Full characterization of measurement-induced transitions of a superconducting qubit

TL;DR

This work investigates how strong, off-resonant drive used for fast dispersive readout of a transmon induces state transitions that degrade QND fidelity. It identifies the dominant leakage mechanism as Raman-like inelastic scattering of readout photons, with rates predicted by a parameter-free theory linked to the environment impedance and the AC Stark shift; it also uncovers resonant leakage via spurious modes and multi-excitation resonances that depend on the qubit frequency. The authors validate the theory with detailed measurements of transition rates across drive powers and qubit frequencies, and propose engineering strategies to suppress leakage. The results provide a comprehensive framework for designing high-fidelity, fast QND readout and have implications for quantum error correction and control of superconducting circuits.

Abstract

Repeated quantum non-demolition measurement is a cornerstone of quantum error correction protocols. In superconducting qubits, the speed of dispersive state readout can be enhanced by increasing the power of the readout tone. However, such an increase has been found to result in additional qubit state transitions that violate the desired quantum non-demolition character of the measurement. Recently, the readout of a transmon superconducting qubit was improved by using a tone with frequency much larger than the qubit frequency. Here, we experimentally identify the mechanisms of readout-induced transitions in this regime. In the dominant mechanism, the energy of an incoming readout photon is partially absorbed by the transmon and partially returned to the transmission line as a photon with lower frequency. Other mechanisms involve the excitation of unwanted package modes, decay via material defects, and, at higher qubit frequencies, the activation of undesired resonances in the transmon spectrum. Our work provides a comprehensive characterization of superconducting qubit state transitions caused by a strong drive.

Figures (4)

• Figure 1: Transmon state transitions caused by inelastic scattering of drive photons. (a) Dispersive readout of a transmon qubit is achieved by elastically scattering microwave photons off the readout resonator coupled to the qubit. Since the resonator frequency depends on the qubit state, the phase of the reflected signal can be used to infer whether the state is $|0\rangle$ or $|1\rangle$. Dispersive shift $\chi$ quantifies the difference of resonator frequencies for the two computational states. (b) Level diagram of the transmon. Colored horizontal lines show computational states while grey lines show non-computational states. (c) Due to the transmon non-linearity, readout photons with frequency $\omega_\mathrm{\rm in}$ can scatter inelastically by giving off part of their energy to the qubit and producing a photon at a lower frequency $\omega_\mathrm{\rm out}$. This process leads to the leakage error where transmon prepared in a computational state $|0\rangle$ excites to state $|2\rangle$. Similar process leads to excitation $|1\rangle\rightarrow |3\rangle$. (d) Schematic of the leakage process mediated by inelastic scattering. (e) Inelastic scattering is governed by a four-wave mixing non-linearity of the transmon. The rate of inelastic scattering is proportional to the drive power, quantified by the number of photons in the resonator $\bar{n}$. It is also proportional to the dissipative part of the impedance $Z[\omega]$ of the transmon island evaluated at the frequency $\omega_\mathrm{\rm out}$. Here, $R_Q = h/e^2\approx 25.8\:\mathrm{k}\Omega$ is the resistance quantum. (f) Dissipative part of the impedance $Z[\omega]$ computed withing lumped-element model of panel (a).
• Figure 2: Transition rates of the transmon in the presence of the drive. The qubit is tuned to frequency $\omega_\mathrm{q}/2\pi = 758\:\mathrm{MHz}$ (working point of Ref. kurilovich_high-frequency_2025). The drive frequency $\omega_\mathrm{\rm in}/2\pi = 9280\:\mathrm{MHz}$ is close to that of the readout resonator, $\omega_\mathrm{res}/2\pi = 9227\:\mathrm{MHz}$. (a) False-color microscope image of the device. Quarter-wavelength readout resonator (blue) is capacitively coupled to the transmon island (red). The resonator is inductively coupled to the transmission line (cyan). (b) Zoom-in on the region of the device containing the Josephson junctions. Two Josephson junctions are arranged in a loop. This allows us to tune the frequency of the device by threading magnetic flux through the loop. (c) Single-shot histogram of resonator measurements after intentionally scrambling the state of the transmon with a $\pi/2$-pulse. The measurement can resolve computational states $|0\rangle$ and $|1\rangle$ as well as the non-computational states $|2\rangle$, $|3\rangle$, $|4\rangle$. All non-computational states higher than $|4\rangle$ are lumped into a single distribution. The population of non-computational states in the histogram stems from transitions caused by inelastic scattering of readout photons. (d) Pulse sequence for rate measurement consists of a pair of measurements separated by a drive pulse of a variable duration and amplitude. We infer the rate by comparing the outcomes for different pulse durations. (e) Transition rates as a function of drive power. The power is quantified by the absolute value of the AC Stark shift experienced by the transmon, $\delta \omega$ (and $\delta f = \delta\omega/2\pi$). The shown range of powers is determined by relevance for qubit readout (see Ref. kurilovich_high-frequency_2025). Vertical dashed lines show the maximum power level reached by the optimal readout pulse in Ref. kurilovich_high-frequency_2025. Left panel: the rate of transitions $|0\rangle\rightarrow |2\rangle$ and $|1\rangle \rightarrow |3\rangle$ linearly increases with power. We attribute this to inelastic scattering of readout photons. Solid lines show the prediction of a parameter-free theory, see Eq. \ref{['eq:gamma_m_m+2']}. Right panel: transition rates $\Gamma_{0\rightarrow 1},\Gamma_{1\rightarrow 0}, \Gamma_{1\rightarrow 2}$ are roughly power independent. Transitions from $|1\rangle$ to $|4\rangle$ and higher states are strongly suppressed for the range of powers relevant for readout. They appear at highest powers, $\delta\omega/\omega_\mathrm{q}\sim 0.1$, but their rate remains small compared to that of other transition channels. The rate of transitions from $|0\rangle$ to states $|4\rangle$ and higher is outside of the plotted range.
• Figure 3: Transition rates $\Gamma_{0\rightarrow 2}$ and $\Gamma_{1\rightarrow 3}$ as a function of qubit frequency controlled with flux bias. Both plots show rates measured at three different values of AC Stark shift $\delta\omega$. Grey dashed line indicates the working point of Ref. kurilovich_high-frequency_2025. Solid lines show the prediction of the inelastic scattering theory, Eq. \ref{['eq:gamma_m_m+2']}, where $Z[\omega]$ is computed within the lumped element model of the device. The deviations between the theory and experiment stem from two sources. The first is our imperfect knowledge of $Z[\omega]$ away from the resonator frequency (impedance mismatches). The second is higher-order non-linear processes involving high-frequency modes, see Section \ref{['sec:full']} for details.
• Figure 4: Full characterization of undesired drive-induced transitions of a transmon qubit. (a-d) Transition rates from $|1\rangle$ to various final states plotted as a function of qubit frequency (controlled with magnetic flux) and drive power (quantified by AC Stark shift $\delta f = \delta\omega/2\pi$). The drive frequency $\omega_\mathrm{\rm in}/2\pi=9280\:\mathrm{MHz}$ is close to that of the readout resonator ($\omega_\mathrm{res}/2\pi=9223-9240\:\mathrm{MHz}$ depending on the flux). (a,b) Rates $\Gamma_{1\rightarrow 0}$ and $\Gamma_{1\rightarrow 2}$. These transitions do not involve the drive photons. The stripey structure is determined by resonances between the AC-Stark shifted qubit frequency and modes in the environment. The frequency shift between the stripes in (a) and (b) is related to difference between $\omega_{1,0}$ and $\omega_{2,1}$ of about $2\pi \cdot 40\:\mathrm{MHz}$. (c) Rate $\Gamma_{1\rightarrow 3}$. Smooth background corresponds to the inelastic scattering process described in Fig. 1. Sharp features corresponds to the higher-order inelastic processes [see panels (e) and (f) for details]. (d) Rate $\Gamma_{1\rightarrow 4+}$ of transitions to $|4\rangle$ and higher states ($|5\rangle$, $|6\rangle$ and higher). These transitions stem either from higher order inelastic processes [see panel (g)] or from activation of multi-excitation resonances in the transmon spectrum [see panel (h)]. The splitting of the transition lines is explained in the main text. (e) The behavior of the sharp feature in $\Gamma_{1\rightarrow 3}$ at $\omega_\mathrm{q}/2\pi=670\:\mathrm{MHz}$ as a function of the drive frequency $\omega_\mathrm{\rm in}$ and power. Black dashed line corresponds to the condition $3\omega_\mathrm{\rm in} = \omega_{3,1}[\delta\omega] + \omega_m$ describing a six-wave mixing process involving a mode with frequency $\omega_{m}/2\pi=26.72\:\mathrm{GHz}$ in the qubit environment. (f) The power-dependence of $\Gamma_{1\rightarrow 3}$ associated with the feature of panel (e). Consistently with the described six-wave mixing process, the rate scales as the cube of power. (g) The behavior of the sharp feature in $\Gamma_{1\rightarrow 4+}$ at $\omega_\mathrm{q}/2\pi = 1070\:\mathrm{MHz}$ as a function of drive-frequency and power. The position of the feature is consistent with condition $3\omega_\mathrm{\rm in} = \omega_{5,1}[\delta\omega] + \omega_{m^\prime}$, where $\omega_{m^\prime} = 24.150\:\mathrm{GHz}$. (h) Reading out the resonator at a lowered frequency allows us to resolve higher excited states of the transmon. The readout histogram shows that the qubit transitions to state $|8\rangle$ after exciting the sharp feature in $\Gamma_{1\rightarrow 4+}$ at $\omega_\mathrm{q}/2\pi = 1500\:\mathrm{MHz}$.