Passive quantum error correction of photon loss at breakeven

TL;DR

Photon loss is a dominant error channel in superconducting qubits; this work demonstrates autonomous quantum error correction at the breakeven point for a cavity qubit by encoding information in a binomial bosonic code and using PReSPA parity recovery. The authors implement two-stage cascaded dissipation via two continuous four-wave-mixing drive combs and a dissipative reservoir, achieving a logical coherence time of about $\tau_{process} \approx 190\ \mu\mathrm{s}$—above the single-photon lifetime of the cavity by roughly $3$–$5\%$ depending on the data set—showing that passive correction can rival active schemes under current hardware constraints. Coherence improvements arise from continuous, probabilistic correction cycles that resemble active QEC in effect but with reduced measurement overhead, while remaining robust to drive-induced heating and residual codeword distortion. This work establishes passive, autonomous QEC as a practical approach to extending bosonic-qubit lifetimes and informs future integration of higher-order dissipation, phase-space stabilization, and logical-gate operations in superconducting circuit QED.

Abstract

Physical qubits in a quantum computer are often represented by superposition states of single particles or excitations. Decay of the excitation itself is a fundamental error channel that is difficult to overcome via external drive or control techniques. Quantum error correcting codes, which encode information in superpositions involving multiple excitations, provide a path to preserve information beyond the capacity of individual excitations, but typically require exquisite active operations on the system. Here, we demonstrate a steady-state driven dissipative quantum system, composed of a superconducting cavity and a transmon ancilla, that preserves a logical qubit beyond the photon-lifetime limit by about 5% using a binomial encoding. This realization of continuous quantum error correction at the breakeven point highlights the quantitative competitiveness of passive correction strategies while circumventing some demanding hardware requirements of its active counterparts.

Figures (7)

• Figure 1: Experimental setup and correction protocol (A) The Bloch sphere representation of the logical qubit, and the measured Wigner function of the pole ($|0_{L}\rangle$, $|1_{L}\rangle$) and the equator ($|+X_{L}\rangle$, $|+Y_{L}\rangle$) states. (B) Cartoon depiction of the experimental device, showing the 3D post cavity storage resonator, transmon ancilla and quasi-planar reservoir/readout mode. One input port is used for transmon drive, transmon readout and the AQEC drives as shown. (C) Level diagram depicting the transition paths of the AQEC scheme, implemented with two combs of simultaneous continuous drives (red arrows and blue arrows) and dissipation from the reservoir mode (green squiggly arrows). Above the drive-tone arrows, four-wave-mixing diagrams denote the conversion of excitations for each drive.
• Figure 2: Characterization of PReSPA performance on even-odd conversion. (A) Starting in even-parity Fock states, we show the cavity populations in converted odd-parity Fock states over pump times. The cavity population in $\ket{n}$ presented here corresponds to $P_{\ket{n,g}}-P_{\ket{n,e}}$, hence a conservative estimate of the success probability of photon addition. (See Supplementary text for discussions.) Dashed lines: fit to the master-equation simulations, with $\Omega_{1} = 55$ kHz and $\Omega_{2} = 160$ kHz. (B) Effective rate of the 2-stage cascaded dissipation process, evaluated from the smallest imaginary component of the eigenvalues of matrix Eq. (\ref{['eq:diss_matrix']}). The $\times$ denotes the drive rates used in the experiment. (C) The transmon $|e\rangle, |f\rangle$ population for the three different cavity initial states as in A over time, using the same master equation fit as A. (D) Wigner tomography for prepared initial even-parity superposition states (top) and the odd-parity states after 15 $\mu$s of PReSPA drives (bottom). There is a deterministic phase-space rotation due to the self-Kerr of the cavity. The coherence preservation factor, defined by the ratio of the corresponding off-diagonal density matrix elements, is stated next to the arrows between the Wigner functions. All initial states are prepared using numerically-optimized control pulses heeres2017implementing.
• Figure 3: Logical qubit lifetime under AQEC. (A) The state and process fidelity lifetimes for the corrected binomial code using PReSPA and the $|0\rangle, |1\rangle$-Fock encoding, as measured over multiple days. Pole and equator state fidelities are fit to an exponential decay lifetime with error bars determined from fit uncertainties. The process fidelity lifetimes for both the encodings are computed as the inverse of its decay slope at at $t\rightarrow0$, or $\tau_{process} = 1/\left(\frac{2}{3} T_{eq}^{-1} + \frac{1}{3} T_{p}^{-1}\right)$. (B) Process fidelity over time of the binomial code with and without correction along with the $|0\rangle, |1\rangle$-Fock encoding, averaged over all data acquired in the last four days of measurement in A. The dashed red and blue curves are sum of two exponential functions two time scales $T_{eq}$ and $T_p$, computed from independent exponential fits of the state fidelity of pole and equator states. The uncorrected binomial code is fit to a sum of two exponential functions, taking into account partial information recovery from the decoding process when the cavity has lost only one photon, with the $1/e$ decay time shown in the plot.
• Figure 4: Comparison of the actions and residual error channels in continuous AQEC vs active QEC. Illustration showing an overview of the passive (top) and active (bottom) QEC schemes. The small circles in the passive case show the correction happening continuously after the error occurs, where each circle can be viewed as a finite probability in the small time step ($t_{\epsilon}$) of the correction process, outlined in the circuit above, occurring. The discrete error correction cycles in the active case are shown, outlined by the orange rectangles. A zoomed in circuit diagram for these measurement based correction cycles is shown below. The uncorrectable error pathways are shown as shaded regions, with specific errors outlined in legend in the upper right corner, where the more opaque regions correspond to larger instantaneous error rates. For example, after an initial error, the probability of a second error decreases over time in passive case thanks to the increasing chance of having completed the correction, shown as a progressively lighter blue shade over time. In the active case, the probability is a constant until reaching the next active QEC operation, thus shown by the constant blue shading, .
• Figure S1: Transmon excitations by a single-tone off-resonant drive. (A) The $|g\rangle-$state populations of the transmon driven with a single parameter ($\epsilon_1$) sweep of an off-resonant tone that is applied for $t_p=75\,\mu$s. Each curve represents a measurement repeated for a drive frequency detuned from the FWM transition, $\omega_{d1}$. The subplot below shows the $\ket{e}, \ket{f}$ and higher excited state populations, averaged across measurements with different drive frequencies. In (B), (C), the measurement is repeated for different drive pumping times $t_p=175\,\mu$s, $t_p=275\,\mu$s, respectively. The shared legend for all the figures is shown in plot (B). (D) The $\ket{g}-$state population is averaged across drive frequencies for the three pump times. The error bars represent the spread of the data across drive frequencies. The cycle 1 and cycle 2 are measurements from two iterations of the same experiment. (E) The excitation rate $\gamma_{\uparrow}$ is analyzed from population data for different drive amplitudes $\epsilon_1$. The Rabi rate $\Omega_1=55$ kHz requires relatively weak drive, $\epsilon_1\approx 26$ MHz that yields a Stark shift $\Delta_{ss}=3$ kHz and corresponds to $\gamma_{\uparrow}\approx 0.5$ ms$^{-1}$.