Tunable Inductive Coupler for High-Fidelity Gates Between Fluxonium Qubits

The fluxonium qubit is a promising candidate for quantum computation due to its long coherence times and large anharmonicity. We present a tunable coupler that realizes strong inductive coupling between two heavy-fluxonium qubits, each with approximately $50$-MHz frequencies and approximately $5$-GHz anharmonicities. The coupler enables the qubits to have a large tuning range of $XX$ coupling strengths ($-35$ to 75 MHz). The $ZZ$ coupling strength is $<3$ kHz across the entire coupler bias range and $<100$ Hz at the coupler off position. These qualities lead to fast high-fidelity single- and two-qubit gates. By driving at the difference frequency of the two qubits, we realize a $\sqrt{i\mathrm{SWAP}}$ gate in 258 ns with fidelity 99.72%, and by driving at the sum frequency of the two qubits, we achieve a $\sqrt{b\mathrm{SWAP}}$ gate in 102 ns with fidelity 99.91%. This latter gate is only five qubit Larmor periods in length. We run cross-entropy benchmarking for over 20 consecutive hours and measure stable gate fidelities, with $\sqrt{b\mathrm{SWAP}}$ drift ($2\sigma$) $<0.02\mathrm{\%}$ and $\sqrt{i\mathrm{SWAP}}$ drift $<0.08\mathrm{\%}$.

Popular Summary

To implement transformative quantum computing algorithms, we need to create carefully controlled interactions between pairs of quantum bits. These two-qubit gates need to consistently perform well, in a metric referred to as gate fidelity. One promising platform for creating quantum processors is the heavy-fluxonium qubit, a circuit that breaks many conventional superconducting circuit paradigms with its low qubit frequencies and large anharmonicities. By leveraging the unique characteristics of the fluxonium qubit, we have realized a two-qubit gate with very high gate fidelities.

Our method for achieving such high-fidelity gates is to implement a tunable interaction between the fluxonium qubits. A key feature of our circuit’s coupler is that we can turn qubit interactions off, allowing us to achieve state-of-the-art single-qubit coherences and gate fidelities. However, when we want to execute a two-qubit gate, we can use an on-demand microwave signal to activate an extremely strong coupling—one that is an order of magnitude higher than previous couplers on such qubits. The combination of these two features enables our high gate fidelities, providing direction for future two-qubit-gate implementations on low-frequency qubits.

Our research demonstrates that heavy-fluxonium qubits can be used for immediate quantum applications and can also be a fundamental component of tomorrow’s scalable quantum information processors.

Figure 1

The device, circuit, and energy-level diagram. (a) Top panel: False-colored scanning electron microscope image of the two fluxoniums and coupler junction loops. Each qubit consists of a small Josephson junction (pink and yellow) and an array of large junctions as an inductor (blue and cyan). The coupler consists of a small Josephson junction (green) and a shorter array of junctions (orange). Bottom panel: optical microscope image of the flux control lines (light blue), readout resonators (purple), qubit shunting capacitors (red), and resonator drive and readout lines (brown). (b) Circuit diagram for the coupled fluxoniums with the tunable coupler. The colors of the components correspond to the colors in the device images. (c) Energy-level diagram of the heavy-fluxonium wave functions at the flux-frustration point (${\mathrm{\Phi }}_{\text{ext}}={\mathrm{\Phi }}_0/2$), plotted using qubit-B parameters. The gray line represents the potential well, and the red line is at the readout resonator frequency. The first six energy eigenstate wave functions are plotted with solid colors. The qubit frequencies, anharmonicities, and resonator detunings for qubits A and B are indicated with black arrows.

Figure 2

Measurements of the $XX$ and $ZZ$ coupling strength. (a) Top panel: the measured $|01⟩$, $|10⟩$, and $|11⟩$ frequencies as a function of the coupler flux as the qubits are tuned to be along the “sweet-spot contour.” The dashed lines are from full numerical simulations with parameters extracted from qubit spectroscopy (see Appendix pp1). Bottom panel: the $XX$ and $ZZ$ coupling strength (diamonds) extracted from the data (dots). Numerical-simulation predictions are plotted as the dashed lines. The $ZZ$ coupling strength is $<3$ kHz across the entire coupler flux range. (b) A Ramsey experiment with qubit B at the coupler “off position,” with the qubits initialized in either $|01⟩$ or $|11⟩$. The frequency difference $(f_{11}-f_{10})-(f_{01}-f_{00})$, extracted using Ramsey fits with different qubit initial states, measures the $ZZ$ coupling strength to be $<100$ Hz. (c) We perform this same measurement on qubit A while sweeping its flux away from the “sweet-spot contour.” We find that the $ZZ$ coupling strength becomes nonzero, which we use as an indicator for fine-tuning qubit flux biases.

Figure 3

The sweeping coupler drive frequencies near the difference ($f_{01}-f_{10}$) and sum ($f_{01}+f_{10}$) of the qubit frequencies generates chevron patterns. (a) The $|01⟩\leftrightarrow |10⟩$ chevron centered at 13.3 MHz. (b) The $|00⟩\leftrightarrow |11⟩$ chevron centered at 110.2 MHz. (c) After initializing the qubit to each of the four basis states, we drive the coupler at the optimal frequency and measure the population of qubit A. When driving at the difference frequency (13.3 MHz), a correlated oscillation occurs between $|01⟩$ and $|10⟩$, where one oscillation period takes 1200 ns. The solid line indicates the best fit to the data. (d) When driving at the sum frequency (110.2 MHz), correlated oscillations occur between $|00⟩$ and $|11⟩$, where one oscillation period takes 400 ns.

Figure 4

The measured single- and two-qubit gate fidelities. (a) Simultaneous single-qubit randomized benchmarking. The average single-qubit Clifford-gate fidelity of qubit A (qubit B) is 99.94% (99.95%). (b) Cross-entropy benchmarking data, where “Reference” consists of layers of single-qubit gates and “Interleaved” includes an interleaved $\sqrt{b\mathrm{SWAP}}$ gate. The depolarizing error is extracted following the procedure in Ref. [39]. The “Reference” gate fidelity is 99.90% and the “Interleaved” gate fidelity is 99.83%. This results in a $\sqrt{b\mathrm{SWAP}}$ gate fidelity of 99.93%. (c) $>25$-h-long consecutive cross-entropy benchmarking runs for $\sqrt{i\mathrm{SWAP}}$ gates, showing an average gate fidelity (dashed line) of 99.72%. The error bars per point indicate the statistical-fit uncertainty. (d) $>20$-h-long consecutive cross-entropy benchmarking runs for $\sqrt{b\mathrm{SWAP}}$ gates, showing an average gate fidelity (dashed line) of 99.91%.

Figure 5

The plasmon-spectroscopy features of both qubits. (a) The $|00⟩\to |20⟩$ and $|00⟩\to |30⟩$ transition frequency versus the qubit-A flux bias. (b) The $|00⟩\to |02⟩$ and $|00⟩\to |03⟩$ transition frequency versus the qubit-B flux bias. (c) An enlargement of the plot around ${\mathrm{\Phi }}_{\text{ext},B}=0$. The data were taken with the coupler flux around 0 and the flux of the other qubit fixed at 0.

Figure 6

The $ZZ$ interaction strength (a) ${\zeta }_s$ obtained from the effective model and (b) ${\zeta }_{\text{tot}}$ from numerical diagonalization of the full model. Both quantities are plotted as a function of the qubit flux shifts ${\mathrm{\Phi }}_s\equiv {\mathrm{\Phi }}_{\text{ext},A}-0.5{\mathrm{\Phi }}_0=0.5{\mathrm{\Phi }}_0-{\mathrm{\Phi }}_{\text{ext},B}$ and the coupler flux ${\mathrm{\Phi }}_{\text{ext},C}$. In both plots, the red circle marks the off position and the solid curves are the “sweet-spot contours.” We find that ${\zeta }_s$ is the main contributor to ${\zeta }_{\text{tot}}$ and that ${\zeta }_s$ can be tuned from positive to negative values to cancel out $\zeta$.

Figure 7

The wiring diagram inside the dilution refrigerator. Outside the dilution fridge, there is 13–20 dB of attenuation and a dc block on the rf-flux line and an ultra-low-pass ($<1$ Hz) $RC$ filter on the dc-flux line. Adding attenuation on the rf-flux lines at room temperature leads to a measured increase in the $T_1$ and $T_2$ coherences of the qubits. MC refers to mixing chamber.

Figure 8

A histogram of the readout data after initializing the qubit in either the ground or the excited state via active reset. The populations are then fitted to a double-Gaussian function and subsequently plotted as red and green solid lines. This fit is analyzed to determine the qubit temperature.

Figure 9

(a) Performing a drive at the sum of qubit frequencies with various lengths while sweeping the phase of a cancellation pulse. The amplitude of this sweep is randomly chosen but it needs to be lower than the correct cancellation amplitude. (b) With the phase calibrated in (a), performing a drive at the sum of qubit frequencies with various lengths while sweeping the amplitude of a cancellation pulse.

Figure 10

(a) Playing the pulse sequence in Fig. 12 while sweeping the phase of qubit-A phase gates ($Z_A$). At ${\varphi }_A={357}^{\circ }$, the phase of ${\varphi }_{01}+{\varphi }_{10}$ is perfectly cancelled; thus ${\varphi }_{01}+{\varphi }_{10}=3$. (b) Playing the pulse sequence in Fig. 13 with various copies of the $\sqrt{b\mathrm{SWAP}}$ gates and measuring the final state of both qubits. With this sequence, qubit A should only see a decay process to thermal equilibrium and the qubit-B state dependence on the number of gates is shown in Eqs. (I3). By fitting the data, we can extract the value of ${\varphi }_{01}$ and ${\varphi }_{10}$.

Figure 11

Direct preparation of the $|gg⟩+i|ee⟩$ Bell state using a $\sqrt{i\mathrm{SWAP}}$ gate on the initialized $|gg⟩$ state. (a) State tomography provides experimental data of the density matrix of this state. On the left is the real part and on the right is on the imaginary part. The purity and state fidelity are 95%, limited by SPAM errors. (b) The density matrix of the ideal $|gg⟩+i|ee⟩$, calculated using the theory, for comparison.

Figure 12

The gate sequence for calibrating the $\sqrt{b\mathrm{SWAP}}$ gate phase ${\varphi }_{11}$.

Figure 13

The gate sequence for calibrating the $\sqrt{b\mathrm{SWAP}}$ gate phases ${\varphi }_{10}$ and ${\varphi }_D$.

Figure 14

(a) The experimental data of the process tomography on the $\sqrt{i\mathrm{SWAP}}$ gate, where the left side shows the real part and the right side shows the imaginary part. (b) The process tomography of an ideal $\sqrt{i\mathrm{SWAP}}$ gate, calculated using the theory.