Electronically-controlled one- and two-qubit gates for transmon quasicharge qubits

TL;DR

This work addresses implementing high-fidelity gates on a protected quasicharge qubit by integrating a topological $4\pi$-periodic Josephson element with a transmon, forming a Majorana-transmon (MT) qubit. By numerically simulating a minimal Kitaev chain junction and deriving an effective two-level MT model, the authors demonstrate gigahertz-speed $R_X$ single-qubit gates controlled purely by DC gate voltages, and extend the framework to a two-qubit $R_{XX}$ gate with analogous dynamics. Charge-noise effects are analyzed via a master equation and Fermi's golden rule, revealing leakage rates that are essentially independent of chain length in the topological phase and yielding fidelities around 0.97 for realistic noise. The results point to a feasible, fully electronic quantum computing paradigm with gating speeds comparable across single- and two-qubit operations, while highlighting practical challenges such as readout and fully electronic architectures for scalable quantum computation.

Abstract

Superconducting protected qubits aim to achieve sufficiently low error rates so as to allow realization of error-corrected, utility-scale quantum computers. A recent proposal encodes a protected qubit in the quasicharge degree of freedom of the conventional transmon device, here referred to as the `quasicharge qubit'. Operating such a protected qubit requires implementing new strategies. Here we show that an electronically-controllable tunnel junction formed by two topological superconductors can be used to implement single- and two-qubit gates on quasicharge qubits. Schemes for both these gates are based on dynamical $4π$-periodic Josephson effect and therefore have gate speeds of the same order. The simulation of the dynamics of a topological Josephson junction in a parameter regime with non-negligible charging energy is the key novelty of this work. We also characterize the robustness of such gate operations against charge noise using Fermi's golden rule. Our results point to a compelling strategy for implementation of quasicharge qubit gates based on junctions of minimal Kitaev chains of quantum dots.

Figures (6)

• Figure 1: (a) A transmon (with degrees of freedom $\hat{n}$ and $\hat{\phi}$) consisting of a capacitor of charging energy $E_{C}$ in parallel with a Josephson junction with Josephson energy $E_{J}$ is coupled in parallel to a topological superconducting junction (crosshatched) that supports Majorana zero modes $\hat{\gamma}_{j}$ (open rectangles) on the boundaries between the topological and the trivial phases. (b) A physical setup realising the circuit of (a). Each of the two Kitaev chains (crosshatched) is proximity coupled to the superconducting islands on the left and right -- each hosting Majorana zero modes $\hat{\gamma}_{I,j}$ and $\hat{\xi}_{I,j}$ at its ends. The grey region represents a bulk superconductor. The pairing amplitude $|\Delta|$ is related to the bulk superconducting gap. A Josephson junction (single-hatched) with gauge-invariant phase difference $\hat{\phi}$ couples the floating superconducting island to the grounded superconductor, defining a transmon. The "twist" in the Kitaev chains indicates the tunable weak link. A gate voltage $V$ controls the single-electron tunnelling amplitude $w$ between the two sides of the wire.
• Figure 2: The first two energy bands of $\hat{H}_{\rm T}$ in an extended Hilbert space in which the restriction on phase eigenstates to be $2\pi$-periodic is lifted. The energy levels of the traditional transmon are at the points $\kappa=0$ in each band. The states proposed by Ref thanh_le_building_2020 to encode a protected qubit, $\ket{\psi_{0}}$ and $\ket{\psi_{1}}$, are at the centre and the edge of the Brillouin zone in the first band. This plot was generated for the parameters $E_{J}/E_{C} = 1$ which is distinct from the transmon regime. The lower band is essentially flat for large $E_{J}/E_{C}$ ratios.
• Figure 3: The spectrum of the effective Hamiltonian in \ref{['eq:tunnelling-hamiltonian-eff']} as $\theta$ is advanced by $4\pi$. The spectrum is $2 \pi$-periodic while the eigenstates are $4\pi$-periodic due to energy level crossings at $\theta = \pi, 3\pi$. There is no hybridisation at the crossing due to conservation of the fermion local to the junction.
• Figure 4: The system was initialised in the state $\rho_{0} = \ket{\tilde{\psi}_{0}}\bra{\tilde{\psi}_{0}} \otimes \ket{\Omega}\bra{\Omega}$ and evolved according to the Hamiltonian in \ref{['eq:hamilonian-jordan-wigner']} with $\tilde{H}_{T}(\hat{\phi})$ expressed in the charge basis with a charge truncation of $n = 5/2$. The solid blue curve is the probability for the system to be in state $\rho_{0}$ at time $t$. The solid orange curve is the probability for the system to be in state $\rho_{1} = \ket{\tilde{\psi}_{1}} \bra{\tilde{\psi}_{1}} \otimes \ket{\Omega} \bra{\Omega}$. The dashed curves indicate the same probability as the solid curve of the matching colour but for a device evolving with charge noise on the single-electron tunnelling junction (modelled by \ref{['eq:noisy-evolution']}). This charge noise was modelled as Gaussian white-noise with power spectrum $S = \frac{0.03}{2\pi} \,\mu$eV$^{-1}$. The solid green curve is the purity of the transmon subsystem under unitary evolution while the dashed green curve is the same purity under charge noise. The physical parameters used in this simulation were $E_{J} = 1\,\mu$eV with $E_{J}/E_{C} = 200$, $w_{F} = 12\,\mu$eV dvir_realization_2023, $w = 3\,\mu$eV.
• Figure 5: Plot of leakage rate out of $\mathcal{H}_{MT}$ with length of the Kitaev chains in the presence of Gaussian distributed white charge noise with power spectrum $S(\omega) = \frac{0.03}{2\pi} \, \mu$eV$^{-1}$. After a transient regime at small chain lengths, the leakage rate for each set of parameters becomes constant after $L = 5$. The "sweet spot" is at $\mu = 0$, $w_{F} = 12 \,\mu$eV. Though shown for completeness, the orange and green curves should not be trusted for $L < 5$ (see \ref{['app:leakage']}).