Microwave calibration of qubit drive line components at millikelvin temperatures

Microwave control in a cryogenic environment is essential to quantum information processing with solid-state qubits.^1,2 The pursuit of scaling quantum systems to the level of 10⁶ physical qubits has motivated studies of a wide range of new microwave technologies for qubit control, qubit readout, and auxiliary systems. Consequently, rapid progress in the field of engineered quantum systems has exposed a general lack of microwave measurement and detection methods that are native to the millikelvin environment that can address its specific measurement constraints or be traced directly to existing metrological rf standards. New measurement techniques must overcome challenges such as the requirement of low operational power dissipation within the cryogenic environment, the ability to correct for the offset from room temperature test equipment by meter-long scale interconnects, temperature-dependent impedance changes, and the practical challenge of the lack of commercially available cryogenically compatible rf components.

In a typical qubit operation scheme, strong microwave control and readout pulses are applied at room temperature and subsequently undergo a reduction in amplitude as they are delivered to the quantum device through lossy coaxial cables and a series of thermalized cryogenic attenuators. The attenuators at each temperature stage of the dilution refrigerator measurement system are selected in order to reduce both the pulse amplitude and accompanying blackbody noise that propagates through the cable.^3,4 Therefore, the cryogenic attenuators and coaxial cables strongly influence the quality of the qubit control and readout pulses. Signal distortions and corresponding qubit errors due to the influence of the measurement lines have been studied using a qubit as an in situ low-temperature sensor.^5–8 However, direct measurement of microwave scattering parameters (S-parameters) of individual microwave components placed at the base temperature of a dilution refrigerator measurement system remains a technical challenge due to the lack of stable and repeatable calibration standards and methodologies for accurate error extraction. The problem has been addressed in specific cases using thru-reflect-line (TRL)^9,10 and one-port short-open-load (SOL)¹¹ methods. Accurate determination of S-parameters in a cryogenic environment has wide applicability to the development of active and passive cryo-electronic devices with immediate impact on filtering,¹² interconnects,¹³ multiplexers,¹⁴ non-reciprocal devices,¹⁵ single-flux-quantum-based technologies,^16,17 cryo-CMOS technologies,¹⁸ quantum hardware packaging,^19–21 microwave impedance and waveform metrology,²² and parametric amplifier development,^23–27 among others. Beyond quantum computing, scientific applications requiring accurate S-parameter characterization at low temperatures include astrophysical observations²⁸ and high-frequency electronic transport experiments.^29–32 

In this Letter, we report direct measurements of the 1-port S-parameter of qubit drive line components over a wide frequency band at 30 mK using the 1-port SOL method. Each of the SOL calibration standards that were used at millikelvin temperatures were pre-characterized at room temperature using an electronic calibration kit (ECal).³³ The pre-characterized standards were subsequently used to define cryogenic data-based SOL calibration at the base temperature of a dilution refrigerator measurement system.^11,34 The motivation for our data-based calibration procedure is described in more detail in the supplementary material (Sec. I D). In particular, this work follows the methods of Ref. 11, where measurement results were corroborated by good agreement between calibrated data of cryogenic resonators to known circuit models. By gating the data in the time-domain and analyzing reflection amplitudes, we can recover the insertion loss of components with sufficient electrical length such as coaxial cables. Time-domain gating refers to the process of selecting a region of interest in the time-domain, filtering responses outside of the region of interest and displaying the result in the frequency domain.³⁵ We interpret the measurement results by calculating how return loss leads to problems such as qubit state preparation infidelity. This Letter is organized as follows. We first present measurements of cryogenic attenuators and coaxial cables to demonstrate the typical values within a dilution refrigerator measurement system. We then discuss the results in terms of state preparation infidelity due to scattering of qubit control pulses at discrete temperature stages within the qubit drive line.

We measured components that are representative of a qubit drive line configuration in recent superconducting qubit experiments.⁴ These include commercial cryogenic attenuators (specified for operation between 4 and 398 K) with 10- and 20-dB constant attenuation across the specified frequency band of operation (dc—18 GHz) and coaxial cables. The attenuators use an on-chip Ni–Cr thin film voltage divider resistor network²⁷ and have SMA male and female interfaces. Attenuation is a scalar quantity that is straightforward to obtain in a cryogenic measurement system by comparing the transmission of an attenuator-under-test with a through-reference cable using a coaxial switch. For example, it was reported in Ref. 27 that a 10-dB cryogenic attenuator deviates by less than < 0.02% from its room temperature value. In contrast, determining the intrinsic reflection of a component requires accurate assignment of the reference plane in the cryogenic environment. We also studied two different types of cryogenic semi-rigid coaxial cables. The first cable was comprised of NbTi metal for both the inner conductor and outer shield. The second cable had a silver-plated cupronickel center conductor with a cupronickel shield. Both coaxial cables have a nominal characteristic impedance of $50 Ω$ with the PTFE dielectric and inner conductor diameter of d = 0.203 mm, dielectric outer diameter of D = 0.66 mm, and soldered SMA connectors.

Figure 1 displays the schematic of the S-parameter measurement setup. The base temperature stage is equipped with two sets of coaxial switches that are used to toggle between calibration standards and each device under test (DUT). Each coaxial switch is connected to its own measurement line that is used to send and receive microwave power during in situ S-parameter characterization. Our measurement scheme enables measurements across a wide frequency band of 10 MHz–18 GHz by removing the added attenuation, directional couplers, circulators, and amplifiers that were present in previous schemes^10,11 with the exception of 3 dB of fixed attenuation on the mixing chamber stage. Signals were sourced and received directly to the test ports of the network analyzer (NA) using a source powers around 1 mW, and the internal electronics of the NA were used to process the data. The SOL calibration standards are installed into the system on the coaxial switches shown in Fig. 1. We can toggle the coaxial switches to measure each standard and establish an in situ reference plane in the dilution refrigerator. During the measurements, we terminate the attenuators with cryogenically compatible loads that are nominally identical to the load standards, and the coaxial cables are terminated with shorts as shown in Fig. 1. In the measurement results below, we use the convention that return loss is the negative of the reflection coefficient, i.e., $−|S11,dB|$⁠.³⁶ 

Measurement results for the frequency-dependent magnitude of the 1-port S-parameters are shown in Figs. 2(a)–2(d). Furthermore, transmission of the cables can be derived from the reflection data using a time-gating technique to analyze the amplitude of reflections in Figs. 2(e) and 2(f) (see the supplementary material, Sec. I B). Pre-characterization of the calibration standards is performed at the ends of the test cables at room temperature (300 K) utilizing ECal calibration. The initial measurements of each DUT are also performed at room temperature and plotted as yellow traces in Fig. 2. For Figs. 2(c)–2(f), data plotted as yellow traces were acquired using a full 2-port S-parameter measurement at the end of the room temperature test cables. To perform the in situ measurements in the dilution refrigerator, we then install the pre-characterized SOL standards into the coaxial switches as shown in Fig. 1 and upload the pre-characterization data onto the NA to define a data-based calibration kit.¹¹ We perform the in situ measurements at 300 K by toggling the coaxial switches to measure each calibration standard and the DUTs (plotted as blue traces in Fig. 2). Finally, after measurements are performed at room temperature, the entire system is cooled down, and the calibration procedure is repeated when the standards and DUTs are at 30 mK (gray traces in Fig. 2).

The data plotted in Fig. 2 have been post-processed by applying gates in the time-domain to remove any post-calibration drift of reflections that originate from higher cryostat stages and room temperature cables (see the supplementary material). The geometry and materials of the coaxial cables define the characteristic impedance at the center of the cable, but the solder connection between the cable and connector is the dominant source or reflection.³⁷ Since coaxial cables have two connectors, the S-parameter data can be processed in two distinct ways: (1) applying the bandpass gate at the connector closest to the reference plane allows one obtaining the return loss of Figs. 2(c) and 2(d), which emphasizes the reflections at the connector or (2) applying it to the connector with the shorting cap at the end of the cable, which isolates a signal proportional to the cable loss of Figs. 2(e) and 2(f). A consequence of the gating procedure is that low frequency information is lost below the cutoff set by the inverse of the gate span 1/t_gate, which is 200 MHz for the attenuators and ∼300 MHz for the cables. Following Ref. 38, we use the original data below cutoff³⁸ for attenuators but due to the multiple reflections within the cables, we only plot the gated data in their case.

Error analysis of the data presented in Fig. 2 is described in detail in the supplementary material. In summary, we have investigated and quantified error sources such as (i) the reflection uncertainty from a root sum squared (RSS) analysis of error sources inherent to the ECal obtained with a calculator from Ref. 39, (ii) switch variability: the difference in the transmission and reflection between microwave switch ports, (iii) switch repeatability: the average change in reflection magnitude for a switch port after four switching events, and (iv) differences in the reflection magnitude of the loads used to terminate the attenuators for measurements of Fig. 2. When calculating the total error as an RSS sum, we find that errors (i) and (ii) are sufficient to represent the error, as (iii) and (iv) would only contribute the maximum of 2% and 0.04% to the total error and are omitted. The resulting error bars, representing 64% confidence intervals, are equal in linear units but look asymmetric when converted to logarithmic units with $RL(dB)=−20 log10(|S11,lin| ∓ σRSS)$⁠. Here, $RL(dB)$ is the return loss, $S11,lin$ is the gated one-port S-parameter data in linear units, and $σRSS$ is the combined RSS error.

When observing both room temperature and cryogenic data in Fig. 2(a), we note that the return loss (⁠$=−|S11|$⁠) of the 20-dB-attenuator is above 40 dB at low frequencies and stays above 30 dB until around the frequency of 10 GHz and reaches its minimum value of 19 dB at 18 GHz. There is not a significant temperature dependence aside from the resonance at 10.4 GHz that only appears at 30 mK. At the frequency of 5 GHz, the in situ 300 K and 30 mK return loss values are $36−3+4$ dB (0.015 ± 0.006) and $35−2+3$ dB (0.019 ± 0.006), respectively, where the corresponding values in linear units are in parentheses. The error bars are always dominated by the switch variability error, which does not depend on the magnitude of reflection and, therefore, at 5 GHz the error is roughly 0.006 for all devices. On the other hand, for the 10-dB-attenuator displayed in Fig. 2(b), the return loss increases at cryogenic temperatures by 3–5 dB with respect to the room temperature measurements up until a frequency of approximately 10 GHz. At higher frequencies of $∼15$ GHz, the measured return loss values at 300 K and 30 mK converge. At the selected frequency of 5 GHz, the in situ room temperature return loss value is $28−1+1$ dB (0.041 ± 0.006), and the 30 mK measurement result is $33−2+3$ dB (0.022 ± 0.006).

As expected, the cryogenic insertion losses of Figs. 2(e) and 2(f) are reduced with respect to the room temperature data. For example, at a frequency of 5 GHz, the in situ loss $1.38−0.04+0.04$ dB of the SCuNi cable decreases to $0.99−0.04+0.04$ dB when only switch and calibration errors are considered. The in situ room temperature measurement (blue) agrees within ±0.2 dB with the data obtained with ECal (yellow), where the dropoff at low frequencies we attribute to the time gate.³⁸ For the NbTi coaxial cable, the loss measured in situ at 5 GHz drops from $4.72−0.03+0.03$ dB to $0.02−0.04+0.04$ dB. The remaining errors are similar to those observed for SCuNi; however, a larger gate-induced difference of 0.4 dB is seen at the lowest measured frequencies. A tabulation of return loss measurement results for the 10- and 20-dB attenuators and the SCuNi and NbTi coaxial cables is shown in Table I.

We now turn to a discussion of the influence of the return loss results of Table I on single-qubit gate fidelity. In microwave pulse driven qubit architectures, single-qubit gate fidelity becomes highly susceptible to any impedance mismatch in the microwave line since such imperfections lead to pulse distortions.⁵ We present a model where a qubit drive pulse travels through a transmission line with two unmatched elements (representing cable connectors or poorly matched attenuators) that partially reflect between these elements in a way similar to the Fabry–Pérot interferometer. The resulting reflections interfere with the original pulse causing distortions in both phase and amplitude.

Numerical simulation is performed using Qutip—an open-source python library for simulating the dynamics of open quantum systems⁴⁰ with additional details reported in supplementary material Sec. I E. Within simulation, we perform an ALLXY benchmarking set.⁴¹ The end result is the fidelity between the state achieved for an ideal case of infinite return loss and the one achieved with a given finite return loss. The results of simulations are presented in Fig. 3. In addition, we plot the return loss threshold of 14.7 dB in Figs. 2(a)–2(d) that corresponds to 99.99% state fidelity for X and Y gates applied to a qubit in sequence with a 5-ns gate-pulse through a 276-mm cable.

As expected, the simulation results reveal that the fidelity deviation $1−F$ depends periodically on the length between the two unmatched elements with the period defined by the pulse frequency and return loss. For any return loss, one can find a suitable length between unmatched elements of the signal frequency, where the fidelity deviation induced by the effect under study is minimal. From the simulation results, we observe that the gate fidelity deviation becomes stronger for shorter drive pulse durations. For example, from Fig. 3, we observe that the fidelity deviation for 60 ns pulses and 5 ns pulses differs by two orders of magnitude. Thus, the effect may become relevant in experimental systems when working with short drive pulses.⁴² 

Fidelity deviations arise due to the phase change within the drive pulse due to interference of the original signal with the delayed reflections. In turn, these phase changes lead to changes in the rotation angle of the qubit state vector on the Bloch sphere during gate evolution. This phenomenon can be described by curving of the trajectory of the qubit state on the Bloch sphere. This curvature can be positive or negative depending on the phase differences between the original signal and the reflections. The effect is most prominent near the beginning and the end of the drive pulse, where multiple phase changes are present.

In conclusion, we presented calibrated 1-port S-parameter measurements of 10- and 20-dB cryogenic attenuators and two different varieties of cryogenic coaxial cables to verify impedance matching. Our measurement technique removes all excess attenuation and frequency-dependent components within the cryogenic environment in order to measure reflections directly using a high-dynamic range NA over a broad frequency band of 10 MHz–18 GHz with pre-characterized data-based calibration standards in situ. We also presented a theoretical analysis of the gate fidelity deviation as a function of the return loss of the attenuators, cable length, and gate-pulse duration. For short cable lengths, larger return loss values can be tolerated, and mismatch-induced pulse distortions will not be a significant source of single-qubit gate errors. Decreasing the gate-pulse duration increases the return loss requirement for maintaining the high gate-fidelity. The results highlight the importance of keeping cables between mismatched elements as short as possible in order to avoid single qubit preparation errors. This point is crucial when designing larger modular multi-chip systems that are connected via coaxial cables in scaled-up quantum processors or systems that have large shielded enclosures that necessitate longer interconnects.

See the supplementary material for detailed descriptions of the calibration procedure, time domain and error analysis, and master equation simulations.

V.Y.M. would like to thank Dr. Aidar Sultanov of Aalto University for helpful discussions regarding master equation simulations.