Single Flux Quantum-Based Digital Control of Superconducting Qubits in a Multichip Module

Single flux quantum (SFQ) digital logic has been proposed for the scalable control of next-generation superconducting-qubit arrays. In the initial implementation, SFQ-based gate fidelity was limited by quasiparticle (QP) poisoning induced by the dissipative on-chip SFQ driver circuit. In this work, we introduce a multichip-module architecture to suppress phonon-mediated QP poisoning. Here, the SFQ elements and qubits are fabricated on separate chips that are joined with $\mathrm{In}$-bump bonds. We use interleaved randomized benchmarking to characterize the fidelity of SFQ-based gates and we demonstrate an error per Clifford gate of 1.2(1)%, an order-of-magnitude reduction over the gate error achieved in the initial realization of SFQ-based qubit control. We use purity benchmarking to quantify the contribution of incoherent error at 0.96(2)%; we attribute this error to photon-mediated QP poisoning mediated by the resonant millimeter-wave antenna modes of the qubit and SFQ-qubit coupler. We anticipate that a straightforward redesign of the SFQ driver circuit to limit the bandwidth of the SFQ pulses will eliminate this source of infidelity, allowing SFQ-based gates with error approaching approximate known theoretical limits, of order 0.1% for resonant sequences and 0.01% for more complex pulse sequences involving variable pulse-to-pulse separation.

Popular Summary

For gate errors below the fault-tolerant threshold, quantum error correction is possible in principle; however, it comes at the cost of huge hardware overhead. For state-of-the-art superconducting quantum circuits, the order of 100–1000 physical qubits will be needed to implement a single logical qubit. The control of a large-scale error-corrected superconducting quantum computer is an open technical problem. An attractive approach leverages classical superconducting digital electronics. Here, instead of resonant microwave drive, quantized flux pulses with picosecond time duration are used to excite the qubits. The approach is analogous to pushing a child on a swing: the flux pulses provide one quick push per oscillation cycle. The trains of flux pulses needed for arbitrary qubit control can be generated in the qubit cryostat using low-power superconducting electronics derived from the single flux quantum (SFQ) digital logic family. The controller can be integrated tightly with the quantum chip, providing for low-latency feedback and stabilization of the quantum array in a compact physical footprint.

In this manuscript, we implement a multichip module for SFQ-based qubit control. Segregation of the quantum and classical control circuits on two separate chips helps to isolate the delicate quantum devices from the dissipation of the flux pulse driver. With this arrangement, we achieve fidelity in excess of $99\mathrm{\%}$ for 90- and 180-degree qubit rotations. We find the residual infidelity is due to excitations out of the superconducting ground state induced by high-frequency transients from the flux driver; we believe that it will be straightforward to suppress this source of infidelity in a subsequent implementation.

Our work shows the potential for scalable control of quantum processors using low-power classical superconducting digital logic, a critical piece that will facilitate development of error-corrected qubit arrays.

Figure 1

The quantum-classical multichip module (MCM). (a) A micrograph of the qubit chip. Two flux-tunable transmons are fabricated on the chip, each with a local quarter-wave coplanar resonator for readout. (b) A micrograph of the SFQ driver chip. Two dc-to-SFQ converters are integrated on the chip, along with the feed line for the readout resonators and flux-bias lines for the qubits. The $\mathrm{In}$ bumps are visible as the regular grid pattern over the continuous ground plane. (c) A photograph showing the assembled MCM stack; the qubit chip is outlined in red and the SFQ chip is outlined in blue. (d) The circuit diagram for one qubit-SFQ pair; here, the quarter-wave readout mode is depicted using its lumped-element equivalent. $\mathrm{In}$-bump bonds between the ground planes provide the only galvanic connection between the two chips; coupling between circuit elements across the chip-to-chip gap is achieved either capacitively or inductively. The colors used in the legend to identify specific circuit elements are matched to the false coloring of the bond pads in the image of (b).

Figure 2

SFQ-based qubit operation at the subharmonic drive frequency $f_{01}/4$. (a) Rabi oscillations as a function of the current bias $I_b$ of the SFQ driver. (b) The Rabi chevron experiment at variable drive frequency in the vicinity of $f_{01}/4$. (c) The generalized Rabi scan at $f_{01}/4$, with variable time $t$ and phase $\varphi$ for the Rabi drive; the inset illustrates the relationship between the phase $\varphi$ and the relative timing of the pulse trains used to implement the $X/2$ and $R_{\mathrm{SFQ}}$ rotations.

Figure 3

(a),(b) Randomized benchmarking (RB) of SFQ-based gates with the drive at frequency $f_{01}/4$ = 1.2264 GHz: pulse sequences for (a) reference and (b) interleaved RB. (c) Depolarizing curves for the reference RB sequence and six interleaved-RB sequences. Each data point is the average of 150 random sequences across 6 h. The inset shows the fidelities of the average Clifford gate [36] and the six interleaved gates.

Figure 4

The characterization of incoherent error. (a) Purity benchmarking of SFQ-based gates at pulse frequency $f_{01}/4=1.2264$ GHz. The inset shows the gate sequence for purity benchmarking. Here, quantum state tomography is applied at the end of the $m$ randomized Cliffords. The incoherent error per Clifford, $r_{\mathrm{Cliff},\mathrm{inc}}$, is extracted from the purity of the final state. (b) The gate error derived from the RB measurements of Fig. 3 versus the gate duration. For the average Clifford gate (black diamond), incoherent errors constitute $80\mathrm{\%}$ of the total error (red circle). The blue dashed line shows the baseline incoherent error of $0.27\times {10}^{-2}$ for the average Clifford gate extracted from the coherence times of the qubit measured with microwave-based gates in the absence of SFQ operation.

Figure 5

QP poisoning induced by operation of the SFQ-pulse driver. (a) Microwave-based energy relaxation curves of the qubit with and without prior application of an off-resonant SFQ pulse train at $f_{\mathrm{SFQ}}=1.21$ GHz. The mean number $⟨n_{\mathrm{qp}}⟩$ of QPs coupled to the qubit is extracted from a fit of Eq. (1) to the data. (b) $⟨n_{\mathrm{qp}}⟩$ versus the SFQ poisoning pulse length. Each point is extracted from eight $T_1$ traces of the type shown in (a).

Figure 6

The dynamics of QP poisoning from the SFQ-pulse driver. A variable idle time follows application of a brief SFQ poisoning pulse; subsequent fast-qubit measurement is used to extract the qubit $|1⟩$-state occupation $P_1$. The inset shows the relative positions of $Q_1$ (red), $Q_2$ (blue), the dc-to-SFQ converter (black rectangle), and the SFQ-qubit coupling capacitor (green square). For both qubits, $P_1$ decays monotonically with the idle time. We observe no time lag between application of the poisoning pulse and the peaking of enhanced excess $P_1$, indicating that poisoning is mediated by pair-breaking photons, as opposed to pair-breaking phonons.

Figure 7

Antenna modeling of the qubit and the SFQ-qubit coupler. Schematics of (a) the qubit and (b) the region opposite the qubit on the classical SFQ driver chip. Here, gray represents $\mathrm{Nb}$ metallization on the two chips; in the white regions, the $\mathrm{Nb}$ has been etched away. The qubit island is a circular pad with radius $75\text{μ}\mathrm{m}$. embedded in a circular ground-plane cavity with radius $87\text{μ}\mathrm{m}$. The $\mathrm{Nb}$ has been removed in a circular region on the classical chip directly opposite the qubit island in order to reduce the capacitance between the qubit and the ground plane of the classical chip. In (a), the position of the qubit junction is indicated by the red cross. In (b), the concentric black dashed lines indicate the position of the gap between the qubit island and ground plane relative to structures on the SFQ driver chip. The patch capacitor that couples SFQ pulses to the qubit is shown at the bottom; the capacitor is realized as a square electrode with dimension $35\text{μ}\mathrm{m}\times 35\text{μ}\mathrm{m}$. embedded in a cavity in the ground plane with dimension $45\text{μ}\mathrm{m}\times 45\text{μ}\mathrm{m}$. The SFQ-qubit coupler is fed from below the image by a microstrip line from the classical SFQ driver circuit (not shown). (c) The numerically calculated free-space coupling efficiency $e_c$ of the antenna modes of the qubit and of the SFQ-qubit coupler [54]. The black curve represents $e_c$ of the qubit with the SFQ chip removed; the dominant feature at 240 GHz corresponds to the fundamental full-wave resonance of the aperture antenna formed by the qubit island embedded in the ground plane. The red and blue traces are simulation results for $e_c$ of the qubit and SFQ-coupler modes calculated in the full MCM architecture.

Figure 8

The temporal response of the qubit antenna mode to a single SFQ pulse. (a) The SFQ pulse and induced transient at the qubit. A single SFQ pulse $V_{\mathrm{SFQ}}$ with $\sigma =2.5$ ps is delivered from the SFQ-qubit coupler (blue trace). Antenna coupling between the SFQ-qubit coupler and the qubit results in an induced voltage $V_{\mathrm{qb}}$ at the qubit port (red trace; voltage scaled by a factor of 50 for clarity). (b) The energy spectral density of the qubit response expressed in units of $\mathrm{\Delta }/\mathrm{Hz}$, where $\mathrm{\Delta }=E_{\mathrm{qp}}=h\times 50\mathrm{GHz}$. The red trace is the energy spectral density calculated from the qubit transient in (a); here, ${\tilde{V}}_{\mathrm{qb}}$ is the Fourier transform of the induced voltage at the qubit and $R_{\mathrm{qb}}$ is the normal-state resistance of the qubit junction. The blue trace shows the product of the photon-coupling efficiency $\eta$ based on the frequency-domain antenna modeling and the energy spectral density of the SFQ pulse. The dominant Fourier component in the qubit response matches the product of the energy spectral density of the SFQ pulse and the photon-coupling efficiency of the coupler-qubit system.

Figure 9

The wiring diagram of the experiments.

Figure 10

The stability of the SFQ-pulse delivery and its effect on qubit control. (a) The scaled Rabi oscillation frequency $f_{\mathrm{Rabi}}$ extracted from Fig. 2 as a function of $I_b$. Over a bias current range from 40 to $120\text{μ}\mathrm{A}$, we find approximately $\pm 3\mathrm{\%}$ variation in $f_{\mathrm{Rabi}}$ around the value 6.25 MHz. A possible explanation for the variation in Rabi frequency is that the number of SFQ pulses delivered per cycle of the trigger wave form is either less than or greater than one. (b) The simulated gate error and leakage to the $|2⟩$ state for a $\pi /2$ gate implemented with an imperfect SFQ-pulse driver. The gate error and leakage both increase in the presence of SFQ-pulse dropouts and double pulses.

Figure 11

The IRB of resonant SFQ-based gates implemented at drive frequency $f_{01}/2$.

Figure 12

The Fourier transform of the Gaussian SFQ pulses with varying standard deviation $\sigma$ in the range $0.5\text{--}5$ ps. In the Fourier-domain representation of the pulse, the standard deviation is ${\sigma }_f=(2\pi \sigma {)}^{-1}$. Narrower SFQ pulses in the time domain involve more spectral weight above the $\mathrm{Al}$ energy gap $2{\mathrm{\Delta }}_{\text{Al}}/h\sim 100$ GHz, where there is the possibility of photon-assisted QP generation.