Parametrically Activated Entangling Gates Using Transmon Qubits

We describe and implement a family of entangling gates activated by radio-frequency flux modulation applied to a tunable transmon that is statically coupled to a neighboring transmon. The effect of this modulation is the resonant exchange of photons directly between levels of the two-transmon system, obviating the need for mediating qubits or resonator modes and allowing for the full utilization of all qubits in a scalable architecture. The resonance condition is selective in both the frequency and amplitude of modulation and thus alleviates frequency crowding. We demonstrate the use of three such resonances to produce entangling gates that enable universal quantum computation: one $i\mathrm{SWAP}$ gate and two distinct controlled-Z gates. We report interleaved randomized benchmarking results indicating gate error rates of 6% for the $i\mathrm{SWAP}$ (duration 135 ns) and 9% for the controlled-Z gates (durations 175 and 270 ns), limited largely by qubit coherence.

Figure 1

Two-qubit circuit consisting of two capacitively coupled transmon qubits, one at fixed frequency ($F$) and one tunable ($T$). (a) Lumped-element circuit corresponding to our coupled qubits, where the tunable transmon is composed of an asymmetric SQUID with an asymmetry of 0.31. (b) Optical image of the chip. Lumped-element readout resonators are labeled $R_F$ and $R_T$. (c) Time-varying flux pulses are actuated by a current $I(t)$ applied through the flux-bias line near $T$. (d) Scanning electron micrograph of the SQUID loop of the tunable transmon. (e) First transition frequency of the flux-tunable transmon (blue) and fixed-frequency transmon (red). Biasing the flux near the maximum or minimum of the band ensures long coherence times by removing first-order sensitivity to flux noise. The effect of flux modulation $\mathrm{\Phi }(t)$ on the transmon frequency $\omega (t)$ is also shown in blue.

Figure 2

Schematic of signal delivery to the two-qubit chip studied in this paper. Qubit drive and readout tones are transmitted, and the readout tone received, by Ettus Research USRP modules running with X300 daughterboards. The combined slow- and fast-flux components are generated by a Yokogawa GS200 dc power supply and another USRP X300, attenuated by 20 dB at 4 K and delivered through a UHF low-pass filter and a custom dissipative Eccosorb filter. The on-chip flux-bias line terminates in an inductive short to ground.

Figure 3

Rabi oscillations driven by flux pulses. With a constant modulation amplitude and state preparation, we sweep pulse frequency and duration. The RF flux drive acts as a pump that compensates the energy detuning between two states. Rabi oscillations are observed when the modulation frequency matches the detuning between a pair of two-qubit states. (a),(b) With $|10⟩$ prepared we observe $|10⟩\leftrightarrow |01⟩$ swapping, and (d),(e) with $|11⟩$ prepared we observe both $|11⟩\leftrightarrow |02⟩$ and $|11⟩\leftrightarrow |20⟩$ swapping, by monitoring the population of the fixed transmon’s $|1⟩$ state. The resonant frequencies depend on the modulation amplitude as shown in thick lines in (c), while the first harmonics of the resonances are shown in thin lines. These curves are the solutions of Eqs. ((6))–((6)) for $n=1,2$ and thus reflect the form of $\mathrm{\Delta }(\tilde{\mathrm{\Phi }})$. This function is calculated using the transmon energies extracted from frequency measurements. Further details on this calculation are available in Ref. [30]. Together these plots show a concordance of theory (c), dynamical simulations (a),(e), and data (b),(d) obtained with a modulation amplitude of $0.245{\mathrm{\Phi }}_0$. Higher harmonics are also visible in this concordance, but they are left unlabeled for clarity. In (f) we show the effective coupling strength $g_{\mathrm{eff}}$ as a function of modulation amplitude for the first harmonic of each interaction [following the same color scheme as in (c),(g)], and in (g) we show the energy-level diagram of the two-qubit system.

Figure 4

Enlarged view of resonant features, similar to those shown in Fig. 3, reveals a “chevron” pattern characteristic of Rabi oscillations. For each type of gate, a modulation amplitude is chosen to provide a suitable $g_{\mathrm{eff}}$. With that amplitude held constant, the flux-pulse duration and frequency are varied and values are chosen to give the roughly the desired gate up to local phases. The patterns observed in the experimental data (left) are in good agreement with the predictions of a dynamical simulation(right), as shown here for transitions $|01⟩\leftrightarrow |10⟩$ at $\tilde{\mathrm{\Phi }}=0.318{\mathrm{\Phi }}_0$ (bottom) and $|11⟩\leftrightarrow |20⟩$ at $\tilde{\mathrm{\Phi }}=0.325{\mathrm{\Phi }}_0$ (top). The only free parameters in the model are the effective relaxation and coherence times in the presence of flux drives.

Figure 5

Two-qubit process tomography parametric cz (top) and $i\mathrm{SWAP}$ (bottom) processes. The corresponding average process fidelities are $F=0.92$ for both cz processes and $F=0.93$ for the $i\mathrm{SWAP}$ process.

Figure 6

Interleaved randomized benchmarking result for the $i\mathrm{SWAP}$ gate. In this experiment, sequence lengths of $\{2,4,6,8,16,24,32,48\}$ are used. For visual clarity, the data points for each sequence length are offset symmetrically about the true length. The fit curves, however, are not offset. The accelerated decay from the expected final state of $|00⟩$ is attributed to the error in the $i\mathrm{SWAP}$ gate.