Autonomous quantum error correction beyond break-even and its metrological application

Abstract

The ability to extend the lifetime of a logical qubit beyond that of the best physical qubit available within the same system, i.e., the break-even point, is a prerequisite for building practical quantum computers. So far, this point has been exceeded through active quantum error correction (QEC) protocols, where a logical error is corrected by measuring its syndrome and then performing an adaptive correcting operation. Autonomous QEC (AQEC), without the need for such resource-consuming measurement-feedback control, has been demonstrated in several experiments, but none of which has unambiguously reached the break-even point. Here, we present an unambiguous demonstration of beyond-break-even AQEC in a circuit quantum electrodynamics system, where a photonic logical qubit encoded in a superconducting microwave cavity is protected against photon loss through autonomous error correction, enabled by engineered dissipation. Under the AQEC protection, the logical qubit achieves a lifetime surpassing that of the best physical qubit available in the system by 18\%. We further employ this AQEC protocol to enhance the precision for measuring a slight frequency shift, achieving a metrological gain of 6.3 dB over that using the most robust Fock-state superposition. These results illustrate that the demonstrated AQEC procedure not only represents a crucial step towards fault-tolerant quantum computation but also offers advantages for building robust quantum sensors.

Figures (4)

• Figure 1: Schematic comparison of the measurement-based QEC and AQEC procedures, and the codewords used in the experiment.a, Conventional measurement-based QEC procedure, with each cycle consisting of three stages: error detection with ancilla measurement, real-time error correction based on the error syndrome measurements, and entropy removal to reset ancilla via classical feedback. This measurement-based QEC risks propagating ancilla measurement errors to the logical qubit. b, AQEC employs a coherent controlled-unitary operation $\mathcal{R}=|\psi_\mathrm{L}\rangle\langle \psi_\mathrm{L}|\otimes |g\rangle \langle g| + |\psi_\mathrm{L}\rangle\langle \psi_\mathrm{E}|\otimes |e\rangle \langle g|$ to perform autonomous error correction without ancilla measurement, and the ancilla is then reset through engineered dissipation to remove the error entropy to a lossy bath. Here $|\psi_\mathrm{L/E}\rangle$ is the code/error state of the logical qubit and $|g/e\rangle$ is the ancilla ground/excited state. c, Wigner function representations of the lowest-order binomial codewords {$|0_\mathrm{L}\rangle = (|0\rangle + |4\rangle)/\sqrt{2}$ and $|1_\mathrm{L}\rangle = |2\rangle$}. The dominant single-photon loss converts the logical state from code space into the error space, spanned by {$|0_{\mathrm{E}}\rangle = |3\rangle$, $|1_{\mathrm{E}}\rangle = |1\rangle$}. The cardinal states on the Bloch spheres within the code and error spaces are defined as $|+X_{\mathrm{L}(\mathrm{E})}\rangle = (|0_{\mathrm{L}(\mathrm{E})}\rangle + |1_{\mathrm{L}(\mathrm{E})}\rangle)/\sqrt{2}$ and $|+Y_{\mathrm{L}(\mathrm{E})}\rangle = (|0_{\mathrm{L}(\mathrm{E})}\rangle + i|1_{\mathrm{L}(\mathrm{E})}\rangle)/\sqrt{2}$.
• Figure 2: Ancilla qubit reset via reservoir engineering.a, Energy level diagram of the system combined with the ancilla and a lossy resonator, which acts as a bath to dump the error entropy into the environment. A resonant drive applied to the ancilla with a Rabi frequency $\Omega_{\mathrm{R}}$ generates dressed qubit states $|\pm\rangle$ with an energy splitting $\Omega_{\mathrm{R}}$ (inset). A sideband drive applied to the resonator with a detuning $\Delta_{\mathrm{r}}=\Omega_{\mathrm{R}}$ induces transition from $|+,n\rangle$ to $|-,n+1\rangle$ (brown arrows), followed by rapid relaxation of the resonator (black arrows), resetting the system to $|-,0\rangle$. b, Experimental sequence for qubit reset characterization. The cavity is prepared in the logical state $|\psi_{\mathrm{L}}\rangle$ and the ancilla in $|g\rangle$ or $|e\rangle$. After applying $M$ reset cycles---each consisting of simultaneous qubit and resonator drives, optimized qubit pulses for basis transformation, and a resonator relaxation delay---the logical or ancilla qubit states are measured through a characterization process. c, Measured ancilla ground-state population versus reset drives' duration $t$ for one reset cycle ($M=1$) with an initial logical state $|+X_{\mathrm{L}}\rangle$. An average ground state probability of $\sim 99.6\%$ is achieved at $t=190$ ns regardless of the initial ancilla states. d, Measured phase coherence of the logical qubit state $|0_{\mathrm{L}}\rangle$ as a function of the number of qubit resets (blue) and the total idling time without reset (red).
• Figure 3: Performance of the AQEC procedure.a, Measured Wigner functions of the logical state $|+X_{\mathrm{L}}\rangle$: immediately after encoding (left), after idling $\sim150\mu$s without AQEC (middle), and after a single AQEC cycle with a duration $\sim150\mu$s (right). b, Bar charts of the magnitudes of the quantum process $\chi$ matrices for three cases: encode/decode only (left); interleaved with an idling process without AQEC (middle); and interleaved with a single AQEC cycle (right). c, Measured process fidelity as a function of storage time for different encodings. Error bars represent one standard deviation obtained from repeated experiments. All curves are fitted to $F_{\chi} = A e^{-t/\tau} + 0.25$ to extract the lifetime $\tau$ of the corresponding encodings, with uncertainties obtained from the fits. The extracted $\tau$ of the corrected binomial code (blue) surpasses the uncorrected binomial code (red) and uncorrected transmon qubit (green), and even exceeds the break-even point (orange) by a factor of 18%.
• Figure 4: Ramsey interference under AQEC protection for quantum sensing.a, Experimental sequence for Ramsey interferometry experiments with and without AQEC protection using superposition of Fock states. b-c, Measured ancilla ground-state population oscillating as a function of the initial phase $\varphi$ with different total interrogation times $t_\mathrm{int}=M\tau_\mathrm{int}$ for the cases without (b) and with (c) AQEC protection. Each AQEC cycle has a fixed duration of $\tau_\mathrm{int}=150~\mu$s. d Extracted normalized Fisher information from (b-c) as a function of the total interrogation time with (blue) and without (red) AQEC protection, as well as that using Fock 0,1, encoding (orange). Error bars are obtained from the error propagation of the fitting parameter uncertainties. A maximum Fisher information gain of 6.3 dB is achieved over the uncorrected Fock 0,1 encoding.