Full control of superconducting qubits with combined on-chip microwave and flux lines

Superconducting quantum processors are one of the leading platforms for near-term applications and large-scale quantum computing due to their flexibility in design, high-fidelity single- and two-qubit gates,^1–4 and fast readout operations.^5–7 An architecture with a square grid of superconducting qubits for the implementation of the surface code requires each qubit to be coupled to four nearest neighbors as well as individual control lines and resonators.^8–10 Routing the control lines on the perimeter of a monolithic device poses some challenges to scaling, and several techniques have been developed to address this problem, including through-silicon vias and 3D coaxial architectures.^11–13 3D wiring is not the only challenge for the development of a scalable architecture. As superconducting devices proceed past the 10-qubit era,^8,9,14,15 it becomes necessary to multiplex the waveguides on the chip. This is already a common practice for readout lines,^16,17 but little work has been reported on the combination of flux (Z) and microwave control (XY) lines because of the different nature of their coupling to the qubit. The combination of these lines into a single “XYZ line” approximately halves the number of on-chip input and output ports for a quantum computer based on tunable qubits, thereby reducing the complexity of the design and the circuitry in general.^8–10 

A technical challenge in combining flux and control lines on a planar geometry comes from the different natures of their coupling to the qubit. The Z line is designed to provide an inductive coupling to the SQUID loop that is high enough to bias the qubit at a specific frequency with a relatively small current such that the overall thermal dissipation does not lead to excessive heating. This can be done by shorting the Z line to ground approximately $10 μm$ away from the SQUID loop. The XY line instead is capacitively coupled to the qubit pads. The capacitive coupling to the $50 Ω$ line must not be too small, otherwise strong microwave signals would be needed to operate single-qubit gates. At the same time, it must not be too high,¹⁸ in order not to limit the qubit relaxation time. Therefore, it is often necessary to position the end of the XY line further away from the qubit pads with respect to the end of the Z line. Combining flux and control lines on the same plane of the qubits can be problematic because of these different interplays.

In this work, we propose and experimentally demonstrate one possible way to combine XY and Z lines into a single XYZ line. By moving the XYZ lines to the surface of the cap using a flip-chip approach,¹⁹ the medium separating the line from the qubit is vacuum whose relative permeability and permittivity are both one. This brings the capacitive and inductive coupling between the XYZ line and the qubit on an equal footing. We show empirical evidence that our approach meets the stringent requirements of qubit applications: keeping a high qubit coherence, supporting microwave drives, and delivering DC and RF flux pulses to the qubits.

The device used for this investigation includes four tunable qubits, each capacitively coupled to a readout resonator. The cap contains two separate readout lines (each capacitively coupled to two resonators) and four XYZ lines, one per qubit. Most of the cap surface is covered with a meshed ground plane. The region of the cap that surmounts the qubits and resonators is characterized by a $24 μm$ deep cavity.²⁰ The device is bonded to a $6×6 mm2$ cap with a flip-chip bonder that provides an alignment precision of a few micrometers. Since the height of the flattened indium bumps is about $3 μm$⁠, the distance between the qubit and the ground plane above is $27 μm$ (see Fig. 1 for an optical image of the qubit and the associated XYZ line on top). The electrical connection between the two chips is tested at room temperature with dedicated test structures. The cap is wirebonded to a printed circuit board and mounted to the coldest plate of a dilution refrigerator with a $10 mK$ base temperature.

The attenuation and filtering of the XY and Z fridge lines connecting the room temperature instrumentation to the device are different. This is because the XY lines must support microwave signals in the 3–7 GHz band, which is the typical qubit frequency band. In addition, these lines must have a strong attenuation to reduce the thermal noise reaching the device.²¹ The Z lines instead provide low frequency pulses (⁠$DC−1.5 GHz$⁠) in order to bias the qubits at specific sweet spots and operate parametric entangling gates.²² The Z lines require a smaller attenuation than the XY lines.

Due to the different filtering requirements, the XY and Z fridge lines are combined at the lowest temperature plate of the dilution refrigerator. To this end, we have developed an in-house cryogenic diplexer as shown in Fig. 2(a). This device allows us not only to combine low and high frequency signals but also to filter the frequency components outside a specific frequency band by using a network of inductors and capacitors. To mitigate the injection of quasi particles generated by high-frequency photons, the diplexer also includes an eccosorb filter on the output line.^23,24 Figure 2(b) shows the measured transmission coefficient of a typical diplexer at room temperature and at $4 K$⁠. The diplexer was designed to have a 3–7 GHz bandpass filter for the XY line (port 1) and a 1.5 GHz low-pass filter for the Z line (port 2). The transmission coefficient from port 1 to port 2 (not shown in the figure) is lower than $−20 dB$ up to $15 GHz$⁠. In our experiment, one diplexer per qubit is thermally anchored to the 10 mK stage of the dilution refrigerator. We have performed several cool downs, and we have not noticed any degradation of their functionalities.

The geometry of the XYZ line has been engineered to obtain an upper limit on T₁ greater than $200 μs$⁠, a mutual inductance with the SQUID loop of approximately $500 fH$⁠, and a capacitive coupling high enough to enable $20 ns$ π-pulses with our fridge setup (see the supplementary material for a schematic of the fridge). To optimize these design parameters, we have performed full wave microwave simulations. The mutual inductance was tuned by varying the distance between the XYZ line and the SQUID as well as the width and length of the inductors shorting the line to ground. The mutual inductance can also be adjusted by changing the area of the SQUID loop. However, its perimeter cannot be too long in order not to limit the qubit coherence.²⁵ With regard to the capacitive coupling, the area of the XYZ line that surmounts the qubit pads affects the coupling strength. As a result, the capacitive coupling can vary depending on the qubit geometry. The geometry illustrated in Fig. 1(b) is the result of an optimization process that takes into account the qubit relaxation time and the capacitive and inductive couplings.

Characterization of the device at base temperature shows that we can tune the qubit frequencies between $3.0 and 3.8 GHz$⁠. The median of the relaxation time for all of the qubits over a day is $T̃1=53 μs$⁠, and transverse relaxation times at the maximum qubit frequency are $T̃2*=10 μs$ and $T̃2E=49 μs$⁠. The relaxation times fluctuate over time (see the supplementary material). This phenomenon has been reported elsewhere.^26,27

We first verify the functionality of the XYZ lines by performing some Rabi experiments with 20 ns microwave pulses. To test the performance of the XYZ lines, we intentionally used longer pulses to separate the contribution from the line itself and leakage to the second level. Using 100 ns DRAG Gaussian pulses, we measured a single-qubit RB fidelity of $99.77±0.02%$ for qubit 1 [see the inset of Fig. 3(a); note that the theoretical limit imposed by the relaxation time of this qubit is 99.84% (Ref. 28)]. We recently cooled down a similar device with higher relaxation times and measured a RB fidelity of $99.96±0.02%$ with 20 ns microwave pulses. This demonstrates that fast high-fidelity single-qubit gates can be implemented with XYZ lines.

We then assess that the XYZ lines can be used to deliver flux signals to the qubits. We measure the qubit frequency $fq$ as a function of the applied DC as shown in Fig. 3(a). The qubit frequency is measured with spectroscopic measurements and Ramsey experiments. The data points are fitted with an analytical transmon model. At each flux bias, we measure the relaxation time T₁ [see Fig. 3(b)]. The value of T₁ does not show a significant flux dependence, and its average value is $75 μs$⁠. Figure 3(b) includes the measurement of $T2*$ as a function of flux close to the DC sweet spot. As expected, the transverse relaxation time increases substantially at the sweet spot where the sensitivity of the qubit frequency to flux noise is the lowest. Close to the DC sweet spots, $T2*$ is above $10 μs$ allowing high-fidelity single-qubit gates. Other devices fabricated on the same wafer show similar performance although they were measured without XYZ lines. We can, thus, conclude that XYZ lines can be used to tune the qubit frequency without compromising their relaxation time.

Next, we validate the ability of the XYZ lines to support RF flux pulses, crucial for the implementation of parametric entangling gates.^22,29 In our experiment, the qubit starts at its maximum frequency $fmax$⁠. A RF flux pulse $Φ(t)=Φdc+Φac cos (ωdt)$ is delivered to the qubit where $Φdc$ is the DC bias, $Φac$ is the flux pulse amplitude, and $ωd$ is the pulse frequency. The flux pulse induces periodic oscillations of the qubit frequency. The time average qubit frequency is measured with a Ramsey type experiment with the parametric modulation applied between two $π/2$ pulses. Figure 4 shows the measured effective qubit frequency as a function of the flux pulse amplitude. The data points are fitted to³⁰ 

where T is the oscillation period, $J0(x)$ is the Bessel function of the first kind, and the constants s_n only depend on the Josephson and charging energies $EJ1, EJ2$⁠, and $EC$⁠. (Their analytical expression is presented in the supplementary material.) In our experiment, $Φdc=0$ since the qubit is initially parked at $fmax$⁠. The flux pulse amplitude can be expressed in terms of the amplitude $Ap$ generated by the room-temperature instrumentation as $Φac=βAp$⁠, where β is a factor that can be extracted from the fit. As shown in the figure, we were able to reach the AC sweet spot, $fmin$⁠, where we operate parametric entangling gates.³¹ Devices with XYZ lines are now used in our lab to run parametric entangling gates routinely.

The combination of the XY and Z fridge lines into a single line by means of a cryogenic diplexer may lead to an undesired effect. When a π pulse is sent through the XY fridge line to excite the qubit, it produces a current that flows through the termination of the XYZ line. This current can inadvertently modulate the qubit frequency. In our setup, a 100 ns π pulse is implemented with a room-temperature amplitude of $Vp=0.3 V$⁠. The signal reaching the device creates a magnetic field through the SQUID of $Φac=1.6×10−4Φ0$⁠. (Here, we assumed that the attenuation of the line is $85 dB$ at the qubit frequency and the mutual inductance between the XYZ line and the SQUID is $M=500 fH$⁠.) This flux does not modulate the qubit frequency by an appreciable amount. Indeed, by approximating Eq. (1) close to $fmax$ up to second order in $Φac$⁠, we obtain

and using the parameters for qubit 0 (⁠$EJ1/h=2140 MHz, EJ2/h=9040 MHz$⁠, $EC/h=182 MHz$⁠, $EJΣ=EJ1+EJ2$⁠), we obtain a frequency shift of $δf=−79 Hz$⁠. This shift is below the qubit linewidth and cannot be detected with Ramsey experiments.

In conclusion, we demonstrated full control of superconducting qubits by combining XY and Z lines into a single XYZ line. We showed that XYZ lines patterned on the surface of the cap can be used to implement fast single qubit-gates and tune the qubit frequency with both DC and fast flux biases. We tested the performance of single-qubit gates with randomized benchmarking achieving a fidelity as high as $99.93±0.04%$ (see Table I in the supplementary information). Realization of combined XY and Z lines can reduce the number of on-chip input ports, an important requirement when scaling superconducting quantum processors. The natural next step is to combine the XY cables and the Z cables into a single cable to further reduce the complexity of the fridge built out for quantum processors with a large number of qubits. This can be accomplished by engineering frequency-dependent attenuators. More sophisticated techniques for the multiplexing of the cables would require optical links.³² 

See the supplementary material for more information about the fridge setup and the device parameters.

This material was based upon work supported by Rigetti Computing and the Defense Advanced Research Projects Agency (DARPA) under Agreement No. HR00112090058.