Balanced cross-Kerr coupling for superconducting qubit readout

TL;DR

The paper introduces junction readout, a nonperturbative cross-Kerr scheme to read out superconducting qubits by coupling a transmon to a readout resonator through a Josephson junction in parallel with a capacitor. The balanced cross-Kerr condition cancels the unwanted exchange (Jaynes–Cummings) coupling, suppressing Purcell decay and multiphoton-induced ionization while maintaining a large, nearly frequency-independent cross-Kerr shift $|\chi_z|/2\pi \sim 2$–$10\ \mathrm{MHz}$ and small resonator self-Kerr $|K_r|/2\pi \lesssim 500\ \mathrm{kHz}$. Numerical simulations show fast, high-fidelity, QND readout with fidelities $>99.99\%$ in $\lesssim 30$ ns, robust against gate-charge fluctuations and compatible with current hardware, with quantum optimal control further reducing readout time to $\sim18$ ns. The work also analyzes readout despite finite $T_1$ and reduced efficiency, demonstrates enhanced Purcell lifetimes, and explores practical circuit variants that remove flux loops, offering flexible, scalable alternatives to dispersive readout for next-generation quantum processors.

Abstract

Dispersive readout, the standard method for measuring superconducting qubits, is limited by multiphoton qubit-resonator processes arising even at moderate drive powers. These processes degrade performance, causing dispersive readout to lag behind single- and two-qubit gates in both speed and fidelity. In this work, we propose a novel readout method, termed "junction readout". Junction readout leverages the nonperturbative cross-Kerr interaction resulting from coupling a qubit and a resonator via a Josephson junction. Furthermore, by adding a capacitive coupling in parallel to the junction, Purcell decay can be suppressed without the need for a Purcell filter. We also show that junction readout is more robust against deleterious multiphoton processes, and offers greater flexibility for resonator frequency allocation. Crucially, junction readout achieves superior performance compared to dispersive readout while maintaining similar hardware overhead. Numerical simulations show that junction readout can achieve fidelities exceeding $99.99\%$ in under $30$ ns, making it a promising alternative for superconducting qubit readout with current hardware.

Figures (15)

• Figure 1: Junction readout circuit (a) with and (b) without flux loop or low-frequency mode. A transmon (green) is coupled to the readout resonator (blue) through a Josephson junction (red) and a capacitor. Unlike in (a), in (b) the transmon nonlinearity is entirely inherited from the coupling junction, see Ref. Supplementary_Information. (c) Jaynes–Cummings induced Purcell lifetime $T_1^{\mathrm{Purcell}}$ of the transmon for varying capacitive coupling strengths $J$. The black dashed line indicates where the cancellation condition of \ref{['balanced_coupling']} is met. There, $J/2 \pi \simeq 32.8 \: \textrm{MHz}$, corresponding to a coupling capacitance around $10 \: \textrm{fF}$. The qubit and resonator frequencies are $\omega_q/2\pi = 5.672$ GHz and $\omega_r/2\pi = 9.375$ GHz, respectively, and the resonator decay rate is $\kappa/2\pi = 8$ MHz, see Sec S2 of Supplementary_Information.
• Figure 2: (a) Cross-Kerr coupling $\chi_z$ and (b) resonator self-Kerr $K_r$ between the transmon and the resonator for varying resonator impedance $Z_r$ and coupling junction energy $E_{Jc}$. The contour lines indicate lines of constant (a) cross-Kerr ranging from $-2$ to $-10 \: \textrm{MHz}$ and (b) self-Kerr ranging from $-100$ to $-800 \: \textrm{kHz}$. The star marks the parameter used in the readout simulations of \ref{['fig:readout_performance']}.
• Figure 3: (a) Critical photon number averaged over gate charge $n_g \in [0,0.5]$ as a function of resonator frequency for junction readout (blue) and dispersive readout (red). In both cases, the dispersive shift is fixed at $|\chi_z|/2\pi \simeq 9$ MHz. Dashed horizontal line indicates $n_{\rm{crit}} = 30$. (b) Minimum critical photon number over $n_g \in [0,0.5]$. Inset: ratio of the minimum critical photon numbers between junction and dispersive readout. Dashed horizontal line indicates where the ratio is $1$.
• Figure 4: (a) IQ-plane of 11,200 single-shot heterodyne readout simulations of a transmon with junction readout. The black dashed line is the optimal discriminator of the two blobs corresponding to the qubit prepared in the ground (blue) or excited state (red). (b) Histogram of the integrated signal using an optimal discriminator. For both (a) and (b) the integrated time is $t_m \simeq 20 \: \textrm{ns}$. (c) Assignment error obtained from the stochastic heterodyne readout simulations compared to coherent state approximated assignment error. Using the coherent state approximated assignment error, we also show the assignment error for readout efficiency of $\eta = 0.5$ (where $0 \leq \eta \leq 1$) as well as when $T_1$ is $30$ or $120 \: \mu s$. Moreover, we show that quantum optimal control (QOC) further improves the readout fidelity. Here, the transmon charging energy is $E_C / 2 \pi = 300 \: \textrm{MHz}$ with $E_{J, \rm{total}}/E_C = 50$ and gate charge $n_g = 0.0$. The resonator frequency is $\omega_r / 2 \pi = 9.375 \: \textrm{GHz}$, with resonator impedance $Z_r = 40 \: \Omega$ and decay rate $\kappa/2\pi = 2 |\chi_z| \simeq 20$ MHz. The junction coupling strength is $E_{Jc} / 2 \pi = 8 \: \textrm{GHz}$. This set of parameter results in a cross-Kerr of $|\chi_z| / 2\pi \simeq 10 \: \textrm{MHz}$, and a critical photon number of $n_{\rm{crit}} = 65$.
• Figure S1: Critical photon numbers for varying resonator frequencies $\omega_r$. The red dots indicate when there is no coupling capacitance, and the blue dots are for where the cancellation condition is met. Here the transmon parameters are the same as in Fig. 4 and the resonator impedance is $Z_r = 35 \: \Omega$. The coupling junction energy is $E_{Jc} / 2\pi = 7.8$ GHz, resulting in $|\chi_z|/2 \pi \simeq 9$ MHz when the cancellation condition is met.