Simulation of a rapid qubit readout dependent on the transmission of a single fluxon

TL;DR

The paper proposes a fast, microwave-free qubit readout mechanism based on a ballistic fluxon scattering at an interface between two long Josephson junctions, strongly coupled to a fluxonium qubit. Through classical circuit simulations and a collective-coordinate model, it demonstrates state-dependent scattering that yields high-contrast, near-deterministic transmission or reflection with sub-nanosecond readout times. A quantum-classical treatment confirms that backaction on the qubit is extremely small (~0.1%), and the results indicate experimental viability for rapid, high-fidelity readout in superconducting qubit platforms. Overall, the work suggests a feasible path to ultra-fast qubit readout that bypasses the need for input microwave tones and leverages ballistic fluxon dynamics.

Abstract

The readout speed of qubits is a major limitation for error correction in quantum information science. We show simulations of a proposed device that gives readout of a fluxonium qubit using a ballistic fluxon with an estimated readout time of less than 1 nanosecond, without the need for an input microwave tone. This contrasts the prevalent readout based on circuit quantum electrodynamics, but is related to previous studies where a fluxon moving in a single long Josephson junction (LJJ) can exhibit a time delay depending on the state of a coupled qubit. Our readout circuit contains two LJJs and a qubit coupled at their interface. We find that the device can exhibit single-shot readout of a qubit -- one qubit state leads to a single dynamical bounce at the interface and fluxon reflection, and the other qubit state leads to a couple of bounces at the interface and fluxon transmission. Dynamics are initially computed with a separate degree of freedom for all Josephson junctions of the circuit. However, a collective coordinate model reduces the dynamics to three degrees of freedom: one for the fluxonium Josephson junction and one for each LJJ. The large mass imbalance in this model allows us to simulate the mixed quantum-classical dynamics, as an approximation for the full quantum dynamics. Calculations give backaction on the qubit at $\leq 0.1\%$.

Figures (4)

• Figure 1: (a) Schematic of the device. Between two LJJs is the circuit interface cell consisting of three JJs connected in a loop: two 'termination JJs’ (with phases $\phi_L, \phi_R$) and a 'rail’-JJ (phase $\phi^B$). A fluxonium qubit is connected in parallel with the rail-JJ. It is composed of a small JJ (with phase $\phi_{q}$) in series with a large 'superinductance' from a large-JJ array. (b) Fluxonium potential $U_q(\phi_q)$ and first energy levels of the isolated fluxonium Hamiltonian $H_q$, Eq. \ref{['eq:Hq_phiq']}, for external flux $\varphi_{\text{ext}}=0.2\pi$, and parameters $I_c^{q}/I_c \approx 0.98$, $C_J^{q}/C_J \approx 0.74$, $L_{q}/L \approx 233$, and $\beta^2 = (4e^2/\hbar) \sqrt{L/C_J} = 0.4$, giving rise to the qubit transition energy of $\hbar \omega_{01} = 2.4 E_c^q = 0.17 E_0$. The characteristic fluxonium energies fulfill $E_J^q > 4 E_c^q \gg E_L^q$. (c) The first three eigenstates of the Hamiltonian for the two coupled phases $(\phi_{q},\phi^B)$, Eq. \ref{['eq:H2_boundstate']}, are strongly localized at $\phi^B \approx 0$, and in three distinct wells of $U_q(\phi_q)$. (d) Spectrum of the isolated fluxonium Hamiltonian $H_q$ under $\phi^B$-variation. Starting from $\phi^B \approx 0$ (solid vertical line), during the fluxon scattering, $\phi^B(t)$ temporarily reaches a maximum amplitude $\lesssim \pi$ (dashed vertical lines), e.g. $\text{max}(\phi^B) \approx 0.73\pi$ if the fluxonium is initialized in state $n=1$. The qubit is thus driven close to the avoided level crossing of states $n=0,1$ at $\phi^B + \varphi_{\text{ext}} = \pi$, but nevertheless unintended state transfer remains very small, cf. panels (iii) in Fig. \ref{['fig2']} and discussion in Sec. \ref{['sec:backaction']}.
• Figure 2: Simulated classical fluxon scattering at interface coupled to fluxonium, for fluxonium parameters of Fig. \ref{['fig1']} and interface parameters $C_J^B/C_J = 11$, $I_c^B/I_c \approx 5.9$, $\hat{C}_J/C_J \approx 2.3$, and $\hat{I}_c/I_c \approx 1.3$. The fluxonium phase $\phi_{q}$ is initialized as a classical variable, in the well of the fluxonium potential corresponding to the qubit state (a) $n=0$ and (b) $n=1$. Panels (i) show the dynamics of the JJ phases $\phi_k$ at positions in the left ($x_k < 0$) and right ($x_k > 0$) LJJ, respectively. Panels (ii) shows the classical evolution of the fluxonium phase $\phi_{q}(t)$ (dashed) and rail-JJ phase $\phi^B(t)$ (solid). Using the data of $\phi^B(t)$ as an external drive to the Hamiltonian $H_q(t):=H_q(\phi_{q};\phi^B(t))$ (Eq. \ref{['eq:Hq1_phiq_phiBBbiased']}), we calculate the time-evolution of the wave function $\psi(t)$ of the driven fluxonium and the overlap coefficients $c_m(t) = \langle \psi(t) | m(t) \rangle$ with the instantaneous eigenstates $|m(t)\rangle$ of $H_q(t)$. Panels (iii) show the weights (infidelities) of the fluxonium states that were not initially excited (orange or green solid and black dashed). Comparable values of the weights $|c_m(t)|^2$ (gray solid and gray dashed, respectively) were also obtained from the mixed quantum-classical dynamics of a collective-coordinate model, see Sec. \ref{['sec:mixedqucl']}. From the final values of $\sum_{m\neq n}|c_{m}(t)|^2 /|c_{n}(t)|^2 = (1 - |c_{n}(t)|^2)/|c_{n}(t)|^2 \ll 1$ we infer a total backaction of the readout on the qubit of $\approx 0.1\%$. In both initial-state cases (a,b), $\phi^B$ almost reaches the avoided crossing between the fluxonium states $n=0,1$ at $\phi^B = 0.8\pi$, cf. Fig. \ref{['fig1']}(d), however, no significant inter-well tunneling occurs and the qubit state remained concentrated in the initial (excited) state. For example, $\max(\phi^B(t)) = 0.73\pi$ in case (b) at $\nu_J t \approx 3.5$ and at this time $\sum_{m \neq 1} |c_m(t)|^2 \approx |c_{0}(t)|^2 = 0.012 |c_{1}(t)|^2$. The expectation value $\langle \phi_{q} \rangle(t) := \langle \psi(t) | \phi_{q} | \psi(t) \rangle$ (dotted) is also shown in panels (ii), confirming $|\langle \phi_{q} \rangle(t)-\langle \phi_{q} \rangle(0)|\ll 2\pi$ due to neglible tunneling. The good correspondence between $\langle \phi_{q} \rangle(t)$ and $\phi_{q}(t)$ (dashed) suggests that the classical approximation is reliable. Case (a) (n=0) exhibits two bounces in the time duration of 3 Josephson periods, and for the achievable period of $2\pi/\omega_J$=1/12.1 $\,\text{GHz}$, the duration is $\ll$ 1 $\,\text{ns}$.
• Figure 3: The CC potentials $U(X_L,X_R,\phi_{q})$, Eq. \ref{['eq:U_CC_cl']}, with $\phi_{q}$ fixed at (a) $\langle \phi_{q} \rangle_{n=0} = 0.0095\pi$ and (b) $\langle \phi_{q} \rangle_{n=1} = 1.917 \pi$, respectively. Equipotential lines (gray) are shown at the initial energy $E_{\text{init}} = E_{\text{fl}} + U_q(\langle \phi_{q} \rangle_{n})$. Panels (a1,b1) show the components $(X_L,X_R)(t)$ and panels (a2,b2) show the component $\phi_{q}(t)$ of the trajectories $(X_L,X_R,\phi_{q},\phi^B)(t)$, which are calculated with three different methods: (i) the classical CC equations of motion, Eqs. \ref{['eq:EOM0_CC']} and \ref{['eq:EOM1_CC']} with the potential \ref{['eq:U_CC_cl']}, (red solid line in (a1,b1) and purple dashed in (a2,b2)), (ii) the mixed quantum-classical equation of motion, Eqs. \ref{['eq:EOM0_CC']} and \ref{['eq:Schroedinger_mixedqucl']} with the potential \ref{['eq:U_CC_mixedqucl']} (green dotted lines), and (iii) the results $\phi_n^{(l,r)}(t)$ of the full circuit simulation, fitted to the form of Eq. \ref{['eq:fluxoncombination']} (blue markers in panels (a1,b1)). Panels (a2,b2) also show $\phi^B=\phi_L-\phi_R$ for the first two methods ($\phi^B$ for the third method is shown in Fig. \ref{['fig2']}).
• Figure 4: Simulated scattering results, including: LJJ phase evolution, trajectory in CC potentials, and dynamics of central gate junction $\phi^B$. The data structure is equivalent to those of Figs. \ref{['fig2']} and \ref{['fig3']}. The fluxonium parameters are changed to $\varphi_{\text{ext}}=1.2\pi$, $I_c^{q}/I_c = 6.0$, $C_J^{q}/C_J = 0.6$, $L_{q}/L = 40$, and $\beta^2 = (4e^2/\hbar) \sqrt{L/C_J} = 0.4$, giving rise to the qubit transition energy of $\hbar \omega_{01} = 2.9 E_c^q = 0.25 E_0$. The interface parameters are changed to $C_J^B/C_J = 11.5$, $I_c^B/I_c = 6.7$, $\hat{C}_J/C_J = 0.75$, and $\hat{I}_c/I_c = 2.0$.