Digital Control of a Superconducting Qubit Using a Josephson Pulse Generator at 3 K

Scaling of quantum computers to fault-tolerant levels relies critically on the integration of energy-efficient, stable, and reproducible qubit control and readout electronics. In comparison to traditional semiconductor-control electronics (TSCE) located at room temperature, the signals generated by rf sources based on Josephson-junctions (JJs) benefit from small device sizes, low power dissipation, intrinsic calibration, superior reproducibility, and insensitivity to ambient fluctuations. Previous experiments to colocate qubits and JJ-based control electronics have resulted in quasiparticle poisoning of the qubit, degrading the coherence and lifetime of the qubit. In this paper, we digitally control a 0.01-K transmon qubit with pulses from a Josephson pulse generator (JPG) located at the 3-K stage of a dilution refrigerator. We directly compare the qubit lifetime T1, the coherence time T2*, and the thermal occupation Pth when the qubit is controlled by the JPG circuit versus the TSCE setup. We find agreement to within the daily fluctuations of ±0.5 μs and ±2 μs for T1 and T2*, respectively, and agreement to within the 1% error for Pth. Additionally, we perform randomized benchmarking to measure an average JPG gate error of 2.1 × 10−2. In combination with a small device size (< 25 mm2) and low on-chip power dissipation (≪100 μW), these results are an important step toward demonstrating the viability of using JJ-based control electronics located at temperature stages higher than the mixing-chamber stage in highly scaled superconducting quantum information systems.


I. INTRODUCTION
Error-corrected quantum computers are projected to require large numbers, O 10 6 , of qubits [1][2][3][4], placing stringent requirements on the per-qubit hardware overhead. Superconducting quantum circuits are a leading technology for scaling existing systems into the noisy intermediate-scale quantum (NISQ) era of ≳ 1000 qubits. In present systems, qubit gates and entangling operations are performed using shaped microwave pulses synthesized using instrumentation at room temperature [5]-here referred to as traditional semiconductorcontrol electronics (TSCE). Signals are routed into a dilution refrigerator (DR) to the approximately 0.01-K qubits and typically attenuated by 40-60 dB to suppress thermal noise on drive lines [6].
Limitations in cryogenic cooling power, TSCE power instability [7], and system complexity mandate a shift to miniaturize and enhance the stability and/or precision of waveform generation in superconducting quantum information systems. Recently, the pulse-shaping DACs and/or mixers used to generate qubit control signals have been successfully integrated at 3 K [8,9] and 100 mK [10] using cryogenic CMOS (cryoCMOS) technology. While this is an impressive step toward miniaturization and large-scale integration, a significant gap exists between these devices and scalable qubit control. Specifically, gate fidelity; power dissipation; and the accuracy, stability, and repeatability of the signals need improvement [7,11,12].
The scalability constraints of physical size and power consumption per channel may be satisfied by superconducting Josephson-junction (JJ) signal-generator circuits or the aforementioned cryoCMOS controllers. Comparable to cryoCMOS devices, JJ circuits have small device sizes (< 1 × 1 cm 2 ) and very low on-chip power dissipation (≪100 μW); while also leveraging the intrinsically calibrated nature of single-flux quantum (SFQ) pulses. This feature provides avenues for improving waveform quality and repeatability beyond what is achievable using semiconductor-based generators. Capitalizing on pulse-area quantization enables the use of JJ arrays to construct exceptionally stable and repeatable voltage sources from dc to a few gigahertz [13][14][15][16]. Similar devices are used to realize intrinsically accurate voltages for the international system of units and are disseminated worldwide as primary dc and ac voltage standards [17,18]. Furthermore, the use of SFQ pulses has been proposed as a scalable paradigm for digitally controlling qubits [19][20][21] and has recently been demonstrated with a SFQ driver and qubit circuit cofabricated on the same chip [22].
The primary limitation of SFQ operation proximal to quantum arrays is degradation of qubit lifetimes from quasiparticles created during pulse generation [23,24]. A solution that mitigates these quasiparticles must be implemented-such as physically separating the SFQ elements and qubits. In this work, we locate the JJ control circuitry on the 3-K DR stage to interrupt quasiparticle-qubit propagation. Similar to Ref. [22], we deliver sparse trains of pulses subresonantly to enact control-giving our device its name of the Josephson pulse generator (JPG). A 0.01-K bump-bonded multichip configuration [25,26] and/or the introduction of normal-metal quasiparticle traps can also be effective for quasiparticle mitigation [27,28].
The location of the cryogenic control electronics at a higher-temperature stage liberates physical volume at 0.01 K-commonly monopolized by the quantum array and readout hardware-and leverages higher cooling powers. This approach may also benefit from integration with cryoCMOS circuits by exploiting the advantages of cryoCMOSimplemented logic and/or memory elements [8][9][10]. The location of the control electronics at 3 K does increase the wiring complexity and the parasitic heat loads to the < 3 K stages; however, solutions are under development that demonstrate low thermal loading and crosstalk [29][30][31].
While the aforementioned merits of JJ-based sources [17] are expected to apply for qubit control, this work is the first validation of using JJ-based pulse generation at 3 K to control a 0.01-K qubit. Here, we show that the JPG does not adversely affect the qubit by separately measuring the qubit energy-relaxation time T 1 , the coherence time T 2 * , and the thermal occupancy P th with both a TSCE setup and the JPG. Our findings show good agreement in all three metrics with each control setup. Additionally, we measure the JPG gate fidelity to be within an order of magnitude of the qubit coherence limit and provide discussion of future devices expected to yield coherence-limited gates.

II. JPG-BASED QUBIT CONTROL
An input current evolving the JJ superconducting phase difference by 2π generates a voltage pulse the time-integrated area of which equals the magnetic flux quantum Φ 0 ≡ ℎ/2e: The duration of this SFQ pulse is approximately τ = Ф 0 /I c R n , where τ is the JJ characteristic time, I c is its critical current, and R n is its normal resistance [32]. SFQ signal amplification can be achieved by connecting a series of N JJ junctions; the pulses of which add coherently. This yields a larger pulse of area N JJ Ф 0 , which we call a JPG pulse. Depending on the qubit coupling to the control line, arrays with N JJ ~ 10 2 -10 4 are required if located at 3 K. In this work, our JPG has N JJ = 4650, I c = 3.05 mA, and R n = 6.93 mΩ, resulting in a characteristic frequency of f c = 1/τ = 10. If f c is much larger than the qubit transition frequency, ω 10 /2π , then during pulse arrival, the qubit undergoes a discrete rotation δθ = N JJ AC c Φ 0 2ω 10 ℏC T , (2) where A is the JPG-qubit amplitude attenuation, C c is the control-line-qubit coupling capacitance, and C T is the qubit capacitance [19]. N JJ , A, and C c may be treated as free design parameters to realize a combination of adequate control-line thermalization and tip angle per pulse, δθ . For three-dimensional (3D) readout-cavity configurations, C c also encapsulates attenuation from pulse transit of the cavity resonance. A train of sharp pulses arriving resonantly at the qubit (ω d = ω 10 ), or at a subharmonic (ω d = ω 10 /k, where k ≥ 2 is an integer), discretely rotate the qubit around the Bloch sphere during pulse arrival, while between pulses the qubit precesses for k periods at fixed θ [see Fig. 1 In our implementation, the JPG must be driven using a sinusoidal signal at k ≥ 2 because there is no isolation between the drive input and the device output. Otherwise, the large drive signal dominates and induces spurious qubit rotations. The generation of an integer number of JPG pulses ℓ is performed by sending an integer number of sinusoidal drive periods, ν.
Under the correct bias parameters, there is a one-to-one correspondence between the number of JPG pulses generated and the number of drive periods (ν = ℓ). Orthogonal axis control, realized by phasing the drive signal relative to a timing reference, is depicted in Fig. 1

III. EXPERIMENTAL DETAILS
In this work, we use a transmon qubit dispersively coupled to a 3D aluminum readout cavity possessing two control lines with different coupling strengths. A simplified experimental schematic is shown in Fig. 2. The JPG is connected to the weakly coupled port of the cavity (0.175-MHz coupling rate) and the TSCE control and readout line to its strongly coupled port (2.01-MHz coupling rate). With this setup, a direct comparison of qubit performance with both control schemes is possible during the same cool-down. This qubit has been measured for a previous publication; for other parameters, see Ref. [34].
Qubit-state readout is performed by probing the qubit-state-dependent frequency shift of the cavity. The dressed cavity frequency is ω |0 , | 1 = ω r ± χ, where ω r is the bare frequency and χ is the shift due to cavity-qubit coupling [35]. A Josephson parametric amplifier (JPA) [36] is operated with a gain of 20 dB (phase insensitive) to enable single-shot measurements.
To minimize measurement-induced transitions, the cavity-probe tone amplitude is typically n r ≈ 6 ≲ n crit /20, where n r and n crit are the readout and critical photon numbers [37]. We perform passive qubit-state reset via relaxation over a period ≥ 15T 1 . The same readout procedure and instrumentation is used for both TSCE and JPG measurements.

A. JPG operation and X π calibration
After characterization with the TSCE setup, we establish operating parameters, specifically the rf drive power and dc current bias I b , for the JPG in which the number of output JPG pulses is equal to the number of input drive periods-called the locking range. Under sinusoidal rf drive at frequency f d , a constant voltage Shapiro step manifests at When the measured voltage is constant and equal to Eq. (3) over a range of I b , then for any I b on the Shapiro step, the device is locked. Thus, we first maximize the locking range by determining the drive power that gives the largest Shapiro steps. Figure  For all measurements in this work, we drive at subharmonic k = 2 (ω d = ω 10 /2) and use the second-harmonic power of the JPG pulse train to control the qubit. As we are restricted to subharmonic drive, k = 2 maximizes the locking range by making f d as close as possible to f c [38] and provides the highest-fidelity (fastest) gates. Next, we measure JPG-induced Rabi oscillations to characterize the JPG-qubit interaction. At the optimal drive power, we measure the number of drive periods ν required for a π rotation, ν π , versus I b . The results of this procedure are shown in Figs. 3(b)-3(e). Fitting these Rabi oscillations at constant I b yields ν π (I b ) and we look for regions where ν π is insensitive to the number of Rabi periods (i.e., the drive time). This demonstrates locking of the JPG, where ν = ℓ, and a stable JPG-qubit interaction as the drive pattern is lengthened.
One may expect the entire locking range in Fig. 3(a) to give a constant ν π ; however, this is not observed in Fig. 3(e). This is because the pulse width for Josephson devices operated at f d ~ f c varies as I b traverses the Shapiro step. Widening of the pulses results in a reduction of δθ which we discuss further in Sec. III B, and in Sec. V of the Supplemental Material [33].
Our simulations for pulse-width variation across the Shapiro step agree with previous work [39,40] and a variation of < 10% is expected. This restricts the region of constant ν π in the Rabi measurements relative to the dc locking-range measurement. Despite these effects, Fig.  3(e) nevertheless demonstrates a range of 150 μA, where ν π is constant to within one pulse.

B. Finite-Width Pulses
The production of perfectly sharp pulses is not possible, so δθ also depends on the JPG pulse width. Indeed, we demonstrate this by broadening the JPG pulses (we heat the JPG to reduce I c ) and observe ν π to increase by approximately the same factor by which I c decreases. To explore the pulse-width dependence of δθ , we perform simulations [41] of a qubit driven at ω d = ω 10 /2 by Gaussian pulses the width of which (the standard deviation σ in units of the qubit period, T q ) we control. Figure 4 shows our simulation results and illustrates a strongly nonlinear dependence on the qubit response for σ > 0.25 T q . For short pulses in the Dirac-delta-function limit σ < 0.01 T q , the qubit response is independent of σ (ν π changes by less than one pulse). Section VII in the Supplemental Material [33] discusses the relationship between JJ τ and σ of a Gaussian fitted to the pulses. For our JPG with τ = 98 ps, the (on-chip) Gaussian-parametrized pulse width is σ = 17 ps.
To measure the pulse width, the JPG output is split (see Fig. S1 in the Supplemental Material [33]) and recorded with an oscilloscope at room temperature. We find that σ = 35 ps which, for our 5.37-GHz qubit, gives σ = 0.19 T q . This is an upper bound for the widths of pulses delivered to the qubit due to added dispersion in the additional 2 m of JPG-oscilloscope cabling compared to the JPG-qubit cable length. From the simulations (after adjusting coupling so ν π = 352) and pulse-width measurements, we obtain a lower bound of our expected JPG X π infidelity of 2 × 10 −3 . While the total infidelity is coherence-limit dominated, subtraction of this contribution gives the infidelity due only to the nature of control via digital pulses, 1 − F pulse . Figure 4(c) demonstrates that, for our current gate times, this digital-pulse-only infidelity is approximately 10% of the total infidelity. Furthermore, this is competitive with state-of-the-art TSCE techniques, reaching approximately 10 −4 infidelity, and shows that there is no fundamental limitation imposed by digital control with sharp pulses [42]. For more discussion, see Sec. VII in the Supplemental Material [33].
When driven at a frequency below f c , the locking range of our prototype JPG decreases by a factor ∝ f d /f c but a lower f c also compromises ideal digital qubit control dynamics. More ideal control may be realized at the expense of the locking range (or vice versa) by tuning f c = I c R n /Ф 0 -with the caveat that σ ≳ 0.3 T q pulses are too wide for efficient digital control.
We keep σ < 0.2 T q to balance the locking range and optimal qubit dynamics. Future devices (see Sec. VI) will not possess this limitation.

IV. COMPARISON OF QUBIT PERFORMANCE
Here, we describe the side-by-side comparison of the TSCE and JPG setups through measurements of T 1 , T 2 * , and P th . For the T 1 comparison, a JPG X π rotation is constructed of ν π = 352 drive periods (131-ns drive time) obtained with the calibration shown in Fig.  3. For the T 2 * comparison, a JPG X π/2 rotation is created with a ν π /2 = 176 period drive waveform. We gather statistics on 500 measurements of T 1 and T 2 * with each setup. The data are compiled in Fig. 5 and show energy decay curves and Ramsey fringes averaged over all measurements, as well as histograms of the extracted T 1 and T 2 * . The small discrepancies in the distribution means are well within the expected variation in T 1 and An important validation of JPG-qubit compatibility is to demonstrate adequate thermalization when controlled with the JPG. State inversions from elevated qubit thermal occupancy can be ≳ 10% in transmon qubits with 3D aluminum readout cavities [47][48][49].
We define the qubit thermal occupancy P th as the probability of incorrect state identification based on the desired preparation.
The measurement of P th is performed in a two-part experiment [50]. First, no qubit rotation is applied and the state is simply measured. Second, we apply an X π rotation to invert the qubit population and then measure. The measurements are single shot and we do not perform heralding. The total state-preparation-and-measurement (SPAM) fidelity is where P(i|j ) is the probability of measuring state |i⟩ when the qubit is intended to be prepared in |j ⟩ . Equation (4) describes the combined preparation fidelity and ability for the single-shot measurement to correctly distinguish between |0⟩ or |1⟩. Ideally, each P(i|j ) only contains contributions from thermal occupancy. In reality, both P(i|j ) include decays from correctly prepared |1⟩ and spuriously excited |0⟩ initial states and the |0⟩ and |1⟩ distribution overlap. The SPAM fidelity thus gives an upper bound on P th and we minimize the effects of overlap infidelity and decays during measurement to improve our estimate of P th .
We limit these decays to < 1%, which becomes the dominant uncertainty in the P th measurement, by shortening the readout pulse to 400 ns and we counter the corresponding reduction of the single-shot SNR using an optimal mode-matching integration weight function [50]. Overlap infidelity is minimized by increasing the cavity drive strength to separate the primary |0⟩ and |1⟩ distribution lobes. For the 400-ns readout pulse, this occurs at n r = 50 ≈ n crit /2.3, which still avoids measurement-induced transitions [37]. Figure 6 shows data from 10 4 measurements with each setup and bimodal Gaussian fits to the data. As the no-X π case (desired preparation in |0⟩) additionally removes decays during preparation, we choose this population to bound P th with the most accuracy. The integration of all spurious |1⟩ outcomes for this case yields excellent agreement in P th of 0.036 ± 0.01 and 0.032 ± 0.01 for the TSCE and JPG setups, respectively. These results demonstrate that qubit thermalization is not affected when the JPG is used-our final compatibility metric of digital control using JJ-based pulses from 3 K.

V. JPG RANDOMIZED BENCHMARKING
We now characterize JPG gate fidelities through a randomized-benchmarking (RB) routine [51][52][53][54][55], where we apply a sequence of m random gates followed by a single sequenceinverting gate. The sequence fidelity is an exponential decay where the constants a and b encapsulate SPAM errors and errors on the final gate. For single-qubit gates, the depolarizing parameter p is related to the per-gate error r by The results of the same routine using the TSCE setup are provided as a reference gate error.
We choose the set of primitive and Pauli gates: {I , ±X π/2 , ±Y π/2 , ±X π , ±Y π }, where the idle I and X π gate lengths are equal [56,57]. Given that π gates are twice as long as π/2 gates and the fact we use a Clifford-group subset, rescaling m by a factor of 1.125 permits comparison with full-Clifford-group RB [22,54]. The JPG gates are constructed as described above, while the TSCE gates use a σ = 35 ns Gaussian pulse truncated at ±2σ to closely match the JPG gate time.
In Fig. 7, we show the results of the RB routine with both setups, giving r TSCE = (4.8 ± 0.5) × 10 −3 and r JPG = (2.1 ± 0.1) × 10 −2 , where the uncertainties are from the Eq. (5) fitted standard error. The JPG r is approximately a factor of 10 higher than the simulated single X π gate error of 2.1 × 10 −3 [58]. A detailed accounting of known possible errors (see Sec. V in the Supplemental Material [33]) from digitization, finite pulse widths, higher-state leakage, and pulse timing jitter [19,59] gives an estimated infidelity of 6 × 10 −3 . This is a factor of 3 below the RB result. We attribute the remaining error to possible systematic or coherent errors, which are presently under investigation. Regardless, these measurements serve as an excellent proof-of-concept demonstration of qubit control using 3-K JJ-based digital pulses.

VI. SCALABILITY AND DISCUSSION OF FUTURE DEVICES
The scalability of digital qubit control using JJ devices at 3 K is promising even with the current device and configuration-which are not optimized for size or power dissipation. The JPG circuit is < 15 mm 2 and the power dissipated at duty cycle η d is approximately For the maximum η d in our experiment of 0.02, this yields P = 1.6 μW. Commercial ). Of greater consideration is dissipation of the large drive signal, which is approximately −6 dBm at the JPG input at 3 K; however, the new devices discussed below offer techniques to circumvent this limitation.
Plans for NISQ systems require approximately 10 m 3 of cryostat volume and numerous cryocoolers. We conclude that neither the power dissipation nor the device size (the volume of silicon) present significant obstacles in scaling the number of 3-K JJ devices to control NISQ-era quantum arrays. Furthermore, our experimental and JPG architectures can both be adjusted in a straightforward manner to reduce the device size and the on-chip power dissipation, each by more than a factor of 10. Thus, the primary obstacle in scaling a 3-K JJbased qubit control architecture, at least in the near term, is one shared by many competing qubit-control technologies: wiring and signal-routing logistics. Multichip modules [61][62][63], high-density and/or bandwidth interconnects [29,31], and out-of-plane coupling [64] make it feasible to overcome these challenges.
The present experiment architecture, where JPG pulses are heavily attenuated by the cavity resonance before reaching the qubit, necessitates the use of a large N JJ to yield appropriate signal levels. This increases the device size and the power dissipation and can reduce qubit coherence by allowing the higher pulse-train harmonics to populate the cavity (not observed here). These limitations can be eliminated using two-dimensional readout-cavity qubit devices and an independent control line. With such a device, we expect to reduce the array factor to N JJ ≈ 500 without sacrificing thermalization or the gate time.
Next-generation devices will implement an SFQ logic shift register and voltage-multiplier pulse amplification with no increase in the JJ count [16,65]. The use of a high speed clock far above the qubit spectrum eliminates the qubit-drive interaction and permits pulse delivery at ω d ≥ ω 10 or at variable timing. The latter has theoretically been shown to reach 99.99% fidelity with under-10-ns gates [20]. Voltage-multiplier amplification minimizes on-chip dispersion and permits narrower output pulses, enabling more ideal digital qubit dynamics. Finally, these devices permit signal routing that eliminates dissipation of the drive signal in cryogenic attenuators, which is a major consideration for the present JPG device and would strongly limit scaling of these prototype devices.

VII. CONCLUSIONS
In this paper, we demonstrate, for the first time, successful digital control of a transmon qubit at 0.01 K using a superconducting Josephson-pulse generator located at 3 K. Through dual characterization of the system, using both TSCE [5] and the 3-K JPG, we see no reduction in intrinsic qubit performance. Specifically, we measure no negative impact on T 1 , T 2 * , or P th -indicative of the fact that quasi-particle propagation is effectively broken by locating the JJ elements and quantum circuits on separate temperature stages. Additionally, we measure an average JPG gate error of r = 2.1 × 10 −2 which, considering the improvements of the future JJ devices discussed in Sec. VI are expected to reach the simulated coherence-dominated infidelity of r min = 2 × 10 −3 .
These results enable scaled quantum information systems that leverage the merits of Josephson-based sources for qubit control: signal stability, reproducibility, SFQ pulse self-calibration, small device size, and low power dissipation. Straightforward alterations in the qubit and JPG architectures enable factors of ≥ 10 reduction in dissipation and size and future devices are expected to bring JJ-based digital gates into competition with contemporary TSCE gates. Such improvements further increase the potential value of 3-K JJ-based qubit control, as current device sizes and dissipation are commensurate with the operation of over 500 000 devices. Integration with cryoC-MOS devices [8][9][10] is also possible; potentially yielding a hybrid cryogenic controller that exploits the advantages of both technologies.

Supplementary Material
Refer to Web version on PubMed Central for supplementary material. Simulations of a qubit digitally driven by a Gaussian pulse train to extract the expected X π fidelity as a function of the pulse standard deviation σ . Here, σ is in units of qubit periods T q . Pulses are delivered at ω d = ω 10 /2, so a pulse arrives at every other qubit period. We use energy-relaxation and dephasing rates consistent with our qubit (see Sec. V). Fits to single JPG pulses measured at room temperature yield an upper bound of σ JPG = 0.19 T q (35 ps) for the width of pulses delivered to the qubit when parametrized as a Gaussian. (a) The qubit excitation probability as a function of the number of pulses for increasing σ and for the first half of the first Rabi oscillation. (b) The extracted value of ν π versus σ and the corresponding total X π total infidelity, 1 − F tot . The pulse-qubit coupling is normalized such that for the Dirac-delta-function limit, ν π = 100. With the coupling readjusted to match our experimental values of ν π = 352 and σ JPG = 0.19 T q , we obtain 1 − F tot = 2 × 10 −3 and 1 − F pulse = 1 × 10 −4 . (c) The digital-pulse-only X π infidelity, 1 − F pulse , as a function of ν π -which is parametrized by σ in (b). 1 − F pulse is calculated by subtracting the coherence-limit contribution to 1 − F tot .

FIG. 5.
A comparison of the measured qubit lifetime T 1 and the Ramsey coherence time T 2 * using a TSCE setup and a JPG at 3 K. For each setup, 500 individual measurements of T 1 and The measurement of the qubit thermal population, P th , to compare control with the TSCE (top) and JPG (bottom). The solid lines are a bimodal Gaussian fit to the data. The P th estimate is obtained by integrating the fit to the no-X π population with voltage levels greater than zero. These data points correspond to instances in which a spurious state inversion occurs, spoiling the desired |0⟩ preparation. The primary uncertainty in P th using this method is due to decays during measurement, i.e., an error of approximately 1%.

FIG. 7.
The depolarizing curve for single-qubit RB using the TSCE and 3-K JPG qubit control setups. The solid lines are a fit to Eq. (5). TSCE rotations are performed with Gaussian pulses with σ = 35 ns and truncated at ±2σ to match the JPG X π gate length of 131 ns. We extract an average error per gate of r TSCE = (4.8 ± 0.5) × 10 −3 and r JPG = (2.1 ± 0.1) × 10 −2 .
The former is just over twice the coherence limit of the qubit, while the latter is a factor of 10 higher than the X π fidelity of 2 × 10 −3 as determined in simulations-which is dominated by qubit coherence. Uncertainties in r are determined as the standard error of the fits to Eq. (5).