Proposal for erasure conversion in integer fluxonium qubits

Abstract

We propose an erasure conversion scheme on the $|e\rangle-|f\rangle$ and $|g\rangle-|f\rangle$ qubits in integer fluxonium qubits (IFQs), which are both first-order insensitive to $1/f$ flux noise. The $|e\rangle-|f\rangle$ transition is identical to that of a usual fluxonium qubit and hence is expected to have excellent coherence time, while the $|g\rangle-|f\rangle$ transition is additionally protected from the energy relaxation by the parity symmetry. The dominant error in both qubits arises due to the energy relaxation: from $|e\rangle$ to $|g\rangle$ in the $e\text{--}f$ qubit and from $|f\rangle$ to $|e\rangle$ in the $g\text{--}f$ qubit. Such errors can be treated as erasure events, and their efficient detection improves the performance of quantum error-correcting codes. We consider a protocol for such erasure conversion based on the dispersive readout. Our main finding is that, with proper circuit parameter choice, carefully designed gate sets, and the integration of erasure conversion, IFQs promise high effective coherence times.

Figures (19)

• Figure 1: (a) Circuit diagram of a fluxonium capacitively coupled to a frequency-tunable resonator. When the resonator is populated with photons from the microwave drive, the output photon from the resonator's decay carries away information about the fluxonium state. (b) Potential landscape and wavefunctions of fluxonium at integer flux bias. The depicted wavefunctions correspond to the example g–f qubit considered in this work. The degeneracy of the two states localized in $m=\pm 1$ potential wells is lifted by consecutive single-well tunneling of amplitude $\epsilon_1$. (c) The absolute value of the charge matrix elements within the lowest four levels of the example g-f qubit in Table \ref{['tab:qubit_parameters']}. The relatively small $\ket{e}-\ket{f}$ element ensures erasure error do not overwhelm the erasure conversion-correction protocol, the relatively large $\ket{g}-\ket{h}$, $\ket{f}-\ket{h}$ transition elements facilitate Raman gate operation via the intermediate state $\ket{h}$. For the e-f qubit, we want to use an even lower $\ket{e}-\ket{f}$ frequency to enhance qubit lifetime, therefore the $\ket{e}-\ket{f}$ charge matrix element will be significantly smaller. (d) In the e-f qubit, the computational level frequency lies below the environmental temperature, while the leakage channel $\ket{g}$-$\ket{e}$ frequency is above it. The primary computational errors are the decay of $\ket{f}$ to $\ket{e}$ and the heating of $\ket{e}$ to $\ket{f}$. Leakage occurs when $\ket{e}$ decays to $\ket{g}$. Symmetry forbids transitions between $\ket{f}$ and $\ket{g}$. (e) For the g-f qubit, direct transitions between computational states are forbidden. The primary error in the computational subspace arises from dephasing induced by frequency curvature with respect to external flux. Leakage occurs when $\ket{f}$ decays to $\ket{e}$.
• Figure 2: (a) Total gate time $t_\text{tot}$ of single qubit X gate on the example e-f qubit versus error rates. Erasure error to state $\ket{g}$ dominates the total error. As the gate becomes shorter, leakage out of the g-e-f manifold becomes larger than the error within the computational subspace. (b) Optimized amplitude (which is not monotonic because $\sin^2$ ramp-up/down ratios (not shown) vary), and detuning $\Delta$ for the e-f qubit X gate.
• Figure 3: (a) Illustration of the g-f qubit Raman gate and its error structure. By applying two drives at frequencies $\omega_1=\omega_{gh}-\Delta, \omega_2=\omega_{fh}-\Delta$ at a relatively large detuning $\Delta$, the decay from the intermediate state $\ket{h}$ is minimized. The large drive amplitude induces some off-resonant excitation into higher levels, which then decay and become trapped in levels outside the computational subspace. The dominant error is decay from $\ket{f}$ to $\ket{e}$. This error is qualitatively equivalent to idling and can be detected via dispersive readout. (b) Error rates of the Raman gate optimized at different detunings, at a fixed total gate time $T_{gate}=50$ ns. Bumps in the error rate indicated unwanted single- or multi-photon resonances with certain transitions. For example, at $0.45$ GHz detuning, an unwanted $\ket{g}\rightarrow\ket{h}\rightarrow\ket{4}$ is resonant, at $0.7$ GHz, the second drive is in resonance with $\ket{g}\rightarrow\ket{e}$. At good detuning choices, such as $1.4$ GHz, the majority of the error is decay that is convertible to erasure (red line); the error of the re-normalized computational subspace (black line) is an order of magnitude lower; and the leakage outside the g-e-f subspace (grey line) is even lower.
• Figure 4: Illustration of dephasing caused by photon number dependent qubit frequency shift, or AC-Stark shift. In a simplified model good enough to model our leakage detection, the amount of dephasing is proportional to the square of photon number frequency dependency $\Lambda$ (Eq. \ref{['eq:smearing_equation_combined']}).
• Figure 5: (a) For the example e-f qubit, we utilize the $\ket{0} - \ket{7}$ plasmon transition to induce large $\chi_0$. In the meantime, $\chi_1\approx\chi_2$ are small because their dispersive shifts are affected by weak transition elements. We drive at the resonator frequency dressed by $\ket{g}$, which is $2\pi \times 2.5$ MHz detuned from the frequencies dressed by $\ket{e}$ or $\ket{f}$. (Plot generated from perturbation theory zhu2013fluxoniumshift. See table \ref{['tab:chi']} for dispersive shifts, coupling strengths, and resonator frequencies from numerical diagonalization that we use in simulation.) (b) Evolution of the coherence state $\alpha$ from Monte Carlo simulation coupled to the example e-f qubit. The Husimi-Q function is shown as contours that encloses $0.1,0.5,0.9,0.99$ of the cumulative probability density. The coherent state trajectory for state $\ket{f}$ almost completely overlaps with the other computational state and is thus not drawn. The blue color stands for the fluxonium state $\ket{g}$, and the red color stands for the state $\ket{e}$. The resonator is driven at $A = 0.063$ GHz, and has photon lifetime $1/\kappa=2\times10^{-7}$ s. (c) AC-Stark shift of the example e-f qubit. It is not entirely linear, which means Eq. \ref{['eq:smearing_equation_combined']} does not accurately predict the amount of dephasing error. (d) The photon number in the resonator when coupled to computational states (black) and the qubit computational subspace error rate (purple) during the dispersive detection on example e-f qubit as a function of integration (measurement) time. The qubit error is computed by evolving the tomographic qubit states coupled with the resonator: $y_0 = \{\pm\ket{X}, \pm\ket{Y}, \pm\ket{Z}\} \otimes \ket{0}$.