Abstract
To build a universal quantum computer from fragile physical qubits, effective implementation of quantum error correction (QEC)^{1} is an essential requirement and a central challenge. Existing demonstrations of QEC are based on an active schedule of errorsyndrome measurements and adaptive recovery operations^{2,3,4,5,6,7} that are hardware intensive and prone to introducing and propagating errors. In principle, QEC can be realized autonomously and continuously by tailoring dissipation within the quantum system^{1,8,9,10,11,12,13,14}, but so far it has remained challenging to achieve the specific form of dissipation required to counter the most prominent errors in a physical platform. Here we encode a logical qubit in Schrödinger catlike multiphoton states^{15} of a superconducting cavity, and demonstrate a corrective dissipation process that stabilizes an errorsyndrome operator: the photon number parity. Implemented with continuouswave control fields only, this passive protocol protects the quantum information by autonomously correcting singlephotonloss errors and boosts the coherence time of the bosonic qubit by over a factor of two. Notably, QEC is realized in a modest hardware setup with neither highfidelity readout nor fast digital feedback, in contrast to the technological sophistication required for prior QEC demonstrations. Compatible with additional phasestabilization and faulttolerant techniques^{16,17,18}, our experiment suggests quantum dissipation engineering as a resourceefficient alternative or supplement to active QEC in future quantum computing architectures.
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Data used in this work is available on reasonable request.
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Acknowledgements
We thank MIT Lincoln Lab for providing the Josephson travelling wave parametric amplifier for our measurement. We thank D. Rosenstock, E. Dogan and X. Deng for assistance with the experiment. This research was supported by the US Air Force Office of Scientific Research (FA95501810092) and the Army Research Office (W911NF1710469 and W911NF1910016).
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J.M.G. carried out the device design, microwave measurements, and data analysis of the experiment under the supervision of C.W. B.B. generated the numerical pulses for unitary control in the experiment under the supervision of J.K. J.L. fabricated the device and contributed to the cryogenic preparation of the apparatus. S.S. carried out numerical simulations for the experiment. B.B., J.K. and C.W. developed the approximate AQEC theory. C.W. conceived and oversaw this project. J.M.G., B.B., J.K. and C.W. wrote the manuscript with input from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Dynamics and intrinsic performance of the approximate AQEC protocol.
a, Diagram illustrating the quantumtrajectory state \({\psi }_{j}^{15}(t)\rangle \) after a time period t under the combined effect of photon loss and instantaneous PReSPA, given a certain number j of jumps. The initial state is \(\psi (0)\rangle ={0}_{{\rm{L}}}\rangle \) in the subspace ℋ\({}_{15}={\rm{span}}\,\{{u}_{0}\rangle ,{u}_{1}\rangle \}\). Both jump and nojump evolution lead to rotation of \({\psi }_{j}^{15}(t)\rangle \) over time within the subspace, and the angle \({\theta }_{j}(t)\) parametrizes this rotation. For each quantum trajectory, \({\psi }_{j}^{15}(t)\rangle \) slowly and continuously rotates clockwise in the absence of jumps and occasionally undergoes stochastic jumps counterclockwise. The diagram and the dynamics for the states \({\psi }_{j}^{37}(t)\rangle \) in the ℋ\({}_{37}={\rm{span}}\,\{{v}_{0}\rangle ,{v}_{1}\rangle \}\) subspace (not shown) follows an analogous pattern. b, Comparing the decay of process fidelities for three cases: T4C encoding using the ideal PReSPA scheme of this section (corrected T4C code, green), T4C encoding without using PReSPA (uncorrected T4C code, teal), and Fock state encoding (uncorrected Fock \(0\rangle \), \(1\rangle \), blue). Experimental values of T_{1A} and K are used, and cavity dephasing is not considered. Exponential curves for the T4C fidelity use the equation ℱ\({}_{{\rm{process}}}(t)=0.75{{\rm{e}}}^{t/\tau }+0.25\) to extract decay rate τ.
Extended Data Fig. 2 PReSPA spectroscopy.
a, b, Control pulse sequence for twodimensional spectroscopy to find the resonance conditions for the PReSPA mixing comb and transmon comb. We prepare an evenparity Fock state (\(0\rangle \), \(2\rangle \), \(4\rangle \), or \(6\rangle \)), apply PReSPA for a fixed time (12 μs) with varying detunings of the transmon comb (Δ_{q}) and the mixing comb (Δ_{m}) in an attempt to activate dissipative photon addition. After a 1 μs wait time for the reservoir to relax, we either selectively πpulse the transmon conditioned on cavity A being in the targeted final state (\(1\rangle \), \(3\rangle \), \(5\rangle \), or \(7\rangle \)) (a) or skip this pulse (for a background measurement, b), and proceed to read out the transmon state. The difference between the two measurements informs the likelihood of successful photon addition. c, Twodimensional PReSPA spectroscopy data: probability of photon addition (colour scale) as a function of the comb detunings (Δ_{q} and Δ_{m}) for the \(0\rangle \) to \(1\rangle \) transition. Note that the linewidth of the fourwavemixing transition is an order of magnitude greater than that of the transmon excitation owing to the short reservoir T_{1R}. We can repeat this procedure to find all four sets of transition frequencies. d, Cartoon spectrum of PReSPA drive frequencies. Four transmon drives, left, and four mixing drives, right, compose PReSPA. The coloured ticks indicate the actual transition frequencies whereas the vertical black bars show the microwave drive frequencies in PReSPA. The transmon drive for the \(0\rangle \) to \(1\rangle \) conversion process is approximately at the Starkshifted transmon frequency, \({\omega }_{{\rm{q}}}{\varDelta }_{{\rm{Stark}}}\), and the \(0\rangle \) to \(1\rangle \) mixing drive is near \({\omega }_{{\rm{A}}}+{\omega }_{{\rm{R}}}{\omega }_{{\rm{q}}}+{\varDelta }_{{\rm{Stark}}}\). Because of the equal frequency spacing η in each comb and the unequal frequency spacing between the transitions with different photon numbers (owing to the 6thorder nonlinearity, \({\chi }_{{\rm{q}}}^{{\prime} }\)), not all drives can be placed exactly on resonance. Experimentally, we settle for η slightly greater than 2χ_{q}, and Δ_{q} = Δ_{m} slightly smaller than Δ_{Stark} to compensate for the effect of \({\chi }_{{\rm{q}}}^{{\prime} }\).
Extended Data Fig. 3 Cavity Wigner and PReSPA Ramsey measurements.
a, Experimental Wigner function W(α) (colour scale, dimensionless) of \({0}_{{\rm{L}}}\rangle \), acquired by applying a cavity displacement operation \({\hat{D}}_{\alpha }=\exp (\alpha {\hat{a}}^{\dagger }{\alpha }^{\ast }\hat{a})\) with variable complex amplitude α followed by an ancillaassisted photonnumberparity measurement (which is composed of two π/2 pulses of the ancilla and a delay time of π/χ_{q} and an ancilla readout^{37,58}). The Wigner function rotates around the origin over time at a rate proportional to the frequency difference between \(1\rangle \) and \(5\rangle \) in the rotating frame of the experiment. b, Measured Wigner function values at a fixed phasespace position (as indicated by the cross in a, at α = 0.75) as a function of time under PReSPA. Analogous to a qubit Ramsey measurement, this cavity PReSPA Ramsey experiment can be used to efficiently track the phase evolution of any twocomponent superposition states using the interference effect enabled by the coherent cavity displacement (\({\hat{D}}_{\alpha }\)) before readout. The exponential envelope of the sinusoidal fit indicates the rate of decay for the coherence between \(1\rangle \) and \(5\rangle \) under the correction of PReSPA. Similar measurements are applied to various superposition states to provide direct calibration of the frequencies and phases of these states under PReSPA. PReSPA enhances the ability to use such Ramsey measurements at high photon numbers because it approximately preserves photon number distributions in the cavity.
Extended Data Fig. 4 Tracking AQEC performance over time.
a, We interleave measurements of the corrected logical qubits under two differently calibrated PReSPA parameter sets (PReSPA1, red and PReSPA2, blue) while also monitoring the cavity T_{2A}. Each circle corresponds to the decay time of process fidelity extracted from measuring all six cardinal points of the logical Bloch sphere as described in Fig. 4. The decay rates for state fidelity are shown in triangles for the two logical pole states (upwards triangles) and the four equator states (downwards triangles). The state fidelity is measured by quantum state tomography of the ancilla after decoding the cavity state as described in the Methods. For ancilla state tomography, we measure all six Pauli operators (\({\hat{\sigma }}_{x}\), \({\hat{\sigma }}_{y}\), \({\hat{\sigma }}_{z}\), \({\hat{\sigma }}_{x}\), \({\hat{\sigma }}_{y}\), \({\hat{\sigma }}_{z}\)) by performing ancilla rotations before readout. The overcomplete measurement set is used for simultaneous calibration of the readout signal contrast, allowing for accurate determination of the transmon state. PReSPA1 is calibrated by adjusting control parameters to achieve matched PReSPA rates and zero conversion phases as discussed in the Methods. For PReSPA2, we employ further empirical parameter optimization to maximize equator state lifetime as described in the Supplementary Information section 5. Cavity A has a twostate switching behaviour of unknown origin (see notes in Extended Data Table 1). For distinct stretches of 2−8 h, cavity A shows fluctuating and abnormally low T_{2A}, and all data recorded during such periods (with example data shown in the shaded region) are excluded in all other parts of the paper. b, Process fidelity averaged over the data, excluding the shaded region, for both PReSPA 1 and 2. Data reported in Fig. 4d for the corrected T4C encoding is a duplication of the blue points here. c, Equator and pole state fidelity for the same time period for both PReSPA 1 and 2. Oscillatory behaviour in the data are caused by the numerical differences between the two decoding pulses discussed in Methods section ‘GRAPE methods’.
Extended Data Fig. 5 Predicted AQEC performance in numerical simulations.
The results are based on masterequation simulations of a T4C qubit (with encoding \(\bar{n}=3.4\) for both basis states) under the Hamiltonian equations (11) and (18), which captures the dynamics under the microwave combs of PReSPA. a, Gain factor (colour scale, dimensionless) of the corrected logical qubit lifetime over the physical photon lifetime (\({T}_{1{\rm{A}}}/\bar{n}\)) in the T4C encoding as a function of ancilla T_{1q}, T_{ϕ} (which are made equal for convenience) and cavity T_{1A}. To illustrate the intrinsic performance of our transmonbased PReSPA pumping scheme, we have assumed no ancilla thermal excitations and other cavity dephasing errors. However, ancilla excitations due to the imperfect frequency selectivity of PReSPA, which is unrelated to photon loss, are reflected in the simulation. Therefore, the gain factor shown is different from the G_{i} defined in the main text, and decreases slightly at long T_{1A}. b, The QEC breakeven ratio (colour scale, dimensionless), defined as the T4C qubit lifetime under PReSPA over the lifetime of the 0/1 Fockstate encoding. Here we use a specific set of achievable coherence times T_{1A} = 1 ms, T_{1q} = T_{q} = 100 μs (refs. ^{46,51}) and show the degradation of AQEC performance in the presence of spontaneous transmon excitation (\({\gamma }_{\uparrow }\)) errors caused by the stray thermal background (horizontal axis) or pumptoneinduced heating from PReSPA (vertical axis). QEC breakeven can be reached if the \({\gamma }_{\uparrow }\) rate is kept reasonably low. The dashed lines in both a and b indicate where the QEC breakeven ratio equals 1. Relevant system parameters: In a, we use λ/2π = 17.5 kHz, Ω/2π = 45 kHz, κ/2π = 227 kHz, χ_{q}/2π = 1.05 MHz, χ_{R}/2π = 1.6 MHz, scaled from the experiment by 50%, 50%, 40%, 80%, 80% respectively. In b, we use λ/2π = 21.6 kHz, Ω/2π = 72 kHz, κ/2π = 364 kHz, χ_{q}/2π = 1.18 MHz, χ_{R}/2π = 1.8 MHz, scaled from the experiment by 80%, 80%, 64%, 90%, 90% respectively. The choice of parameters here is guided by the scaling laws of various error channels (Extended Data Table 3) but did not go through optimization of individual parameters.
Extended Data Fig. 6 Cavity heating effect caused by spurious transmon excitations.
a, Schematic mechanism of sequential twophoton gain triggered by spurious excitation of the transmon. Starting from \(1g0\rangle \), following the transitions labelled with bluecircled numbers, the system is excited to \(1e0\rangle \) by a transmon \({\gamma }_{\uparrow }\) jump, and then driven unintentionally to \(2g1\rangle \) by mixing tones that are offresonance by only \(\pm {\chi }_{{\rm{q}}}/2{\rm{\pi }}=\pm 1.3\,{\rm{M}}{\rm{H}}{\rm{z}}\) (which does not provide strong enough frequency selectivity relative to the reservoir linewidth κ/2π = 0.58 MHz), and then relaxes to \(2g0\rangle \) following reservoir decay. Once a photon is added in this spurious oddtoeven conversion process, the PReSPA scheme by design will add a second photon, driving the system ultimately to \(3g0\rangle \). Similarly, a transmon \({\gamma }_{\uparrow }\) jump can add two photons to \(3\rangle \) and \(5\rangle \) states (but not \(7\rangle \)). b, Schematic illustration of the steadystate photon number distribution established between the ancilla\({\gamma }_{\uparrow }\)induced photon addition and the natural photon loss of the cavity. The figure corresponds to the configuration of the test experiment in d when only the mixing comb (but not the transmon excitation comb) is applied. c, Steadystate (\(t\to \infty \)) cavity photon number distribution under PReSPA, as probed by ancilla qubit spectroscopy. In this measurement, PReSPA is applied nearly at all times, only briefly interrupted by spectroscopy probes once every 2 ms. The peak amplitudes confirm that odd photon number parity is permanently stabilized, and also reveals the presence of spurious excitation processes in the cavity. d, Steadystate cavity photon number distribution when only the mixing comb of PReSPA is applied, as in b. Under this configuration, a transmon \({\gamma }_{\uparrow }\) jump may add just one spurious photon in the cavity, leading to an effective cavity heating rate out of its vacuum state \({\gamma }_{01}\approx {\gamma }_{\uparrow }\) (when the transmon decay rate through the reservoir \({\gamma }_{{\rm{m}}}\gg 1/{{\rm{T}}}_{1{\rm{q}}}\)). The relative probability of \(0\rangle \) versus \(1\rangle \) informs the balance between the cavity decay rate 1/T_{1A} = 1.9 ms^{−1} and the heating rate \({\gamma }_{01}\). e, Blue triangles show the cavity heating rate \({\gamma }_{01}\), as measured with the technique described in d, as a function of the Rabi amplitude of a single \(0e0\rangle \leftrightarrow 1g1\rangle \) mixing tone. \({\gamma }_{01}\) initially follows the expected values corresponding to transmon \({\gamma }_{\uparrow }\) = 1.4 ms^{−1} (green dashed curve), but additional heating effects due to the microwave pump is observed at higher fourwave mixing (FWM) rate, Ω.
Extended Data Fig. 7 Process χ matrix block for 25 μs of PReSPA.
The matrix converts elements of input density matrices (top axis) to output density matrix elements (left axis) expressed in the Fock state basis. a, Amplitude values of the χ matrix elements. The upper left block (\({\chi }_{nn,mm}\)) describes the conversion of diagonal elements of the input and output density matrices, which is associated with transfer of photon occupation probabilities calculated from transmon spectroscopy experiments (as in Fig. 2d, e). The lower right block (\({\chi }_{nm,kl}\)) describes the conversion of relevant offdiagonal elements of density matrices, which is calculated from Wigner tomography and density matrix reconstruction^{59} (as in Fig. 3). The greyed blocks are assumed to be zero owing to the absence of interference between the four conversion paths in PReSPA. b, Phase values of the χ matrix elements for the lower right block in a. For best illustration of the PReSPA process, the phases are reported in the full rotating frame where all Fock states have zero energy. Values in grey are measured but not statistically significant because the corresponding amplitude value is not large enough. In this frame, as prescribed by equation (2), PReSPA requires zero phase for the six \({\chi }_{nm,(n+1)(m+1)}\) elements representing the coherence of the eventoodd conversion process, which is accomplished by our PReSPA calibration. The diagonal elements \({\chi }_{nm,nm}\) representing the preservation of oddstate superpositions should have zero phase by definition. Their systematic deviation from zero was caused by parameter drift in the experiment as that block of data was acquired at a later time than the earlier rotating frame calibration.
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Gertler, J.M., Baker, B., Li, J. et al. Protecting a bosonic qubit with autonomous quantum error correction. Nature 590, 243–248 (2021). https://doi.org/10.1038/s41586021032570
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