Fast High-Fidelity Gates for Galvanically-Coupled Fluxonium Qubits Using Strong Flux Modulation

Long coherence times, large anharmonicity, and robust charge-noise insensitivity render fluxonium qubits an interesting alternative to transmons. Recent experiments have demonstrated record coherence times for low-frequency fluxonium qubits. Here, we propose a galvanic coupling scheme with flux-tunable $\mathit{\text{XX}}$ coupling. To implement a high-fidelity entangling $\sqrt{i\mathrm{SWAP}}$ gate, we modulate the strength of this coupling and devise variable-time identity gates to synchronize required single-qubit operations. Both types of gates are implemented using strong ac flux drives, lasting for only a few drive periods. We employ a theoretical framework capable of capturing qubit dynamics beyond the rotating-wave approximation as required for such strong drives. We predict an open-system fidelity of $F>0.999$ for the $\sqrt{i\mathrm{SWAP}}$ gate under realistic conditions.

Popular Summary

Quantum computers today are small, noisy, and prone to errors. To begin tackling interesting problems, it is necessary to scale up the number of qubits. However, for devices with increased qubit numbers to be useful, stringent requirements on gate fidelities must be met. In this work, we introduce a framework for constructing and analyzing quantum gates that breaks from the confinement of the rotating-wave approximation (RWA) that underlies most standard techniques. By lifting the artificial constraints imposed by the RWA, the parameter space available to explore in pursuit of high-fidelity gates grows. This freedom is particularly advantageous for the superconducting qubit called fluxonium. Recent work has shown that the fluxonium circuit is currently the least-noise-sensitive superconducting qubit and thus an interesting candidate as a competitor to the ubiquitous transmon and as a robust building block of a future quantum computer.

We propose a scheme for coupling multiple fluxonium qubits that has two distinct features: first, the coupling is tunable and can explicitly be turned off; second, the coupling scheme is particularly well suited for heavy-fluxonium qubits, which are the specific type of fluxonium qubits with the best noise insensitivity. We apply our non-RWA analysis to quantum gates in this system and predict gate fidelities greater than the current state of the art.

Future work may apply this framework to gates in larger fluxonium arrays or even different qubit platforms.

Figure 1

Galvanically-coupled heavy-fluxonium qubits. (a) A schematic of the qubit-coupler interaction. The fluxonium qubits $a$ and $b$ are each coupled to a harmonic coupler mode ${\theta }_{+}$ and a flux-tunable coupler mode ${\theta }_{-}$ but do not interact directly. (b) The circuit diagram of the device. Qubit $a$ (dark blue) and qubit $b$ (light blue) are galvanically linked to the two coupler modes ${\theta }_{\pm }$ (orange). Each loop can be threaded by an external flux ${\mathrm{\Phi }}_{\mu }$.

Figure 2

Bare wave functions and energy spectra. (a) The bare wave functions and potential of qubit $a$, located at the half-flux sweet spot. The bare-qubit transition frequency is ${\omega }_a/2\pi =62$ MHz, while the energy of the next excited state is $4.3h\mathrm{GHz}$ above the energy of the first-excited state. (b) The spectrum of qubit $a$ as a function of the qubit flux ${\mathrm{\Phi }}_a$. The wave functions and spectra of qubit $b$ are qualitatively similar to those of qubit $a$. (c) The bare wave functions and potential of the coupler ${\theta }_{-}$ mode, located at ${\mathrm{\Phi }}_c/{\mathrm{\Phi }}_0=0.27$. The energy of the first excited state is more than $10h\mathrm{GHz}$ above the coupler ground-state energy. (d) The spectrum of the coupler ${\theta }_{-}$ mode as a function of the coupler flux ${\varphi }_c$. See Table 1 for device parameters as well as Table 2 for a summary of the frequencies and anharmonicities of the bare modes.

Figure 3

The sweet-spot contour. As the coupler flux is tuned, the qubit fluxes must be simultaneously adjusted according to $\delta {\mathrm{\Phi }}_{\mu }/{\mathrm{\Phi }}_0=(-1{)}^{\mu +1}⟨0_{-}|{\theta }_{-}|0_{-}⟩/4\pi$ in order to keep the qubits at their sweet spots ${\mathrm{\Omega }}_{\mu }=0$. The qubit fluxes are measured as deviations away from the bare sweet-spot locations. The sweet-spot contour $\mathcal{C}$ lies in the semitransparent gray plane, which bisects the angle between the two qubit-flux axes. The off position where ${\mathrm{\Omega }}_{\mu }=J=0$ is marked by a black star.

Figure 4

The flux sensitivity and dispersiveness as a function of the coupler parameters $E_{Jc}$ and $E_{C-}$. The coloring indicates $|{\mathrm{\partial }}_{{\mathrm{\Phi }}_c}{J}_{-}|$, i.e., the linear sensitivity of the coupling strength with respect to the flux. The contour lines quantify the dispersiveness of the qubit-coupler interaction via the Lamb shift ${\chi }_a$. A dispersive interaction and suitable flux sensitivity are achieved in the parameter regime $E_{C-}>E_{Jc}\gtrsim E_{Lc}$. The star marks the chosen parameters (see Table 1).

Figure 5

(a) The low-energy spectrum of the coupled system. As the coupler flux is tuned, the qubit fluxes are adjusted to remain on the sweet-spot contour $\mathcal{C}$, ensuring that ${\mathrm{\Omega }}_{\mu }=0$. The full lines correspond to the exact spectrum calculated from the full-model $H$, while the dashed lines correspond to the spectrum as calculated from the effective Hamiltonian $H_{\mathrm{eff}}$. The eigenenergies are labeled according to the bare state $|ij⟩\equiv |i_a,j_b,0_{-},0_{+}⟩$, with the largest overlap with the corresponding dressed state when the coupler is in the off state (at ${\varphi }_c\approx 0.27\times 2\pi$, marked by the gray dashed line). (b) The strength $J$ of the effective $\mathit{\text{XX}}$ coupling. The device parameters can be found in Table 1.

Figure 6

A variable-time single-qubit identity pulse. We plot the fidelity of an identity operation under a single-period sinusoidal qubit-flux drive. The lines mark locations in amplitude and frequency space where $A=j_k{\omega }_d/2$ for the first five zeroes $j_k$ of $J_0$. Multiple other lines of high fidelity corresponding to larger zeros of $J_0$ are visible in the numerics. These variable-time identity gates are ultrafast, with gate times that can be small compared to the Larmor period. The pointlike region of high fidelity at $\delta \varphi =0$ corresponds to the passive identity operation. The marked points label example drive parameters used for visualizing Bloch-sphere trajectories in Fig. 7. The numerical simulations are performed using the parameters of qubit $b$, where ${\omega }_q/2\pi =37\mathrm{MHz}$.

Figure 7

(a)-(c) The Bloch-sphere trajectory in the laboratory frame of the initial states $|0⟩$ and $|1⟩$ subject to the pictured identity gates. The pulse parameters utilized here are marked in Fig. 6. Each gate achieves an identity operation with fidelity $F\ge 0.9998$.

Figure 8

A $\sqrt{i\mathrm{SWAP}}$ gate composed of a $\sqrt{\varphi \mathrm{SWAP}}$ gate and corrective $\mathit{\text{Z}}$ rotations. (a) A quantum circuit representation of the flux pulses shown in (b). The qubit flux $\delta {\varphi }_a(t)$ (blue) is modulated to achieve the required identity operation, while the coupler flux $\delta {\varphi }_c(t)$ (orange) is activated to entangle the qubits. Single-qubit $\mathit{\text{Z}}$ rotations are obtained during the time spent idling. (c) The time evolution with the initial state $|\overline{00}⟩$. The population transfer to the $|\overline{11}⟩$ or other states is negligible during the $\sqrt{\varphi \mathrm{SWAP}}$ time window. (b) The time evolution for $|\overline{01}⟩$ as the initial state. The $|\overline{01}⟩$ and $|\overline{10}⟩$ states exchange population during the segment when the coupler flux is nonzero. The closed-system fidelity of the $\sqrt{i\mathrm{SWAP}}$ gate is $F=0.9996$.

Figure 9

A schematic of the full circuit, accounting for parameter disorder.

Figure 10

A schematic of the unwanted transitions in the computational subspace. These transitions are due to virtual excitations of the higher-lying states $|\overline{{\ell }_a,m_b,1_{-},0_{+}}⟩,\ell ,m\in \{0,1\}$ mediated by the drive operator $h_c$. As an example, we show the four perturbative paths contributing to the undesired transition $|\overline{1_a,0_b,0_{-},0_{+}}⟩\leftrightarrow |\overline{1_a,1_b,0_{-},0_{+}}⟩$.