of the always-on drive is far smaller than that of the former case, we thus expect that the contribution of the always-on drive to the excess QPs is negligible.

(ii) when the always-on shared (global) drive is shared by multiple qubits, fluctuations in the drive can result in errors on multiple qubits. At first glance, this can cause correlated errors. However, as discussed in Ref. Fowler2014 , provided the fluctuation is small and quantum error correction is frequent, the fluctuation in the shared drive can result in a correlated probability of error on multiple qubits, but the errors themselves will not be correlated. Additionally, as suggested in Fig.5(b) and Eq.(7), by increasing the qubit-drive detuning, the fluctuation-induced dephasing (error) can be heavily suppressed at the system idle time. And during single-qubit gate operations, with the currently available control electronics, single-qubit gates with gate errors below 0.0001 have been demonstrated Somoroff2021 ; Li2023 . Thus, we argue that fluctuations in the shared drive itself can indeed cause errors, but probably a rather small one ($<0.0001$). In this case, the control-noise-induced error may still be handled by the error correction schemes Fowler2014 .

As mentioned in Sec. I and Sec. II.1, due to the presence of the always-on drive, in this work, the microwave dressed states are defined as the computational states. Here, since the available control over the always-on drive is limited, the previous method Zhao2022 ; Huang2021 , in which the dressed state is first mapped back to the corresponding bare state, and then the traditional dispersive readout is employed for inferring the qubit information Wallraff2005 , cannot be directly utilized. However, as discussed in Sec. II.1, when the drive detuning is far larger than the drive amplitude, i.e., $|\Delta_{d}|\gg\Omega_{d}$, the difference between dressed states and bare states can be neglected. Therefore, we expect that by keeping a large ratio of the drive detuning to the drive amplitude, the qubit information can be directly inferred using the traditional dispersive readout scheme.

To explore the possible impact of the always-on drive on the qubit dispersive readout, we numerically simulate the system dynamics during the dispersive readout. By applying a 250-ns square readout pulse with frequency $\omega$ and amplitude $\Omega$ to the readout resonator with decay rate $\kappa$, the full system dynamics are governed by the Hamiltonian

where $H_{q}$ denotes the qubit Hamiltonian given in Eq. (6), $\omega_{r}$ is the frequency of the readout resonator, $a_{r}\,(a_{r}^{\dagger})$ is the annihilation (creation) operator of the resonator, and $g$ denotes the strength of the qubit-resonator coupling. In this following, the qubit information is encoded into single quadrature, i.e., $I$-quadrature, by choosing the readout frequency to be $\omega=(\omega_{r0}+\omega_{r1})/2$ Wallraff2005 . Here, $\omega_{r0}$ and $\omega_{r1}$ denote the dressed resonator frequencies with the qubit in states $|0\rangle$ and $|1\rangle$, respectively. The other system parameters are: $\omega_{q}/2\pi=6.0\,\rm GHz$, $\alpha_{q}/2\pi=-250\,\rm MHz$, $\omega_{d}/2\pi=6.1\,\rm GHz$, $\omega_{r}/2\pi=5.0\,\rm GHz$, $g/2\pi=100\,\rm MHz$, $\kappa/2\pi=5\,\rm MHz$, and $\Omega/2\pi=7\,\rm MHz$.

According to Eq. (8), we simulate the system dynamics based on solving the stochastic master equation Johansson2012 . Then, following Ref. Walter2017 , we further caulate the integrated readout quadrature with an optimal weight function (see Appendix C for details). With 5000 repetitions of the simulation for each qubit basis state, i.e., $|0\rangle$ and $|1\rangle$, Figure 6(a) shows the two histograms of the integrated readout quadrature with the qubit prepared in states $|0\rangle$ and $|1\rangle$, respectively. Here, the drive magnitude is $20\,\rm MHz$. For easy comparison, we also present the result for the global drive is absent, as shown in Fig. 6(b).

Fitting the histograms to Gaussian functions gives the state-decision threshold at the intersection point of the two fitted distributions. Accordingly, the readout fidelity can be calculated as $F=1-[P(0|1)+P(1|0)]/2$, where $P(0|1)$ ($P(1|0)$) denotes the error probability that the qubit initialized in state $|1\rangle$ ($|0\rangle$) is identified as in state $|0\rangle$ ($|1\rangle$). Accordingly, Figure 6(c) shows the readout error $1-F$ versus the drive strength. One can find that when increasing the drive amplitude from $0$ to $20\,\rm MHz$, while the error shows an upward trend, the increased error is below $1\%$. Moreover, the upward trend also suggests that by further increasing the ratio $|\Delta_{d}|/\Omega_{d}$, the increased error should be heavily suppressed.

Following the scheme introduced in Sec. II.2, here, Z-controlled single-qubit gates are realized by using the fast-adiabatic flat-top pulse. Note here that in the present two-qubit system, single-qubit gate operations for one qubit are tuned up and characterized with the other qubit in its ground state $|0\rangle$.

Figure 8 shows the leakage versus the ramp time of the pulse with the qubit initialized in its excited state $|1\rangle$. One can find that while for $Q_{0}$, the result is in line with our theory discussed in Sec. II.2, i.e., by increasing the ramp time, the leakage can be further suppressed, the result of $Q_{1}$ seems unreasonable at first glance. However, during gate operations applied to $Q_{1}$, $Q_{1}$ is tuned from its idle point at $5.9\,\rm GHz$ to the working point at about $6.1\,\rm GHz$, according to the fast-adiabatic pulse, while $Q_{0}$ is fixed at its idle point at $6.0\,\rm GHz$. Therefore, during the pulse ramp, $Q_{1}$ will sweep through a tiny avoided crossing formed by the residual resonance XY coupling between $Q_{0}$ and $Q_{1}$ at $6.0\,\rm GHz$. On contrast, during single-qubit gate operations, $Q_{0}$ will not sweep through $Q_{1}$. As shown in Fig. 7(a), the strength of the residual XY coupling is about $0.1\,\rm MHz$. Thus, sweeping through this avoided crossing slowly will generally cause more leakage into the nearby qubit $Q_{1}$, as shown in Fig. 8(b), where the orange solid line denotes the population of $Q_{0}$ in state $|1\rangle$. One can find that for $t_{r}\geq 10\,\rm ns$, the leakage into $Q_{0}$ gives the leading contributions to the total leakage error. These results suggest that there exists a trade-off between gate error resulting from the qubit itself and error from spectator qubits, i.e., suppressing leakage into $|2\rangle$ favors longer gate times, while mitigating the leakage into the $Q_{0}$ favors short gate times. This observation is in agreement with that in previous work Zhao2022b .

To address the above issue, one can change the idling coupler frequency, the XY coupling can thus be further suppressed at the resonance point, i.e., $6.0\,\rm GHz$. By biasing the coupler at $11.25\,\rm GHz$, the residual XY coupling is suppressed below $0.01\,\rm MHz$. Accordingly, the leakage into $Q_{0}$ is indeed suppressed heavily, as shown in Fig. 8(b), where the dashed lines show the results with the coupler biased at $11.25\,\rm GHz$.

Here, we turn to evaluate the gate performance of the baseband controlled single-qubit gates, and use the metric of the state-average gate fidelity Pedersen2007 in the following discussion (details on the fidelity calculation can also be found in Ref. Zhao2022b ). As mentioned in Sec. II.1, in the present work, we focus on the implementation of $\sqrt{X}$ gates. From the inset of Figs. 8(a) and  8(b), one can find that for both qubits, $\sqrt{X}$ gates can be realized with a ramp time of about $10\,\rm ns$, giving rise to the total gate time of about $30\,\rm ns$. Moreover, even by biasing the coupler at $11.35\,\rm GHz$, Figure 8 shows that when the ramp time is about $10\,\rm ns$, the leakage error can still be suppressed below $5\times 10^{-5}$ for both qubits. This is to be expected, since sweeping through the tiny avoided crossing with fast speed could suppress leakage. By optimizing the ramp times, we find that for both qubits, up to single-qubit Z rotations, $\sqrt{X}$ gates can be achieved with gate fidelity exceeding $99.999\%$ (for $Q_{0}$, the gate fidelity is $99.9998\%$ and the optimal gate time is $30.2\,\rm ns$, while for $Q_{1}$, are the $99.9996\%$ and $29.4\,\rm ns$). As mentioned before, Z gates can be easily realized by choosing suitable time delays between flux pulses. In this way, universal single-qubit gates can be achieved by combining Z gates and $\sqrt{X}$ gates.

Having discussed the single-qubit control, we now turn to the two-qubit case. Here, we consider the implementation of CZ gates in the two-qubit system with an always-on drive. During the gate operations, $Q_{0}$ is fixed at its idle point, i.e., $6.0\,\rm GHz$, $Q_{1}$ is tuned from its idle point ($5.9\,\rm GHz$) to the working point, where a complete oscillation between states $|101\rangle$ and $|200\rangle$ can occur. Meanwhile, the coupler is tuned from its idle point at $11.35\,\rm GHz$ to a working point at about $7\,\rm GHz$, giving rise to the CZ coupling strength of $20\,\rm MHz$ (see Appendix D).

Figure 9(a) shows the typical control pulse, i.e., the pulse with a flat middle part and cosine-shaped ramps (see Appendix A for details), with a pulse length of $30\rm ns$ for the CZ implementation. By numerically optimizing the working points of $Q_{c}$ and $Q_{1}$, the gate fidelity of the implemented CZ gate (up to single-qubit Z phases) is $99.82\%$. After inspecting the qubit dynamics during the gates, one can find that the leading error source is the population swap between two qubits, as shown in Fig. 9(b). Following the fast-adiabatic scheme discussed in Sec. II.2 and the previous work Martinis2014b ; Sung2021 , here, the fast-adiabatic flap-top pulses, as shown in Fig. 9(c), with a hold time of $10\,\rm ns$, is used to suppress the population swap. Accordingly, the population swap is indeed largely suppressed, as shown in Fig. 9(d), improving the CZ gate fidelity to $99.94\%$ with a gate time of $30.7\,\rm ns$. Additionally, we note that generally, by increasing the gate length, the residual gate error can be further suppressed (see also in Appendix E).

The above results show that although there exists an always-on global drive, high-fidelity two-qubit gates can still be achieved in a short time. This success is mainly based on the fact that during the gate operations, the global drive is far detuned from both qubits and the coupler.

While our theoretical study shows that baseband controlled gate operations can be realized with high fidelity and fast speed, we note that besides the qubit decoherence, there exist several practical experimental issues that will limit the available gate performance:

(i) Flux pulse distortion. Flux pulse distortion has been demonstrated as a critical issue faced by baseband flux-controlled gate operations Jerger2019 ; Rol2020 ; Foxen2019 . Moreover, the above-demonstrated high-fidelity gate operations are achieved by using pulse shaping technologies, thus, the impact of flux pulse distortion can become more prominent in our setting.

(ii) Stray coupling beyond nearest neighbors. Generally, in our setting, the always-on drive is shared by multiple qubits. When performing single-qubit gates in parallel, multi-qubit will be tuned on-resonance with the same drive. This means that any stray coupling between these qubits will cause population swaps among these qubits, as discussed in Sec. III.1, leading to additional gate errors compared to isolated gates. While near-neighbor couplings between qubits can be controlled well in the tunable coupling architecture, parasitic coupling beyond nearest neighbors can still exist due to, such as stray capacitive coupling, in multi-qubit systems Barends2014 ; Zajac2021 ; Yanay2022 ; Zhao2022b . This will degrade the efficiency of the baseband control strategy in large-scale quantum processors.

(iii) Defect modes, such as TLSs Muller2019 . Same as in (ii), when performing single-qubit gates, the working frequencies of multi-qubit are almost limited to a fixed one, i.e., the frequency of the shared drive. This will limit the ability to mitigate the impacts from defect modes by tuning the qubit away from the defects Klimov2018 .

(iv) Keeping track of the single-qubit phase accumulation. In our setting, the qubit frequency at its idle point is detuned from the always-on drive. Thus, the single-qubit phase will accumulate at the speed of the drive detuning $\Delta$ during the idle time. While the accumulated phase can be employed to realize single-qubit Z gates, on the other hand, when performing gate sequences or quantum circuits, the accumulated phase should be tracked carefully over the whole time domain. Compared with the traditional microwave control, this could complicate the implementation of quantum circuits.

In addition, we note that owing to the great flexibility of Z-controlled $\sqrt{X}$ gates, as discussed in Sec.II.1, the issues, related to (ii) and (iii), may be addressed. From the results shown in Fig. 2, we can find that given a fixed drive amplitude or a fixed pulse length, $\sqrt{X}$ gates can be achieved with a small drive detuning, for which its magnitude can even be compared with that of the always-on drive. Thus, when implementing isolated or paralleled single-qubit gates, the working frequencies of qubits can be biased intentionally at different frequency points, thus impacts of sub-MHz stray coupling can be mitigated. Similarly, the defect’s impact can be suppressed by biasing qubits away from the leading defect modes.

Currently, in small-scale superconducting quantum processors, each qubit has its dedicated control lines, such as XY lines and flux (Z) lines. Moreover, due to the non-uniformity of qubit parameters, such as qubit frequency and anharmonicity, the coupling efficiency between qubits and control lines, and the signal attenuation and the distortion in control lines, control pulses can differ from qubit to qubit. Thus, generally, microwave control pulses differ with each other in their amplitudes, frequencies, and phases, while for the baseband flux pulse, their amplitudes could be different. When scaling up to large-scale quantum computing, such strategy is not scalable.

Given the recent progress in the pursuit of scalable spin-based quantum computing with multiplexing technologies and crossbar technologies Hill2015 ; Vandersypen2017 ; Veldhorst2017 ; Li2018 , we may also consider how to utilize these technologies for solving the above-mentioned challenges toward large-scale superconducting quantum processors. One possible example is schematically illustrated in Fig. 10(a), where both the XY and Z lines are shared by multiple qubits in a square lattice of frequency-tunable qubits. Similarly, in qubit architectures with tunable couplers, the Z-line shared scheme could be employed for controlling tunable couplers, thus enabling the implementation of baseband-controlled two-qubit gates in parallel.

Note that in principle, the shared control scheme should be applicable for driving all qubits with a single continuous microwave source. As a practical application, we expect that the shared scheme could be feasible for tens of superconducting qubits. With this strategy, the number of needed microwave drive lines is, at least, an order of magnitude less than that in the traditional setting. Additionally, to integrate with the widely used flip-chip technology, the shared drive may be applied to multiple qubits through a shared XY line. According to the discussion given in Ref. Krinner2019 , we estimate that when the strength of the always-on microwave drive is about 10 MHz ($\sqrt{X}$ gates with a gate time of 30 ns), the required power is about -14 dBm at the room temperature, and through a series of attenuators and filters (giving rise to the total attenuation of 60 dB), the actual power delivered to the qubit chip is about -74 dBm. As in the traditional setting, the average required power per qubit drive line is about -78 dBm (to the qubit chip) Krinner2019 , we expect that the heating issue of the present shared control scheme can be addressed.

Compared with the spin qubit, it seems that the superconducting qubit can provide more flexible control over its physical size and qubit parameters Martinis2020 ; Barends2014 ; Zhao2020N ; Mamin2022 ; Zhao2022c ; Chow2015 , yet, it can also show prominent non-uniformity. Unfortunately, the success of the multiplexing technologies and crossbar technologies highly hinges on the uniformity of qubit parameters. This can be more prominent for superconducting quantum processors based on individual microwave control. In the context of the implementation of multiplex control of superconducting qubits, baseband flux control may alleviate this issue of non-uniformity. Within our baseband control setup, and applying the multiplexing technologies shown in Fig. 10(a), there are three main leading challenges from the qubit non-uniformity:

(i) The non-uniform amplitude of the shared microwave drive or (ii) flux pulse felt by qubits (caused by, such as the different coupling efficiency between qubits and the global XY/Z line and the signal attenuation in control lines);

(iii) Independently calibrated parameters of flux pulse, including pulse length and pulse shape, for implementing accurate control on individual qubits.

However, as discussed in Sec.II.1, for two-level systems, given a fixed pulse length and pulse shape (i.e., square shape, see Appendix A for results with smooth pulses), owing to the flexibility of Z-controlled single-qubit gates (based on $\sqrt{X}$ gates), the available parameter ranges (i.e., the drive amplitude and the drive detuning) can be explored for compensating the non-uniform of drive amplitude. This exciting feature is illustrated in Fig. 2(b). Furthermore, similar results can also be obtained for superconducting qubits, such as transmon qubits. Figures 11(a) and 11(b) present the results for transmon qubits with $50$-$\rm ns$ square pulses and smooth pulses (i.e., cosine-decorated square pulse with a hold time of $50\,\rm ns$ and a ramp time of $10\,\rm ns$), respectively. Here, the qubit anharmonicity is $-250\,\rm MHz$ and the other used parameters are same as in Fig. 2. In addition, we note that to achieve uniform control pulses, the Z gate scheme based on time-delay and the proposed fast-adiabatic scheme cannot be employed. Here, we can instead use cosine-decorated square pulses for implementing Z gates by tuning the qubit from the idle point, and for suppressing leakage. From Figs. 11(a) and 11(b), one can find that by adding cosine-shaped ramps, the leakage error can be suppressed below $10^{-4}$, while for the square pulse the leakage can approach $10^{-3}$. To further suppress the leakage error, one can increase the ramp time or decrease the drive amplitude. However, this will increase gate length and thus cause more decoherence errors.

Considering the above results, the above three challenges can be effectively overcome by only addressing the non-uniformity issue of flux pulse amplitudes. This non-uniformity issue could be removed by developing on-chip programmable memory, such as the one demonstrated by using Single Flux Quantum (SFQ) logic Johnson2010 ; McDermott2010 , which could be used to compensate for the remaining non-uniformity. In this way, combined with the word lines for digital addressing, as shown in Fig. 10(b), it is possible using only a few global XY and Z lines to achieve parallel control of large numbers of qubits Hill2015 ; Vandersypen2017 ; Veldhorst2017 ; Li2018 . Nevertheless, given the practical experimental limitations, such as the limited cooling power, the realization of the local programmable memory, which is compatible with superconducting qubits, is still rarely explored Johnson2010 , and undoubtedly, will be one of the most crucial challenges for implementing multiplexing control technologies. Last, but not least, we must stress that before solving the issue of pulse distortion, the efficiency of the above-discussed scheme could be limited for implementing gate-based quantum computing.