Digital Coherent Control of a Superconducting Qubit

High-fidelity gate operations are essential to the realization of a fault-tolerant quantum computer. In addition, the physical resources required to implement gates must scale efficiently with system size. A longstanding goal of the superconducting qubit community is the tight integration of a superconducting quantum circuit with a proximal classical cryogenic control system. Here we implement coherent control of a superconducting transmon qubit using a single-flux quantum (SFQ) pulse driver cofabricated on the qubit chip. The pulse driver delivers trains of quantized flux pulses to the qubit through a weak capacitive coupling; coherent rotations of the qubit state are realized when the pulse-to-pulse timing is matched to a multiple of the qubit oscillation period. We measure the fidelity of SFQ-based gates to be approximately 95% using interleaved randomized benchmarking. Gate fidelities are limited by quasiparticle generation in the dissipative SFQ driver. We characterize the dissipative and dispersive contributions of the quasiparticle admittance and discuss mitigation strategies to suppress quasiparticle poisoning. These results open the door to integration of large-scale superconducting qubit arrays with SFQ control elements for low-latency feedback and stabilization.

Figure 1

Coherent control of a qubit using SFQ pulses. (a) A phase slip across a JJ induces a voltage pulse with the time integral precisely quantized to $h/2e\equiv {\mathrm{\Phi }}_0\approx 2.07\mathrm{mV}\times \mathrm{ps}$, the superconducting flux quantum. For typical parameters, pulse amplitudes are of order 1 mV and pulse durations are of order 2 ps. (b) Block diagram of the experiment. An external microwave tone with frequency ${\omega }_d$ is used to trigger a dc-SFQ converter, which generates a train of pulses with interpulse timing of $2\pi /{\omega }_d$; the resulting pulse train is coupled capacitively to a transmon qubit. (c) Trajectory of the qubit state vector on the Bloch sphere due to a resonant train of SFQ pulses. Each SFQ pulse induces an incremental rotation; the green parallels represent free precession of the qubit in between SFQ pulses. (d) Simulated qubit Rabi oscillations for SFQ drive at various subharmonics of the qubit transition frequency ${\omega }_{10}$. Discrete steps in excited-state population $P_1(t)$ are induced by the arrival of single SFQ pulses; in between pulses, the qubit undergoes free precession with no change in $P_1(t)$. In (c) and (d), a large, nonoptimal value of $\delta \theta =\pi /10$ is chosen for the purpose of display.

Figure 2

Overview of the quantum-classical hybrid circuit. (a) False-color optical micrograph of the SFQ driver circuit (green) with capacitive coupling (red) to the superconducting transmon qubit (blue). The qubit is coupled to the voltage antinode of a $\lambda /4$ coplanar waveguide resonator (yellow) and is biased by a dc flux line (blue). The SFQ driver is triggered by a microwave tone delivered via a $Z=50\mathrm{\Omega }$ CPW transmission line (purple). The distance between the SFQ driver and qubit is approximately 3.5 mm. (b) Circuit diagram for the device of (a). Each JJ in the classical circuit is shunted by a thin-film $\mathrm{Pd}$ resistor (not shown). (c) False-color cross-section SEM micrograph showing layer stack in the vicinity of a ${\mathrm{Nb}\text{/}\mathrm{Al}\text{−}\mathrm{Al}\mathrm{O}}_x\text{−}\mathrm{Al}\text{/}\mathrm{Nb}$ JJ of the SFQ driver. The $\mathrm{Nb}$ wiring layers (blue) are separated by ${\mathrm{Si}\mathrm{O}}_x$ dielectric layers (brown); the driver junctions are formed in vias in the dielectric layer separating the second and third $\mathrm{Nb}$ metal layers. $\mathrm{Pd}$ resistors (red) shunt the junctions of the driver circuit. (d),(e) Conventional microwave-based characterization of qubit energy relaxation (d) and dephasing (e) with extracted characteristic times $T_1=23.6(6)\mu \mathrm{s}$ and $T_2^{\ast }=24.4(8)\mu \mathrm{s}$, respectively.

Figure 3

Basic qubit operations driven by SFQ pulses. (a) SFQ-based Rabi oscillations as a function of bias current $I_b$ to the SFQ driver circuit. Here, ${\omega }_d={\omega }_{10}/3\approx 1.65$ GHz. (b) Rabi chevron experiment and (c) Ramsey fringe experiment where the SFQ pulse frequency ${\omega }_d$ is varied slightly in the vicinity of ${\omega }_{10}/3$. (d) Time trace of a SFQ-based Rabi flop obtained with a pulse rate ${\omega }_{10}/41=120.90$ MHz. Inset: An enlargement of the discrete steps in $P_1(t)$ occurring every $8.3$ ns, the SFQ pulse-to-pulse timing interval.

Figure 4

(a) Illustration depicting SFQ pulse sequences for orthogonal qubit control. The resonant $X$ sequence establishes a timing reference; a change in the relative timing of a second resonant sequence can be used to access different orientations of the qubit control vector in the equatorial plane of the Bloch sphere. A $Y_{\mathrm{SFQ}}$ sequence is shifted in time by one quarter of the qubit oscillation period with respect to the $X_{\mathrm{SFQ}}$ sequence. (b) Generalized Rabi scans for resonant SFQ pulse trains triggered at subharmonic frequency ${\omega }_{10}/3$ (left) and ${\omega }_{10}/41$ (right), respectively. The pulse sequence is shown below the figure. The radial coordinate is swept by incrementing the duration of $R_{\mathrm{SFQ}}(t,\varphi )$ from 0 to 60 ns (left) or 660 ns (right), while the polar angle corresponds to the phase $\varphi$ of the trigger waveform used to generate the pulse train. The length and phase of each $X_{\mathrm{SFQ}}/2$ pulse are fixed. Adjustment of the phase $\varphi$ of the trigger waveform by $\pi /2n$ provides access to an orthogonal control axis; here, $n$ is the subharmonic drive factor. The $2n$-fold symmetry of the plots is explained by the fact that subharmonic drive automatically yields an $n$-fold increase in the number of (time-shifted) resonant sequences that access the same control vector on the Bloch sphere, and by the fact that our sequence does not distinguish between positive and negative rotations (yielding an additional doubling of the symmetry of the plot). On the left panel, we label rays corresponding to trigger waveform phases that yield $\pm X_{\mathrm{SFQ}},\pm Y_{\mathrm{SFQ}}$ rotations.

Figure 5

Randomized benchmarking depolarizing plot for all Clifford gates and for the interleaved gate sequence designed to probe the $X_{\mathrm{SFQ}}/2$ gate as a function of the number of Clifford operations. The (blue, orange) traces correspond to SFQ sequences with pulse rate ${\omega }_{10}/3$, while the (yellow, purple) traces correspond to pulse rate ${\omega }_{10}/41$. For long sequence lengths, operation of the dissipative SFQ driver produces significant incoherent excitation of the qubit, which prevents faithful return to the ground state following the final pulse of the RB sequence. This results in the observed suppression of the RB saturation level to below 50%. Inset: Table summarizing RB data for SFQ-based gates with ${\omega }_d={\omega }_{10}/\{3,41\}$.

Figure 6

QP poisoning of the qubit induced by operation of the SFQ driver. (a) Energy decay curves with and without a prior applied poisoning pulse from the SFQ driver. The enhanced relaxation rate can be attributed to an increase in the mean number of QPs $⟨n_{\mathrm{QP}}⟩$ coupled to the qubit. (b) $⟨n_{\mathrm{QP}}⟩$ versus number of phase slips in the SFQ poisoning pulse. We find $1.6(2)\times {10}^{-3}$ QPs couple to the qubit per JJ phase slip. The lengths of single $X_{\mathrm{SFQ}}$ (blue) and $X_{\mathrm{SFQ}}/2$ (orange) gates are shown for reference. (c) QP recovery experiment for a fixed poisoning pulse (approximately 2 $\times {10}^4$ phase slips) as a function of the time between the poisoning pulse and the $T_1$ experiment.

Figure 7

Parametric plot of qubit frequency shift $\delta {\omega }_{10}$ versus QP-induced decay rate $\mathrm{\Gamma }$. Here, QP density at the qubit is controlled by incrementing the recovery time following an intentional QP poisoning pulse prior to the $T_1$ or Ramsey experiment.

Figure 8

Device layer stack described in Appendix A. During the entirety of the fabrication process of the SFQ driver circuit, the area that will support the quantum circuit is left unpatterned and unetched and is protected by the first deposited layer of ${\mathrm{Si}\mathrm{O}}_x$.

Figure 9

Experimental wiring diagram. Component symbols are defined in the panel above the wiring diagram. All components within the $\mu$-metal shield (gray box) are made of either nonferric or certified nonmagnetic materials.