Converting qubit relaxation into erasures with a single fluxonium

Abstract

Qubits that experience predominantly erasure errors offer distinct advantages for fault-tolerant operation. Indeed, dual-rail encoded erasure qubits in superconducting cavities and transmons have demonstrated high-fidelity operations by converting physical-qubit relaxation into logical-qubit erasures, but this comes at the cost of increased hardware overhead and circuit complexity. Here, we address these limitations by realizing erasure conversion in a single fluxonium operated at zero flux, where the logical state is encoded in its 0-2 subspace. A single, carefully engineered resonator provides both mid-circuit erasure detection and end-of-line (EOL) logical measurement. Post-selection on non-erasure outcomes results in more than four-fold increase of the logical lifetime, from $193~μ$s to $869~μ$s. Finally, we characterize measurement-induced logical dephasing as a function of measurement power and frequency, and infer that each erasure check contributes a negligible error of $7.2\times 10^{-5}$. These results establish integer-fluxonium as a promising, resource-efficient platform for erasure-based error mitigation, without requiring additional hardware.

Figures (4)

• Figure 1: Erasure conversion in a single fluxonium.a, The logical qubit is encoded in the $\{|{0}\rangle,|{2}\rangle\}$ subspace of an integer fluxonium. The lowest three energy eigenstates are connected by spontaneous transitions $|{i}\rangle\rightarrow |{j}\rangle$ at rate $\Gamma_{ij}$. To facilitate erasure conversion, the direct transition rate $\Gamma_{20}$ should be ideally zero, such that relaxations from $|{2}\rangle$ always proceed through $|{1}\rangle$ as an intermediate state. This can be achieved by operating at the zero-flux position. b, The vanishing charge-matrix element $|\langle{0}|\hat{n}|{2}\rangle|$ at $\Phi_\text{ext} = 0$ testifies to a symmetry-induced suppression of the direct transition, which establishes the basis for erasure conversion. c, The qubit spectrum likewise reflects this symmetry-induced suppression: the spectral feature corresponding to the $|{0}\rangle\leftrightarrow|{2}\rangle$ transition vanishes at $\Phi_\text{ext} = 0$. d, False-colored scanning electron micrograph of the device. The insets provide closer views of the overlay junctions (blue box), which form the junction loop, and the Manhattan junction (red box), which provides the Josephson energy, at locations indicated by the boxes of the corresponding colors.
• Figure 2: Erasure detection and end-of-line (EOL) measurement.a, The qubit dressed resonator frequency $\omega_i$ is measured as a function of $\Phi_\text{ext}$ for the lowest three energy eigenstates of the qubit. We exploit the resulting flux dependence of the dispersive shifts $\chi_{ij} = \omega_i - \omega_j$ to enable erasure detection at $\Phi_\text{ext} = 0$ (purple arrow) and EOL measurement at $\Phi_\text{ext} = 1.3~m\Phi_0$ (pink arrow). b, c, With an average intra-cavity photon number of $n = 2.3$ and an integration time of $t_\text{meas} = 1.6~\mu$s, the erasure detection cannot distinguish $|{0}\rangle$ from $|{2}\rangle$ in the IQ-plane (b), but resolves $|{1}\rangle$ clearly. Rather than using the midpoint (dotted vertical line) between the state distributions, we intentionally bias the assignement threshold to the right (solid vertical line) to reduce the probability of misidentifying $|{1}\rangle$ as a computational state (false-negative), at the cost of increasing the chance of erroneously labeling a computational state as an erasure event (false-positive). With this choice, the histogram (c) reveals a false-negative rate of $4.9\%$ and a false-positive rate of $56.7\%$. d, At $\Phi_\text{ext} = 1.3~m\Phi_0$, with $n = 5.6$ and the same $t_\text{meas} = 1.6~\mu$s, the EOL measurement resolves all three states clearly.
• Figure 3: Converting decay errors into erasures. a, The dynamics of the qubit relaxation are measured by idling the qubit at a fixed $\Phi_\text{ext}$ for a variable wait time $t_\text{wait}$ after initialisation. The time-dependent populations of the lowest three energy states are then acquired via an EOL logical measurement. b, c, At $\Phi_\text{ext} = 0$, for example, the spontaneous transition rates $\Gamma_{ij}$ can be extracted by jointly fitting the measured dynamics for the qubit initially prepared in $|{1}\rangle$ (b) and $|{2}\rangle$ (c). d, Repeating this procedure for different $\Phi_\text{ext}$ maps out the transition-rate landscape of $\Gamma_{ij}$. Importantly, close to $\Phi_\text{ext} = 0$, the direct transition rate $\Gamma_{20}$ between $|{2}\rangle$ and $|{0}\rangle$ is suppressed by orders of magnitude compared to all other rates. This ensures that logical decays within the computational space predominantly occur via $|{1}\rangle$ as an intermediate state, thereby enabling the conversion of decay events into detectable erasures. e, To characterise erasure conversion, a qubit initialized in $|{2}\rangle$ is subjected to a series of $m$ erasure detections, each lasting $t_\text{meas} = 1.6~\mu$s and separated by a variable interval $t_\text{EC}$, resulting in a total delay of $t_\text{tot}$, before an EOL measurement is performed. f, A sample of erasure-detection outcomes for $t_\text{EC} = 5~\mu$s and $t_\text{tot} = 153.4~\mu$s is shown, where each row corresponds to a single experimental shot, and a purple-colored block on the $i$-th column indicates the detection of an erasure event during the $i$-th detection. g, Compared to ignoring the erasure information (green), post-selecting on traces where no erasure event is detected (purple) extends the characteristic logical lifetime by more than a factor of four, from $193~\mu$s to $869~\mu$s.
• Figure 4: Dephasing of the logical qubit. a, We measure the decoherence of the logical qubit using a Ramsey sequence, and extract a coherence time $T_\text{2L} = 70.4~\mu$s. b, Applying continuous erasure detections during the Ramsey free-evolution period, we extract the dephasing rate $\Gamma_\varphi = 1/T_\text{2L} - 1/(2T_\text{1L})$ as a function of readout frequency $\omega_\text{RO}$ and intra-cavity photon number $n$. The measured data (dots) are in good agreement with the theory of measurement-induced dephasing (solid lines). For erasure detections used in \ref{['fig:Fig2']} and \ref{['fig:Fig3']}, with $n = 2.3$, $t_\text{meas} = 1.6~\mu$s, and $\omega_\text{RO} = 2\pi\times 6.993~$GHz (vertical arrow), we find a dephasing error of $\epsilon_\varphi \approx 7.2\times 10^{-5}$ per check.